Efficient democratic decisions via nondeterministic proportional consensus
EE ffi cient democratic decisions via nondeterministicproportional consensus One sentence summary:
Novel, nonmajoritarian voting methods based on conditionalcommitments can achieve fairness without major welfare costs.
Jobst Heitzig ∗ and Forest W Simmons FutureLab on Game Theory and Networks of Interacting Agents,Potsdam Institute for Climate Impact Research,PO Box 60 12 03, D-14412 Potsdam, Germany Liberal Arts & Mathematics Division, Portland Community College,Cascade Campus, 705 N. Killingsworth Street, TH 220, Portland, OR 97217, USA ∗ To whom correspondence should be addressed; E-mail: [email protected].
Abstract: Are there voting methods which (i) give everyone, including minori-ties, an equal share of e ff ective power even if voters act strategically, (ii) pro-mote consensus rather than polarization and inequality, and (iii) do not favourthe status quo or rely too much on chance?We show the answer is yes by describing two nondeterministic voting meth-ods, one based on automatic bargaining over lotteries, the other on conditionalcommitments to approve compromise options. Our theoretical analysis andagent-based simulation experiments suggest that with these, majorities cannotconsistently suppress minorities as with deterministic methods, proponents ofthe status quo cannot block decisions as in consensus-based approaches, theresulting aggregate welfare is comparable to existing methods, and averagerandomness is lower than for other nondeterministic methods. Majority rule, considered a cornerstone of democracy, allows the oppression of minorities —Tocqueville’s ‘tyranny of the majority’ ( ) — which may lead to separatism or violent conflict(
2, 3 ). One way to address this is the fundamental fairness principle of proportionality (e.g.,(
3, 4 )). But if proportionality is only used to elect a representative body that then uses majorityvoting after all, the problem remains ( ). Why? Proportional representation does not implyproportional power: even a 49 percent faction may not be able to influence any decision. For1 a r X i v : . [ ec on . GN ] J un xample, given the strong polarization in the US Senate ( ), the Democratic Party currentlyappears to have zero e ff ective power in it, according to the Banzhaf and Shapley–Shubik powerindices ( ). But can power be distributed proportionally at all?Smaller groups often try to overcome the problem by seeking consensus, but that is di ffi cultin strategic contexts ( ). Status quo supporters may block consensus indefinitely, or, if thefallback is majority voting, a majority can simply wait for that to be invoked. Hence commonconsensus procedures are either not neutral about the options or e ff ectively majoritarian likemost common voting methods if voters act strategically.Judging from social choice theory, the formal science of group decision making, such non-proportional e ff ective power distribution seems unavoidable ( ), whether in a political or every-day context. But this is only so if the employed decision methods are required to be essentially deterministic, only using chance to resolve ties. In fact, nondeterministic methods cannot onlydistribute power proportionally, which is obvious, but at the same time support consensus andthus lead to e ffi cient outcomes ( ).It may seem outlandish to use a decision method that employs chance on a regular basis,producing uncertain outcomes. But real-world problems typically involve quite some unavoid-able stochastic risk and other forms of uncertainty anyway, e.g., due to lacking information,complexity, or dependence on others ( ). Also, routine use of nondeterministic procedures incontexts such as learning ( ), optimization ( ), strategic interactions ( ), or the allocation ofindivisible resources as in school choice ( ) shows that using chance can be quite beneficial,e ffi cient, and acceptable. Those examples also demonstrate that carefully using chance mustnot be confused with outright randomness.In this article, we adopt the working hypothesis that at least in everyday situations in whichpeople often say “let’s have a vote”, many groups might try a nondeterministic voting methodif that has clear advantages. For such situations, we study two such methods, one of whichis novel, that achieve fairness by distributing power proportionally and increase e ffi ciency bysupporting not just full but also partial consensus and compromise.As an illustrative test case (Fig. 1), consider a group of three factions F , F , F with sizes S , , (in percent), each of which has a favourite option, X , , , not liked by the other twofactions, respectively. Assume there is a fourth option A not liked by F but liked by F , almost as much as X , . We call A a potential ‘partial consensus’ for F , together. Whilee ffi ciency requires that A gets a good chance of winning, proportionality requires that also X gets some chance of winning. Accordingly, our methods will assign winning probabilities of S + S % to A and S % to X , even if voters vote strategically. If we add a fifth option B which F , like slightly less than A , and which F likes almost as much as X , both our methodswill pick this potential ‘full consensus’ B for sure. In contrast, if S > S + S and voters actstrategically, virtually all existing voting methods will either pick X with certainty, or willassign probabilities of S , , % to X , , , in both cases ignoring A (and B ) and thus producingmuch less overall welfare. No deterministic voting method can let F , together make sure that A gets a chance without allowing them to render F ’s votes completely irrelevant. This can only2 action F F F X X X X AX X X X A B X p r e f e r e n c e Figure 1:
Archetypical group decision problem with potential for suppression of minorities, partial,or full consensus. Each of three factions of di ff erent size (column width) has a unique favourite (top-most). There might also be a potential ‘partial consensus’ option A and / or a potential ‘full consensus’ B . With strategic voters, common deterministic methods pick X for sure. Our methods Nash Lottery and
MaxParC pick B for sure if present (green); they pick one of X , , with probabilities (colored area)proportional to faction size if neither A nor B is present (orange); and they pick A or X with proportionalprobabilities if A but not B is present (blue). be achieved by employing a judicious amount of chance.But how exactly? How to design a nondeterministic voting method that is both e ffi cientand proportional, even when voters act strategically, and also fulfills other basic consistency re-quirements like those typically studied in social choice theory — such as anonymity, neutrality,monotonicity, and clone-proofness — that make it plausible and hard to manipulate?Our first method, the Nash Lottery (NL) , is basically what is known as ‘Nash Max Product’or ‘Maximum Nash Welfare’ in the literature on fair division of resources. As suggested in ( ),we translate it to our voting context by interpreting winning probability as a “resource” to bedivided fairly, and study the strategic implications of this. NL can be interpreted as a formof automatic bargaining by means of the Nash bargaining solution. Similar to score-basedmethods such as Range Voting (RV) ( ), it asks each voter, i , to give a rating, r ix ≥
0, foreach option x . It then assigns winning probabilities, p x , that maximize a certain function, f ( r , p ). RV maximizes f ( r , p ) = (cid:80) i (cid:80) x r ix p x , resulting in a very e ffi cient majoritarian methodthat is deterministic (usually p x = x ) but neither distributes power proportionally norsupports consensus when voters are strategic. NL instead maximizes f ( r , p ) = (cid:88) i log (cid:88) x r ix p x , (1)resulting in a nondeterministic method that supports both full and partial consensus. In the Supplementary Text, we prove that in situations similar to Fig. 1, a full consensus will be the3
Option COption E (receiving Alice’s “vote”)
Option BOption A (Alice’s favourite)
Option D (always approve) (never approve) voter Alice’s view Alice all ratings for C
80% approval all ratings for E
60% approval doesn’t approve approvescutoff cutoff
Alice
Figure 2:
Voting method MaxParC from the view of some voter Alice (left). Rating 13 for option Cis interpreted as saying that Alice approves of C if less than 13 percent of voters do not approve of C.Resulting approval scores can be found graphically in a way similar to ref. ( ) (right). sure winner, and that using the logarithm rather than any other function of (cid:80) x r ix p x is the uniqueway to achieve a proportional power distribution.NL is conceptually simple and has some other desirable properties shown in Fig. 3 suchas being immune to certain manipulations, e.g., cloning options or adding bad options. Butit has three important drawbacks. Its tallying procedure is intransparent, requiring numericaloptimization. It lacks certain intuitive ‘monotonicity’ properties: when a new option is addedor a voter increases some existing option’s rating, some other option’s winning probability mayincrease rather than decrease. And NL often employs much more randomness than necessary.All three drawbacks are overcome by our second method, the novel Maximal Partial Con-sensus (MaxParC), which is conceptually more complex, but strongly monotonic, much easierto tally, and produces less entropy. Based on the idea of conditional commitments, it lets eachvoter safely transfer “their” share of the winning probability to potential consensus options ifenough other voters do so as well. That is done in a way inspired by Granovetter’s famous‘threshold model’ (
18, 19 ). Again, voters assign numerical ratings, 0 ≤ r ix ≤ i will approve of x if strictly less than r ix percent of all voters do not approve of x .” To solve this recursive definition of ‘approval’ for anygiven option x , MaxParC sorts the ballots ascendingly w.r.t. their rating of option x , then finds4 monotonicclone-proofreveals preferencesdistributes power prop.supports partial cons.supports full consensusMaximal Partial Consensus (MaxParC)Nash LotteryFull Consensus/Random Ballot/RatingsFull Consensus/Random Ballot ballot complexitytally complexityRandom Ballottypical Condorcet methodInstant Runoff VotingRange VotingApproval VotingPlurality Voting
43 3
02 2
45 465 d e t e r m i n i s t i c n o n d e t e r m i n i s t i c independent from losing options Figure 3:
Properties of common group decision methods, Nash Lottery, and MaxParC. Solid and dasheddiamonds indicate full and partial fulfillment, numbers are qualitative complexity assessments by theauthors, color only distinguishes di ff erent groups of criteria (see Supplementary Text details and proofs). the first ballot i in this ordering such that strictly less than r ix percent of the ballots precede it(i.e., have r jx < r ix ). This ballot i and all later ballots j (those with r jx ≥ the cuto ff r ix ) are saidto approve of x . After thus determining which ballots approve which options, MaxParC thenproceeds like the ‘Conditional Utilitarian Rule’ from (
16, 20 ): one ballot is drawn at random,and from the options approved by this ballot, that with the largest overall approval wins. Ifone compromise option is rated positive by everyone, it will win for sure. Fig. 2 illustrates theMaxParC procedure.Fig. 3 summarizes our theoretical analysis of the formal properties of NL and MaxParC ascompared to typical voting methods from the literature, validating that the latter perform wellin terms of these qualitative criteria.To assess the potential costs of achieving fairness and supporting consensus in more quanti-tative terms of welfare, voter satisfaction, and entropy, we finally performed a large agent-basedsimulation experiment. In over 2.5 million hypothetical group decision problems, we comparedNL and MaxParC’s performance to that of five deterministic majoritarian and three nonde-terministic proportional methods: Plurality Voting (PV), Approval Voting (AV), RV, Instant-Runo ff Voting (IRV), Simpson–Kramer (a simple Condorcet method, SC); ‘Random Ballot’5RB), and two methods from ( ) (FC and RFC). To generate the decision problems, we usedrandom combinations of the number and compromise potential of options and the number, in-dividual preference distributions, and risk-attitudes of voters. For each combination of decisionproblem and voting method, we simulated several opinion polls, a main voting round, and aninteractive phase where ballots could be modified continuously for strategic reasons. In this,we assumed various mixtures of behavioural types of voters: lazy voting, sincere voting, indi-vidual heuristics, trial and error, and coordinated strategic voting. For each decision problem,we computed several metrics of social welfare, randomness, and voter satisfaction for all votingmethods, and which voters would prefer which voting methods (see Materials and Methods fordetails).As can be expected, typically a majority of the simulated voters preferred the results ofthe majoritarian methods over those of the proportional ones. On average, voters preferredMaxParC over the other proportional methods; among the majoritarian methods, there wasno predominant preference. Individual voters’ satisfaction, normalized to zero for their least-preferred option and one for their favourite, averaged around 67 % for PV, AV, RV, and IRV;61 % for SC, NL, MaxParC; and still 57 % for RB, FC, RFC.MaxParC produced about 60 % of the entropy of RB, NL about 80 %. In MaxParC, thelargest winning probability was about 65 % on average, in NL only about 53 %.The deterministic methods produced somewhat higher welfare on average, but for somepreference models and welfare metrics, the nondeterministic methods matched or outperformedthem (Fig. 4). In more than 75 % of cases, the utility di ff erence between the average and theworst-o ff voter under RV was at least seven times the di ff erence in average voter utility be-tween RV and MaxParC. This can be interpreted as saying that the welfare costs of fairness andconsensus are small compared to the inequality costs of majoritarianism.On most results, preference distributions had a larger e ff ect than behavioural type or theamount of interaction. Surprisingly, strategic voters gained no clear advantage over lazy voters,and also risk-attitudes played a minor role.In 2007, one of us (Heitzig) asked the election methods electronic mailing list ( ) what methodwould elect the compromise rather than the majority option in a situation similar to Fig. 1, evenwhen voters acted strategically. Soon it became obvious that no deterministic method would do,but several lottery methods were quickly found that elected the compromise with certainty. Sowhy do election methods experts show little enthusiasm for nondeterministic methods? Perhapsbecause their primary interest is in periodic high-stakes public elections every several years. Theproportional fairness of lottery methods is due to their average proportionality over many indi-vidual decisions. Few would suggest deciding which of two newlyweds shall be the householdsdictator by flipping a coin. Using coin flips for their many everyday decisions would be better— because stakes are lower and advantages level out over time — but would still not lead toa single consensus. Using the two methods presented here would likely make them agree onsome compromise in most situations and toss a coin only rarely. Both the splitting-up into manydecisions and the incentives for agreement lower the resulting overall entropy.6 BM Plurality VotingApproval VotingRange VotingInstant RunoffVotingSimple CondorcetRandom BallotFull Consensus/Random BallotFull Consensus/Random Ballot/RatingsNash LotteryMaxParC 0.2 0.3 0.4 0.5 0.6 unif
Plurality VotingApproval VotingRange VotingInstant RunoffVotingSimple CondorcetRandom BallotFull Consensus/Random BallotFull Consensus/Random Ballot/RatingsNash LotteryMaxParC GA Plurality VotingApproval VotingRange VotingInstant RunoffVotingSimple CondorcetRandom BallotFull Consensus/Random BallotFull Consensus/Random Ballot/RatingsNash LotteryMaxParC
40 20 0 QA Plurality VotingApproval VotingRange VotingInstant RunoffVotingSimple CondorcetRandom BallotFull Consensus/Random BallotFull Consensus/Random Ballot/RatingsNash LotteryMaxParC
15 10 5 0 LA Plurality VotingApproval VotingRange VotingInstant RunoffVotingSimple CondorcetRandom BallotFull Consensus/Random BallotFull Consensus/Random Ballot/RatingsNash LotteryMaxParC
Figure 4:
Distribution of final Gini-Sen welfare across 2.5 mio. agent-based simulations by votingmethod (rows), for five di ff erent models of how voter preferences might be distributed (columns). See Supplementary Text for definitions and more results.
While this seems to imply that such consensus-supporting proportional methods are bestused for everyday decisions only, they might even be applied to larger decisions such as allocat-ing some budget or electing a parliament. This is because the asset distributed by these methodsneed not be ‘winning probability’ in a single-outcome decision as in this article. For example,suppose NL or MaxParC instead of one of the common simple proportional methods was usedfor allocating parliamentary seats to party lists, based on voters’ ratings of all parties. Would notthis method take better advantage of opportunities for consensus without sacrificing proportion-ality? Since the seat distribution would on average have a lower entropy than usual, would it notavoid unnecessary balkanization or fragmentation of parliament without sacrificing representa-tion of minorities? We hope that this discussion serves to stimulate the reader’s imagination tosome of the possibilities of application, as well as avenues for further exploration.
References
1. D. Lewis,
Direct democracy and minority rights: A critical assessment of the tyranny ofthe majority in the American states (Routledge, 2013).2. P. Collier,
Oxford Economic Papers , 563 (2004).3. L. Cederman, A. Wimmer, B. Min, World Politics (2010).7. F. S. Cohen, Comparative Political Studies , 607 (1997).5. F. Zakaria, Foreign a ff airs , 22 (1997).6. N. McCarty, K. T. Poole, H. Rosenthal, Polarized America: The dance of ideology andunequal riches (MIT Press, 2016).7. P. Dubey, L. S. Shapley,
Mathematics of Operations Research , 99 (1979).8. J. H. Davis, Organizational Behavior and Human Decision Processes , 3 (1992).9. K. O. May, Econometrica: Journal of the Econometric Society pp. 680–684 (1952).10. J. Heitzig, F. W. Simmons,
Social Choice and Welfare , 43 (2012).11. R. Carnap, The Journal of Philosophy , 141 (1947).12. J. G. Cross, The Quarterly Journal of Economics , 239 (1973).13. D. P. Kingma, J. Ba, arXiv preprint arXiv:1412.6980 (2014).14. J. C. Harsanyi, International journal of game theory , 1 (1973).15. P. Troyan, Games and Economic Behavior , 936 (2012).16. H. Aziz, A. Bogomolnaia, H. Moulin, ACM EC 2019 - Proceedings of the 2019 ACMConference on Economics and Computation pp. 753–781 (2019).17. J.-F. Laslier, M. R. Sanver,
Handbook on approval voting (Springer Science & BusinessMedia, 2010).18. M. S. Granovetter, Threshold Models of Collective Behavior (1978).19. M. Wiedermann, E. K. Smith, J. Heitzig, J. F. Donges,
Scientific Reports (in press) (2020).20. C. Duddy,
Mathematical Social Sciences , 1 (2015).21. R. Lanphier et al., https: // electorama.com / em, last accessed 14 April 2020 (1996).22. H. Moulin, Fair division and collective welfare (MIT Press, 2004).23. J.-F. Laslier,
Handbook on Approval Voting (2010), pp. 311–335.24. R. Carroll, J. B. Lewis, J. Lo, K. T. Poole, H. Rosenthal,
American Journal of PoliticalScience , 1008 (2013).25. A. Bruhin, H. Fehr-Duda, T. Epper, Econometrica , 1375 (2010).86. J. Behnke, S. Hergert, F. Bader, Stimmensplitting Kalkuliertes Wahlverhalten unter denBedingungen der Ignoranz (2004).27. J. Sommer, Wer w¨ahlt strategisch und warum? Eine Analyse strategischen Wahlverhaltensbei der Bundestagswahl 2013, Phd dissertation, Heinrich-Heine-Universit¨at D¨usseldorf(2015).28. G. Gigerenzer, W. Gaissmaier, Annual review of psychology , 451 (2011).29. K. van der Straeten, J. F. Laslier, N. Sauger, A. Blais, Social Choice and Welfare , 435(2010).30. J. S. Bower-Bir, N. J. D’Amico, A Tool for All People, but Not All Occasions: How VotingHeuristics Interact with Political Knowledge and Environment (2013).31. J. Laslier, Journal of Theoretical Politics , 113 (2009).32. R. Myerson, R. Weber, American Political Science Review , 102 (1993).33. A. Dellis, Handbook on Approval Voting , J.-F. Laslier, M. R. Sanver, eds. (Springer, 2010),chap. 18, pp. 431–454.34. H. Moulin,
Econometrica , 1337 (1979).35. P. K. Bag, H. Sabourian, E. Winter, Journal of Economic Theory , 1278 (2009).36. A. Sen,
Journal of Public Economics , 387 (1974).37. E. Koutsoupias, C. Papadimitriou, Lecture Notes in Computer Science (including subseriesLecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Springer Ver-lag, 1999), vol. 1563, pp. 404–413.38. D. R. Woodall,
Discrete Applied Mathematics , 81 (1997).39. T. N. Tideman, Social Choice and Welfare , 185 (1987).40. M. Schulze, Social Choice and Welfare , 267 (2011).41. B. D. Bernheim, B. Peleg, M. D. Whinston, Journal of Economic Theory , 1 (1987). Acknowledgements:
We’d like to thank Marius Amrhein, Markus Brill, Pascal F¨uhrlich,Anne-Marie George, Ulrike Kornek, Fabrizio Kuruc, Adrian Lison, E. Keith Smith, Lea Tam-berg, and the members of the election-methods list ( ) for fruitful discussions and comments.Funding: this work received no external funding. Author contributions: J.H. and F.W.S. con-ceptualized the study, developed theory and methodology, performed the formal analysis, andwrote the manuscript. J.H. developed the software and performed the simulations. The authors9eclare no competing interests. Data and materials availability:
All data is available in the manuscript or the supplementarymaterials. An online voting tool based on the MaxParC method presented here is under open-source development at https: // github.com / mensch72 / maxparc-ionic. Supplementary Materials:
Materials and Methods, Supplementary Text, References (22–44),Table S1–S3, Fig S1–10. 10 upplementary Materials ff ective power . . . . . . . . . . . . . . . . 392.1.5 Consensus supporting properties . . . . . . . . . . . . . . . . . . . . . 402.2 Results of simulation experiments . . . . . . . . . . . . . . . . . . . . . . . . 482.2.1 Social welfare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.2.2 Randomization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.2.3 Satisfaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.2.4 Preferences over methods . . . . . . . . . . . . . . . . . . . . . . . . 562.2.5 Further conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.3 Derivation of heuristics and strategies . . . . . . . . . . . . . . . . . . . . . . 582.3.1 Sincere voting under MaxParC . . . . . . . . . . . . . . . . . . . . . . 582.3.2 Heuristic Nash Lottery strategy . . . . . . . . . . . . . . . . . . . . . 612.3.3 Factional unanimous best response in IRV . . . . . . . . . . . . . . . . 6311 Materials and Methods
A finite group of voters must collectively pick exactly one winning option out of a given fi-nite number of options of any kind (e.g., certain kinds of objects, places, time-points, actions,strategies, people, etc.). The menu of options is already given at the beginning of the situationwe consider, and all options are mutually exclusive and feasible, i.e., each one could be imple-mented without violating any relevant constraint (e.g., budgets, applicable laws, basic rights,time constraints etc.). Although in the real world, the option menu might sometimes changeduring a group decision procedure, we consider the composition of the option menu as a sep-arate process here which has been completed before the situation we consider. We assume thevoters will apply some formalized method to pick one option that may be described as someform of protocol or game form which requires the voters to provide some kind of information ina step we term voting and then determines a winning option from this information in a step weterm tallying, using some kind of algorithm that may or may not involve some form of random-ization. We call the information a voter provides this voter’s ballot and the method that turnssets of ballots into winning options a voting method. In addition, we assume each voters possesses some form of preferences regarding the op-tions and regarding possible probability distributions of options. Again, even though in the realworld, preferences might sometimes change during a group decision procedure, in particular incertain forms of deliberation or consensus finding, we consider here also the formation of pref-erences as a separate process which has been completed before the voting. This assumption isin line with the established approach taken in social choice theory. We do not, however, assumethat there is a simple deterministic relationship between voters’ preferences and their ballots,but rather assume that voters may use di ff erent kinds of heuristics or strategies to decide how tofill in their ballots. In this context, we aim at finding a voting method that fulfills certain consistency, fairness, ande ffi ciency criteria as stated in the main text and detailed further in Section 2 of this document. To this end, we study the qualitative and quantitative properties of di ff erent voting methods,some well-known from the social choice literature, one adapted from the theory of fair budgetallocation, and one designed newly. To study the qualitative properties of voting methods, weapply a mixture of methods from social choice theory and game theory. For the quantitativeproperties we apply large-scale (Monte-Carlo) numerical simulations of an agent-based model12hose assumptions are partially based on the spatial theory of voting from political science andon insights from the study of risk attitudes and bounded rationality in behavioural economics.We analyze simulation results by means of metrics adapted from welfare economics and infor-mation theory. Note on terminology.
Because we assume preferences have been determined before voting,we do not in this study distinguish linguistically between the terms ‘consensus’, ‘consent’, and ‘compromise’, but rather use a very pragmatic working definition of potential consensus here.In this study, consensus does not mean that all voters consider the exact same option from theoption menu their favourite option. Informally, we rather say there is (full or partial) potentialconsensus whenever there is some option which (all or some group of) voters would prefer tohaving one randomly drawn voter make the choice.
In this section we introduce a formal mathematical framework for comparing quite di ff erentvoting methods and then use it to define our versions of a number of common group decisionmethods, at which point the motivations for the various abstract notions should become clear.We assume an infinite universe of potential voters I and an infinite universe of potentialoptions X , leading to a universe of potential finite electorates E = { E ⊂ I : 1 ≤ | E | < ∞} and auniverse of potential finite choice sets C = { C ⊂ X : 1 ≤ | C | < ∞} . For each choice set C ∈ C , let L ( C ) = { (cid:96) ∈ [0 , C : (cid:80) x ∈ C (cid:96) ( x ) = } be the set of all lotteries on C , and let (cid:96) x ∈ L ( C ) with (cid:96) x ( x ) = sure-thing lottery that picks x ∈ C for sure. In most of what follows we deal with fixedsets E and C and denote their sizes by N and k , but for some proofs we have to consider all of E and C . Since we will deal with voting methods using quite di ff erent types of ballots, some only lettingthe voter mark a single option, others many, still others requiring a strict ranking or asking forquantitative ratings or the like, we need a formal framework general enough to cover all relevantcases and make them comparable in those respects important for the assessment of the method’sproperties. In particular, we need to make clear for each ballot type what it means when we saythat a ballot states a preference for one option over another or a ballot results from another ballotby advancing an option to a certain degree. The following abstract definition will prove usefulin these tasks:A ballot type is a tuple ( B , P , Q ) with the following properties: • B is a function such that for all choice sets C ∈ C , B ( C ) is a nonempty set representingthe di ff erent ways b in which a voter might fill in a ballot for the choice set C , and B ( C ) ∩ B ( C (cid:48) ) = ∅ if C (cid:44) C (cid:48) . 13 P is a function such that for all C ∈ C and all filled-in ballots b ∈ B ( C ), P b is a strict partialordering relation on C (i.e., irreflexive, asymmetric, and transitive, but not necessarilycomplete) representing that part of ballot b that will be interpreted as stated preferences ,with x P b y meaning that x is put in a strictly “better place” (ranking, rating, threshold,etc.) on b than y is. • Q is a function such that for all C ∈ C and all options x ∈ C , Q Cx is a strict partial orderingrelation on B ( C ) with b Q Cx b (cid:48) meaning that the two filled-in ballots b , b (cid:48) ∈ B ( C ) only di ff erin the fact that b puts x in a strictly “better place” than b (cid:48) while each other option is treatedthe same on b and b (cid:48) , so that b can be seen as resulting from b (cid:48) by “advancing” x in someway and changing nothing else.Note that P b = P b (cid:48) does not imply b = b (cid:48) in general since some ballot types (e.g., ratingsballots) also contain other information than just a binary preference relation.If C ∈ C , b ∈ B ( C ), and x P b y for all y ∈ C \ { x } , we call x the stated favourite on b and write F ( b ) = x . Since P b may be incomplete but is asymmetric, a ballot contains either no statedfavourite (in which case we write F ( b ) = ∅ ) or exactly one. If F ( b ) = x and ¬ y P b z for any y , z (cid:44) x , we say that b is a bullet vote for x . A ballot profile of type ( B , P , Q ) for electorate E ∈ E and choice set C ∈ C is a function β : E → B ( C ) specifying a filled-in ballot β i for each voter i ∈ E .A (potentially probabilistic) voting method is a tuple ( B , P , Q , M ) such that ( B , P , Q ) is aballot type and M is a function such that for all E ∈ E and C ∈ C , and all ballot profiles β oftype ( B , P , Q ) for E and C , it specifies a winning lottery M ( β ) ∈ L ( C ). Of course, M ( β ) may bea sure-thing lottery with M ( β ) x = x ∈ C . Plurality Voting (PV).
We formalize the ballot type of
Plurality Ballot as follows: • B ( C ) = C , i.e., each voter has to vote for exactly one option x ∈ C by putting b = x . • x P b y i ff b = x (cid:44) y , i.e., voting for x is interpreted as stating a strict preference for x overall other options. • b Q Cx b (cid:48) i ff b = x (cid:44) b (cid:48) , i.e., “advancing” x means converting a vote for a di ff erent optioninto a vote for x .Using this ballot type, the voting method of Plurality Voting now puts M ( β ) x = A ( x ) / | A | ,where 1 A is the indicator function of the set A = { x : p ( x ) ≥ p ( y ) for all y ∈ C } of pluralitywinners, and p ( x ) = |{ i ∈ E : β i = x }| is x ’s plurality score .Note that for simplicity, in our version one cannot abstain under plurality voting, and henceevery ballot is a bullet vote. Generally | A | = pproval Voting (AV). We formalize the ballot type of
Approval Ballot as follows: • B ( C ) = { , } C , i.e., each voter can either approve (by putting b ( x ) =
1) or disapprove (byputting b ( x ) =
0) of each option x ∈ C individually. • x P b y i ff b ( x ) > b ( y ), i.e., i ff b ( x ) = b ( y ) =
0, meaning that approving x is interpretedas stating a strict preference for x over all non-approved options. • b Q Cx b (cid:48) i ff b ( x ) > b (cid:48) ( x ) and b ( y ) = b (cid:48) ( y ) for all y ∈ C \ { x } , i.e., “advancing” x means con-verting a non-approval of x into an approval of x .Using this ballot type, the voting method of Approval Voting now puts M ( β ) x = A ( x ) / | A | , where1 A is the indicator function of the set A = { x : a ( x ) ≥ a ( y ) for all y ∈ C } of approval winners and a ( x ) = (cid:80) i ∈ E β i ( x ) is x ’s approval score .There are two equivalent ways to “abstain” under approval voting: putting b ( x ) ≡ x ∈ C or putting b ( x ) ≡ x ∈ C . Bullet voting for x means putting b ( x ) = b ( y ) = y . Range Voting (RV).
In our version of range voting, the ballot type of
Range Ballot has: • B ( C ) = [0 , C , i.e., one can assign any real-valued rating ≤ b ( x ) ≤
100 to each option x ∈ C individually. • x P b y i ff b ( x ) > b ( y ) as before, meaning that a higher rating states a strict preference. • b Q Cx b (cid:48) i ff b ( x ) > b (cid:48) ( x ) and b ( y ) = b (cid:48) ( y ) for all y ∈ C \ { x } , i.e., “advancing” x means raisingits rating.Using this ballot type, the voting method of Range Voting puts M ( β ) x = R ( x ) / | R | , where 1 R is the indicator function of the set R = { x : r ( x ) ≥ r ( y ) for all y ∈ C } of range winners and r ( x ) = (cid:80) i ∈ E β i ( x ) is x ’s range score .There are infinitely many equivalent ways to “abstain” under range voting: choose some α ∈ [0 , b ( x ) ≡ α for all x ∈ C . There are also infinitely many ways to “bullet vote”for x under range voting, and they are not (!) equivalent: choose 0 ≤ α < γ ≤
100 and put b ( x ) = γ and b ( y ) = α for all other y ∈ C .In addition to the above three “scoring methods”, we consider the following two more compli-cated “ranking methods”. Voter i ’s choice of b ( x ) is what was denoted r ix in the main text. nstant-Runo ff Voting (IRV).
In our version of instant-runo ff voting, we use the followingballot type of Truncated Ranking Ballot : • B ( C ) = (cid:110) b ∈ ( N ∪ {∞} ) C : b [ b − [ N ]] = { , . . . , | b − [ N ] |} (cid:111) . In other words, one has to assignconsecutive and distinct integer ranks , , . . . to any empty or nonempty subset of theoptions, leaving the other options unranked (here formally encoded by “rank” ∞ ). • x P b y i ff b ( x ) < b ( y ) (!) since a smaller rank number indicates a “better place”. • b Q Cx b (cid:48) i ff P b | C \{ x } = P b (cid:48) | C \{ x } , x P b y whenever x P b (cid:48) y , y P b (cid:48) x whenever y P b x , but P b (cid:44) P b (cid:48) . In other words, advancing x means changing the ranks so that the resulting ordering P doesn’t change except that some options ranked better than x before are now eitherranked lower than x or not ranked at all, and / or x was not ranked before and is nowranked better than at least one option.Using this ballot type, our simple version of the voting method of Instant-Runo ff Voting (akaSingle Transferable Vote, or Alternative Vote) runs like this: Initialize D = C and repeat thefollowing as long as | D | >
1: For each option x ∈ D , calculate the score s ( x ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:110) i ∈ E : β i ( x ) <β i ( y ) for all y ∈ D \ { x } (cid:111)(cid:12)(cid:12)(cid:12)(cid:12) . From the set of worst-scored options, W = { x ∈ D : s ( x ) ≤ s ( y ) for all y ∈ D } , draw a random member and remove it from D . The remaining member of D wins.Abstention means not ranking any option (i.e., putting b x = ∞ for all x ). A bullet vote ranksexactly one option (i.e., puts b x = x and b y = ∞ for all other y ). Simple Condorcet (SC).
In our version of the Simple Condorcet method, we use the follow-ing ballot type of
Weak Ranking Ballot: • B ( C ) = ( N ∪ {∞} ) C . In other words, one can assign arbitrary integer ranks , , . . . toany empty or nonempty subset of the options, leaving the other options unranked (againencoded by ∞ ). • x P b y i ff b ( x ) < b ( y ) as in IRV. • b Q Cx b (cid:48) i ff P b | C \{ x } = P b (cid:48) | C \{ x } , x P b y whenever x P b (cid:48) y , y P b (cid:48) x whenever y P b x , but P b (cid:44) P b (cid:48) . In other words, advancing x means changing the ranks so that the resulting ordering P doesn’t change except that some options ranked better than x before are now eitherranked equal to or lower than x or not ranked at all, and / or some options ranked equal to x before are now ranked lower than x or not ranked at all,Note that we allow both that some options are ranked equal and some rank numbers are skippedsince the only information our version of the Simple Condorcet method (aka the Minimax Con-dorcet or Simpson–Kramer method with pairwise opposition as score) is the ordering P b . We If R ⊆ X × X is a binary relation on some set X and S ⊆ X is a subset of X , then R | S = R ∩ ( S × S ) is therestriction of R to S . M ( β ) x = O ( x ) / | O | , where 1 O is the indicator function of the set O = { x : o ( x ) ≤ o ( y ) forall y ∈ C } of weak condorcet winners , o ( x ) = max y ∈ C o ( x , y ) is x ’s worst opposition value , and o ( x , y ) = |{ i ∈ E : y P β i x }| for all x , y ∈ C .There are infinitely many equivalent ways to “abstain” under the Simple Condorcet method:choose some α ∈ N ∪ {∞} and put b ( x ) ≡ α for all x ∈ C . The most straightforward is to not rankany option at all. There are also infinitely many equivalent ways to “bullet vote” for x under theSimple Condorcet method: choose ∞ ≥ α > γ ∈ N and put b ( x ) = γ and b ( y ) = α for all other y ∈ C .After these five “deterministic” methods, we now turn to five “non-deterministic” methods,beginning with three from the literature. Random Ballot (RB).
The
Random Ballot (aka Random Dictator) method uses Plurality Bal-lots but puts M ( β ) x = p ( x ) / N . The interpretation is that one ballot is drawn uniformly at randomto decide. Full Consensus / Random Ballot (FC). An FC Ballot is basically a combination of twoPlurality Ballots: • B ( C ) = C × C , i.e., each voter specifies one “proposed consensus” b ∈ C and one “fall-back” option b ∈ C . • x P b y i ff b = x (cid:44) y , i.e., only the fall-back part of the ballot is interpreted as stating astrict preference for b over all other options, while the consensus part is interpreted asinherently strategic. • b Q Cx b (cid:48) i ff b (cid:44) b (cid:48) and ( b = x (cid:44) b (cid:48) or b = b (cid:48) ) and ( b = x (cid:44) b (cid:48) or b = b (cid:48) ), i.e., “ad-vancing” x means advancing it in at least one of the two ballot parts.The method of Full Consensus / Random Ballot (FC) is now defined as in ( ) (there called“Voting method 1”): M ( β ) x = β i = x for all i ∈ E ; otherwise M ( β ) x = p ( x ) / N for all x ∈ C (“fall-back lottery”), where p ( x ) = |{ i ∈ E : β i = x }| is x ’s fall-back score . The interpretation isthat if all voters propose the same consensus, that option wins, otherwise the fall-back lotteryapplies.A bullet vote is a bullet vote on both ballot parts. There is no way to abstain. Full Consensus / Random Ballot / Ratings (RFC).
Similarly, an
RFC Ballot is a combina-tion of two Plurality Ballots and a Range Ballot: • B ( C ) = C × C × [0 , C , i.e., each voter specifies one proposed consensus b ∈ C , onefall-back option b ∈ C , and a vector of ratings b ( x ) ∈ [0 , x ∈ C .17 x P b y i ff b ( x ) > b ( y ), i.e., only the ratings part of the ballot is interpreted as statingpreferences, while the other two parts are interpreted as inherently strategic. • b Q Cx b (cid:48) i ff b (cid:44) b (cid:48) and ( b = x (cid:44) b (cid:48) or b = b (cid:48) ) and ( b = x (cid:44) b (cid:48) or b = b (cid:48) ) and b ( x ) ≥ b ( y ), i.e., “advancing” x means advancing it in at least one of the three ballot parts.The method of Full Consensus / Random Ballot / Ratings (RFC) is also defined as in ( ) (therecalled “Voting method 2”). For all j ∈ E , let r j = (cid:80) y ∈ C p ( y ) β j ( y ) / N be the rating of the fall-backlottery by voter j . Then put M ( β ) x = | A x | N + (1 − | A | N ) p ( x ) N , where A x is the set of all i ∈ E for which β i = x and β j ( x ) ≥ r j for all j ∈ E (i.e., whose proposed consensus is x and is preferred to thefall-back lottery by all voters according to their ratings), and A = (cid:83) y ∈ C A y . The interpretationis that a voter i is drawn uniformly at random, and if i ’s proposed consensus β i is unanimouslypreferred to the fall-back lottery p ( x ) / N , it wins, otherwise the fall-back lottery is applied.A bullet vote is a bullet vote on all three ballot parts. There is no way to abstain. Nash Lottery (NL).
The
Nash Lottery method uses Range Ballots. Given β , i ∈ E , and (cid:96) ∈ L ( C ), let r i ( (cid:96) ) = (cid:80) x ∈ C (cid:96) ( x ) β i ( x ) and S ( (cid:96) ) = − (cid:80) i ∈ E log r i ( (cid:96) ). If there is a unique (cid:96) ∈ L ( C ) with S ( (cid:96) ) > S ( (cid:96) (cid:48) ) for all (cid:96) (cid:48) ∈ L ( C ) \ { (cid:96) } , we put M ( β ) = (cid:96) . In the rare cases where arg max (cid:96) ∈ L ( C ) S ( (cid:96) )is not a singleton, we use that (cid:96) which our numerical optimizer (the minimize function fromthe scipy.optimize Python package with method ‘SLSQP’) returns.For formal theoretical analyses, one can use the following tie-breaker instead. Put r ki = (cid:80) x (cid:96) ( x ) k (cid:112) β i ( x ) for all k = , , , . . . , and S k ( (cid:96) ) = (cid:80) i log r ki ( (cid:96) ) ∈ [ −∞ , ∞ ). Note that all S k arecontinuous, continuously di ff erentiable, and weakly concave functions of (cid:96) . Hence S has aglobal maximum that is attained on a non-empty compact convex set T ⊆ L ( C ), and for all k ≥ S k restricted to T k − has a global maximum that is attained on a non-empty compactconvex set T k ⊆ T k − . Then also T = (cid:84) ∞ k = T k is non-empty compact convex, hence Lebesgue-measurable, and hence has a well-defined unique centre of mass (cid:96) = (cid:82) T (cid:96) d (cid:96)/ (cid:82) T d (cid:96) with (cid:96) ∈ T because of the convexity. We now put M ( β ) = (cid:96) . The rationale for using concave functions ofratings to break ties is that in this way lotteries with lower entropy are preferred. The rationalefor using the k -th square roots for this task is that in this way the tie-breaking is complete exceptin the case of clones (see below).A bullet vote is to rate one option at > > ) under the name “Nash Max Product”. The same idea is also common in theliterature on fair division ( ). Maximal Partial Consensus (MaxParC, MPC).
For our simulations, we use this version of
MaxParC ballots: • B ( C ) = [0 , C , i.e., one can assign any real-valued willingness to approve ≤ b ( x ) ≤
100 to each option x ∈ C individually. 18 x P b y i ff b ( x ) = > b ( y ) or b ( x ) > = b ( y ), i.e., we only interpret the special values100 and 0 as “stated preferences” and treat all intermediate values as inherently strategicsince their interpretation relates to other voters’ willingnesses. • b Q Cx b (cid:48) i ff b ( x ) > b (cid:48) ( x ) and b ( y ) = b (cid:48) ( y ) for all y ∈ C \ { x } , i.e., “advancing” x means raisingthe willingness to approve of it.The Maximal Partial Consensus (MaxParC, MPC) method now works as follows. A voter ap-proves of an option if enough other voters do so as well; a non-abstaining voter i is drawnuniformly at random; then from the highest-scoring options approved by i , one is drawn uni-formly at random. Once it is decided who approves of what, the procedure corresponds to whatis described in ( ), page 4 (last paragraph of section 3), and analysed in ( ) under the name‘Conditional Utilitarian Rule’.To define this formally, we will introduce the following mathematical objects. The set A ( x )will be the set of voters i that turn out to approve of x since their willingness to approve of x , β i ( x ), is properly larger than 100 × (1 − | A ( x ) | / N ). The quantity a (cid:48) ( x ) will be x ’s approval scoreplus a fractional part used for tiebreaking. The set A i will be the set of options approved by i , and A (cid:48) i the set of highest-scoring options in A i . Finally, A ( ∅ ) will be those voters who don’tapprove of any option and thus “e ff ectively abstain”.Formally, their definition is this: For all x ∈ C , let A ( x ) be the largest subset A ⊆ E suchthat | A | / N + β i ( x ) / > i ∈ A . Let a (cid:48) ( x ) = | A ( x ) | + (cid:80) i ∈ E β i ( x ) / N . For all i ∈ E ,put A i = { x ∈ C : i ∈ A ( x ) } and A (cid:48) i = arg max y ∈ A i a (cid:48) ( y ). Finally, put A ( ∅ ) = E − (cid:83) x ∈ E A ( x ). Then M ( β ) x = (cid:80) { / | A (cid:48) i | : i ∈ E with x ∈ A (cid:48) i } / ( N − | A ( ∅ ) | ).A bullet vote is to rate one option at 100 and all others at 0, while abstention means ratingall options at 0.Note: since A ( x ) can be found in N log N time, the total tallying complexity is O ( kN log N ). We simulate voter preferences by using one of several di ff erent models to generate a profile ofindividual utility functions over options, u i : C → R , and then derive individual utility functionsover lotteries depending on each voters risk attitude type. Our utility models are the following. Uniform model (Unif)
In this simplest non-spatial utility model, each value u i ( x ) is drawnuniformly at random from the unit interval [0 , impartial culture model ( ). Block model (BM)
In this non-spatial utility model, there are r ≥ voter blocks whose ex-pected relative sizes s , . . . , s r are drawn independently from a log-normal distribution such that19n s j ∼ N (0 , h ), where h ≥ block size heterogeneity parameter. In particular, if h =
0, allblocks are of similar size, while larger values of h will lead to ever smaller minorities.For each voter i independently, the probability to belong to block j is then s j / (cid:80) rj (cid:48) = s j (cid:48) . Let J ( i ) be i ’s block. Then the utility u i ( x ) that voter i would get from option x is now a sum of ablock-dependent component and an individual component, u i ( x ) = U J ( i ) ( x ) + ιε i ( x ) , (2)where all U J ( i ) ( x ) and ε i ( x ) are independent standard normal variables and ι > individu-ality parameter. Spatial preference models
In the spatial theory of voting ( ) (also called “spatial cultures”in ( )), voters i and options x are represented by ideal points (or bliss points) η i and positions ξ x in a low-dimensional policy space R d , d ≥
1, and the utility u i ( x ) that voter i would get fromoption x depends in a monotonically decreasing fashion on the distance between η i and ξ x . Wedistinguish the following spatial voting models: Linear homogeneous (LH) model.
Utilities are decreasing linearly with distance, u i ( x ) = −|| η i − ξ x || (3)where the ξ x are distributed independently and uniformly on the cube [ − , d , and the η i aredistributed independently and uniformly on the cube [ − ω, ω ] d , where ω > voter hetero-geneity parameter. Quadratic homogeneous (QH) model.
Utilities decrease quadratically with distance, u i ( x ) = −|| η i − ξ x || , (4)the ξ x are distributed independently according to the multivariate standard normal distribution,and the η i according to the symmetric multivariate normal distribution with zero mean andstandard deviation ω . Gaussian homogeneous (GH) model.
As in the quadratic homogeneous model, but withGaussian utilities u i ( x ) = e −|| η i − ξ x || / σ (5)for some σ > ) calls “distributive cultures”:20 aussian allotment (GA) model. As in the Gaussian homogeneous model, but with eachoption having a di ff erent standard deviation σ x >
0, so that u i ( x ) = e −|| η i − ξ x || / σ x / ( √ πσ x ) d . (6)The interpretation is that each option x allots a unit amount of total utility to all potential idealpoints of voters using a symmetric multivariate normal distribution whose standard deviation σ x represents the broadness of option x ’s “platform”. Because of the normalization factor σ − dx ,if two options have very close positions but di ff erent broadness, voters close to their positionwill prefer the “narrower” option and voters farther away will prefer the “broader” option. Quadratic allotment (QA) model.
As in the Gaussian allotment model, but with log-transformed utilities, resulting in a quadratic functional form: u i ( x ) = −|| η i − ξ x || / σ x − d ln( √ πσ x ) . (7)The interpretation is that here option x allots a unit amount of total wealth instead of a unitamount of total utility, and voters’ utility is logarithmic in wealth. Linear allotment (LA) model.
As in the quadratic allotment model, but with ξ x and η i distributed on cubes as in the linear homogeneous model, and with a linearly decreasing utility: u i ( x ) = −|| η i − ξ x || /σ x − d ln(2 σ x ) . (8)The interpretation is that each option x allots a unit amount of total wealth to all potentialideal points of voters using a symmetric multivariate exponential distribution with density e −|| η i − ξ x || /σ x / (2 σ ) d , and that utility is logarithmic in wealth. Distribution of options’ broadnesses.
In the three allotment models, we draw the op-tions’ broadnesses σ x independently from a log-normal distribution such that ln σ x ∼ N (ln σ , (cid:37) ),where σ > median broadness and (cid:37) ≥ broadness heterogeneity parameter. Thethree homogeneous models are then equivalent to the case (cid:37) = Utility of uncertain prospects
Regarding their preferences over uncertain prospects, repre-sented as proper lotteries (cid:96) ∈ L ( C ) over options, we assume each voter is of one of three riskattitude types that determine how they derive utility functions over lotteries from their utilityfunctions over options. Expected utility theory.
We assume voters of expected utility theory (EUT) type evaluatethe utility of a lottery of options (cid:96) ∈ L ( C ) by taking the expected value of the individual options’utilities, u i ( (cid:96) ) = (cid:80) x ∈ C (cid:96) ( x ) u i ( x ). To see a major qualitative di ff erence between the above linear,21uadratic, and Gaussian models, consider the one-dimensional case of three options placedsymmetrically at ξ A = − ξ B = ξ C = σ x ≡
1, and compare the utility a non-centralvoter at η i ≥ C and to the “polar” lottery (cid:96) = ( A + B ) / A and B ). In the LH model, u i ( C ) = u i ( (cid:96) ), i.e., the voter isindi ff erent between the compromise option and the polar lottery. In the QH model, u i ( C ) > u i ( (cid:96) ),i.e., the voter prefers the compromise. In the GH model, u i ( C ) < u i ( (cid:96) ), i.e., the voter prefers thepolar lottery. More generally, this means that the quadratic / Gaussian models tend to have alarger / smaller number of potential consensus options than the linear models, respectively. Cumulative prospect theory.
Motivated by recent empirical evidence ( ), we assumethat only about 20 percent of voters are of EUT type, while the remaining 80 percent evaluate (cid:96) instead as follows.40 percent are of low-expectations cumulative prospect theory (LCP) type. Such a voter i treats all options she prefers to her least-desired one as “gains”, hence sorts the optionsby descending utility, u i ( x ) ≥ u i ( x ) . . . ≥ u i ( x k ), calculates the cumulative probabilities c j = (cid:80) jj (cid:48) = (cid:96) ( x j (cid:48) ), so that c = c k =
1, and then evaluates (cid:96) as u i ( (cid:96) ) = k (cid:88) j = w j u i ( x j ) , (9)where w j = W ( c j ) − W ( c j − ), W ( c ) = δ c γ / ( δ c γ + (1 − c ) γ ) is the probability weighting function with W (0) = W (1) =
1, and we choose δ = .
926 and γ = .
377 following the pooled groupestimates for gains from ( ).The remaining 40 percent are of high-expectations cumulative prospect theory (HCP) type.Such a voter i treats all options except her favourite one as “losses”, hence sorts the optionsby ascending utility, u i ( x ) ≤ u i ( x ) . . . ≤ u i ( x k ), calculates the cumulative probabilities c j = (cid:80) jj (cid:48) = (cid:96) ( x j (cid:48) ), so that c = c k =
1, and then evaluates (cid:96) as u i ( (cid:96) ) = k (cid:88) j = w j u i ( x j ) , (10)where w , W are as above, but now with δ = .
991 and γ = .
397 following the pooled groupestimates for losses from ( ). Generic utilities.
Note that due to the involvement of independent continuously distributedutility components, in all our utility models the resulting utility functions u i will be generic withprobability one in the following sense. Di ff erent rational-valued lotteries (cid:96) (cid:44) (cid:96) (cid:48) ∈ L ( C ) ∩ Q C will have di ff erent utilities u i ( (cid:96) ) (cid:44) u i ( (cid:96) (cid:48) ), so that each voter will have strict preferences over allpairs of rational-valued lotteries. In particular, each voter will have a unique favourite option f i = arg max x ∈ C u i ( x ) ∈ C . 22 .3.2 Voting behaviourPolling and final voting rounds. In our simulation model, the actual voting round is precededby a number R > favourite polling scores f p ( x ) and approval polling scores a p ( x ) are published so that voters can base their voting behaviour inlater polling rounds and the final voting round on this information. The actual voting roundis then assumed to consist of an initial ballot that can then be changed for some time in aninteractive phase as a response to the current ballots’ tallying results, so that our setup allowsfor the simulation of the emergence of strategic equilibria.We assume that each voter i is of either of five behavioural types τ ( i ): sincere, lazy, heuristic,trial-and-error, or factional. Heuristic and factional voters together form the set of “strategic”voters, which we assume to make up about half the electorate. Trial-and-error and factionalvoters together form the set of “interactive” voters who will potentially change their ballotsduring the interactive phase. For lack of empirical data on interactive voting systems, we assumethat these also make up about half of the electorate, while the rest will stick to their ballotsduring the interactive phase. Lazy voters, filling in their ballots in the simplest possible andnon-strategic way, are assumed to make up about one sixth of the electorate. We assume thatabout another sixth is sincere and fills in their ballots non-strategically to best represent theiractual preferences. Together with the above assumptions, this implies that also about one sixthis heuristic, using a simple form of strategic reasoning, about one sixth is of trial-and-error type,starting sincerely but testing simple modifications during the interactive phase, and about onethird is of factional type, starting heuristically and following best-response strategies proposedby their faction leaders during the interactive phase. Although some studies also suggest thatsome voters do not su ffi ciently understand elections in order to vote sincerely or at least properlylazily, but will rather vote more or less erratically, we do not include an erratic type here sinceit would only increase the noise in the data. One of the few countries in which the election outcome can be used for a rough assessment of the percentageof voters who take into account strategic reasoning is Germany because of the strategic incentive to “split vote”by voting for di ff erent parties with your first and second votes. Several studies show that in recent parliamentaryelections about half of the voters who had an incentive to “split vote” because their favoured party had no chance ofwinning a direct mandate actually did split their vote ( ( ), p.17). Using an elaborate methodology, ( ) classifiedvoters in Germany’s 2013 parliamentary elections into several behavioural types and found that about 15.9 per centhad an incentive to split and did split, while 11.9 per cent had an incentive to split and didn’t split ( ( ) Table 1),i.e., 57 per cent of those with an incentive to vote strategically did so. One of the few systems in which the election outcome can be used for a rough assessment of the percentage ofvoters who supply less information on their ballot than would be advisable is the Single Transferable Vote systemused in Ireland’s parliamentary elections because voters may keep their submited ranking so short that during theiterative tallying process their vote gets “exhausted” and thus essentially wasted. Election outcomes suggest thatbetween 10 and 25 per cent of voters are “lazy” in this sense (the number of exhausted votes can be calculatedeasily from public data, e.g., on https://en.wikipedia.org/wiki/Dublin_Central_%28D%C3%A1il_%C3%89ireann_constituency%29 , by comparing the elected candidates total vote turnout with the number of castballots)
23n addition to this “middle”scenario, we test two further scenarios, one “strategic” and one“lazy”, with behavioural types distributed according to the probabilities listed in Table 1.We now specify our behavioural assumptions for the five types. Sincere voters ( τ ( i ) = S ). A sincere voter fills in her ballot b in a certain way that representsher “true” preferences, in particular so that the stated preferences P b are compatible with hertrue preferences in the sense that x P b y implies u i ( x ) > u i ( y ). Since many ballot types allowfor more than one way of sincere voting, we make the following explicit assumptions for thedi ff erent voting methods. Several of these make use of the benchmark lottery (cid:96) ∈ L ( C ) whosewinning probabilities are proportional to the latest favourite polling scores, (cid:96) ( x ) = f p ( x ) / N , andon its expected utility u i ( (cid:96) ) = (cid:80) x ∈ C (cid:96) ( x ) u i ( x ). The MaxParC sincere strategy also makes use ofan estimate α ∈ [0 ,
1] of the proportion of lazy voters in the electorate. • In the first polling round, she names her true favourite and approves of all x with above-average utility, using equal weights for all options. • In later polling rounds, she names her true favourite and approves of all x with above-average utility, using weights based on the latest favourite polling scores, as in ApprovalVoting. • In the actual voting round, her initial and final ballot is determined like this: – In Plurality and Random Ballot, she marks her true favourite: b = f i . – In Approval Voting, she marks all x with at-least-average utility, where the averageis weighted with the latest favourite polling scores so that “approval” is with respectto the benchmark of the currently most relevant seeming options: b ( x ) = ff u i ( x ) ≥ u i ( (cid:96) ). – In Range Voting and the Nash Lottery, she assigns ratings from 0 to 100 proportionalto utility: b ( x ) = u i ( x ) − min y ∈ C u i ( y )max y ∈ C u i ( y ) − min y ∈ C u i ( y ) . – In IRV and Simple Condorcet, she ranks all x with at-least-average utility as inApproval Voting, in correct order of preference: b ( x ) = |{ y ∈ C : u i ( y ) ≥ u i ( x ) }| i ff u i ( x ) ≥ u i ( (cid:96) ), else b ( x ) = ∞ . – In FC, she marks her true favourite as “favourite” and marks that option as “consen-sus” which has the highest approval polling score among those options she herselfapproves of: b = (arg max x ∈ C u i ( x ) , arg max x ∈ C , u i ( x ) ≥ u i ( (cid:96) ) a p ( x )). – In RFC, she combines a sincere FC ballot with a sincere Range Voting ballot. In many US elections, voters may use a so-called “straight ticket” which might be interpreted as indicating acertain level of lazyness. As there are often up to or even more than half of all voters using straight ticket voting,we assume that in the “lazy” scenario half of the voters are lazy. In the rare cases where several a p ( x ) are equal, we use f p ( x ) as a first-order tie-breaker and u i ( x ) as a second-order tie-breaker. In MaxParC, she assigns a willingness of 0 to all non-approved options, and will-ingness values from 100 α to 100 scaling linearly with utility for all other options: b ( x ) = u i ( x ) < u i ( (cid:96) ), else b ( x ) = (cid:18) α + (1 − α ) u i ( x ) − u i ( (cid:96) )max y ∈ C u i ( y ) − u i ( (cid:96) ) (cid:19) . Lazy voters ( τ ( i ) = L ). A lazy voter marks or ranks (only) her true favourite in Plurality, Ran-dom Ballot, Approval Voting, IRV, and Simple Condorcet, marks the same option as consensusin FC and RFC, and gives it a rating / willingness of 100 and all others a rating / willingness ofzero in Range Voting, the Nash Lottery, RFC, and MaxParC (“bullet voting”). Heuristic voters ( τ ( i ) = H ). Heuristic voters try to adjust their voting behaviour to that ofthe other voters in order to increase the chances of preferred options and avoid “wasting theirvote”. But since their information is restricted to polling scores, they can only act boundedlyrational. In addition, we assume they do not employ full optimization given that data but ratheruse more or less simple or moderately complex “heuristic” strategies ( ) mainly based onthe idea of “exaggerating” their stated preferences regarding the two options between which anip-and-tuck race seems most likely (
29, 30 ), and possibly taking into account next-most likelynip-and-tuck races as well. For the more complex voting methods, we do however allow forheuristics that require basic computational tasks such as forming sums, products and ratios andfollowing simple decision trees. • In polling rounds, she acts as in Plurality and Approval Voting, while in the actual votinground, her initial and final ballot is determined as follows. • In Plurality, she marks her preferred option among the two best-placed in the latestfavourite polling scores: b = y if u i ( y ) > u i ( z ), else b = z , where y = arg max x ∈ C f p ( x )and z = arg max x ∈ C \{ y } f p ( x ). • In Approval Voting, she marks all x she prefers to the option y leading the latest approvalpolling scores, and marks y i ff she prefers y to the runner-up z in the latest approval pollingscores: b ( x ) = ff ( u i ( x ) > u i ( y ) or x = y and u i ( y ) > u i ( z )), where y = arg max x ∈ C a p ( x )and z = arg max x ∈ C \{ y } a p ( x ). This is the simplest sincere voting heuristic for MaxParC that (i) guarantees that my share of winning prob-ability goes to an option which I prefer to the benchmark lottery, that (ii) leads to full consensus if applied by alland if a potential full consensus exists, and that (iii) otherwise leads to partial consensus with a high probability ifthe utility-by-distance curves are rather concave (as in the LH, QH, LA and QA models) than convex (as in the GHand GA models). See Section 2.3.1 for a more detailed discussion of this and for alternative heuristic formulas forsincere voting under MaxParC. The rationale is that your vote is most relevant in a nip-and-tuck race, and the most likely nip-and-tuck race isbetween the favourite poll’s leader and runner-up, so that you should vote for your preferred one among those two. This is called the “leader rule” in ( ), see also (
32, 33 ). Since the most likely nip-and-tuck race is betweenthe approval poll’s leader and runner-up, you should approve only your preferred among those two. Since the nextlikely nip-and-tuck race is between the leader and some other option, you should also approve of all options youprefer to the leader. In Range Voting, she applies the same strategy as in Approval Voting to find her “ap-proved” options, then assigns a rating of 100 to approved options and a rating of 0 to theother options. • In IRV, she denotes the options in descending order of their latest approval polling scoresas x , x , . . . , x k − and then constructs her ranking as follows: In rank 1 she puts either x or x depending on which she prefers, and labels the other option as y . Then, for eachrank r = , , . . . , k −
1, she puts either y or x r in rank r , depending on which she prefers,and labels the other option as the new y . • In Simple Condorcet, she finds y , z as in Approval Voting, assigns a tied rank of one toher preferred option among y , z and all options she prefers to both, assigns sincere ranksto those other options she prefers to at least one of y , z , and doesn’t rank the less preferredoption among y , z and all she considers even less desirable. • In Random Ballot, FC and RFC, she acts like a sincere voter. • In the Nash Lottery, she first uses the latest favourite polling scores to compute the utility υ of the benchmark lottery with probabilities f p ( x ) / N . She then computes her ratingfor any x based on x ’s apparent chances as estimated by f p ( x ) / N and on the di ff erencebetween u i ( x ) and υ as follows. If she is of EU type, she has υ = (cid:80) x ∈ C u ( x ) f p ( x ) / N anduses b ( x ) = + f p ( x )( u i ( x ) − υ )max y ∈ C f p ( y )( υ − u i ( y )) , (11)where the denominator is chosen so that the smallest resulting rating is exactly zero. Ifshe is LCP or HCP type, she similarly uses b ( x j ) = + w j ( u i ( x j ) − υ )max kj (cid:48) = w j (cid:48) ( υ − u i ( x j (cid:48) )) , (12)with x j , w j as described in the LCP and HCP models above. • In MaxParC, she applies the same strategy as in Approval Voting to find her “approved”options, then assigns to an approved option x a willingness that is at least as large as Following the same rationale. The rationale here is that the most likely nip-and-tuck race is between x and x , in which case her vote mustgo to the better of those from beginning on. Her 2nd ranked option only becomes relevant when the 1st ranked getseliminated during the tally, in which case the most likely race is between the other ( y ) and x , so she should rankthe better of those two 2nd. Her r -th ranked option only becomes relevant when all higher ranked get eliminated,in which case the most likely race is between the one option among x , . . . , x r − not yet ranked, which is the current y , and x r , so she should rank the better of those two next. See also ( ), Fig. 3, and ( ). The rationale is that if some options appear to have higher chances than others, exaggerating one’s preferencesregarding these options will increase one’s influence (see Section 2.3.2 for a formal derivation of this heuristicstrategy). In the special case where all f p ( x ) are equal, the strategy reduces to voting sincerely. x and large enough to make sure she is counted asapproving of x should x ’s approval score be as predicted by a p ( x ). More precisely, sheputs b ( x ) = max { b s ( x ) , − a p ( x ) / N } , (13)where b s ( x ) = max { , u i ( x ) − u i ( (cid:96) )max y ∈ C u i ( y ) − u i ( (cid:96) ) } . To a non-approved option x , she assigns awillingness at most her sincere willingness and small enough so that she is counted as notapproving of x : b ( x ) = min { b s ( x ) , − a p ( x ) / N } . (14) Interactive voting.
We assume that in the final voting round voters have to submit a ballot butcan then still change their ballots continuously over some time interval during which they canobserve the resulting tally statistics in real-time, thus introducing the possibility to interactivelytest voting strategies and react on others’ strategies. We assume there are two additional types ofvoters who will change their votes during this interval: “trial-and-error” voters and “factional”voters, while all other voters submit an initial ballot as described above and don’t change it af-terwards. This interactive phase is simulated long enough so that in typical situations a strategicequilibrium can emerge. Before the interactive phase starts, these voters behave like heuristicvoters. In the interactive phase, they vote as follows:
Trial-and-error voters ( τ ( i ) = T ). The interactive phase consists of a large number of con-secutive time points, at each of which some percentage of the trial-and-error voters will updatetheir ballots. When a trial-and-error voter i updates her ballot, she picks a random modificationout of a set of elementary modifications that depend on the ballot type (see below), submitsthe modified ballot, observes the resulting change in utility u i due to all simulateneous modifi-cations, and either sticks to or undoes the modification. She undoes the modification if either u i has decreased or if u i has stayed constant and the modification was towards a strictly less“sincere” ballot (see below).We assume these elementary modifications: • On a Plurality, FC, or RFC Ballot, either the favourite or the consensus option may bereplaced by any other option. • On an Approval Ballot, one can add or remove approval for a single option. • On a Range, MaxParC, or RFC Ballot, one can replace the rating or willingness value fora single option by any value in [0 , • A Truncated or Weak Ranking Ballot b may be replaced by another such ballot b (cid:48) if forsome option x , P b | C \{ x } = P b (cid:48) | C \{ x } , i.e., a single option x may be moved to an arbitrarynew position in the ranking, making place for it by shifting the other ranks if necessary.27n Random Ballot, trial-and-error voters will always vote sincerely since that is a dominantstrategy.A modified ballot b (cid:48) is strictly less sincere than b if: • In Plurality and FC: u i ( b (cid:48) ) < u i ( b ) (resp. u i ( b (cid:48) ) < u i ( b ) for FC). • In Approval Voting, IRV, Simple Condorcet, and MaxParC: e ( b (cid:48) ) > e ( b ) with e ( b ) = |{ ( x , y ) ∈ C : u i ( x ) > u i ( y ) but y P b x }| (number of wrongly stated binary preferences). • In Range Voting and the Nash Lottery: || b (cid:48) − b s || > || b − b s || , where b s is the sincere ballotdescribed above. • In RFC: u i ( b (cid:48) ) ≤ u i ( b ) and || b (cid:48) − b s || ≥ || b − b s || , but not both equal.Trial-and-error voters behave as sincere voters during polling and also start the interactivephase with a sincere ballot. Factional voters ( τ ( i ) = F ). Since strategic voting can be much more e ff ective when coordi-nating with other voters having similar preferences, we assume that voters of this type changetheir ballots as follows during the interactive phase, starting it with a heuristic ballot as describedabove, and after voting as heuristic voters in the polling rounds, too.For each x ∈ C , we consider the “faction” F x of all voters i with τ i = F favouring x , F x = { i ∈ E : τ i = F , f i = x } . Each faction F x is assumed to possess enough information and computingcapabilities to calculate a best unanimous response to all other voters’ current ballots, whichis a voting behaviour where all i ∈ F x submit the same filled-in ballot and no other unanimousvoting behaviour of all i ∈ F x would generate a strictly higher total utility U = (cid:80) i ∈ F x u i giventhat all other voters j ∈ E \ F x submit the same ballots as before. The assumption that factionscannot coordinate their members to vote di ff erently even if that might be better than all votingthe same way can be interpreted as a form of bounded rationality.During the interactive phase, each faction, whether small or large, has the same constantprobability rate for updating their ballots, leading to a Poisson process of updates by randomlypicked factions. When a faction F x updates their ballots, they replace their current ballots by abest unanimous response to all other voters’ current ballots as follows: • In Plurality, they find the plurality scores p ( y ) resulting from all other voters’ ballots,find the set A of options less than | F x | many votes behind the leader, A = { y ∈ C : p ( y ) + | F x | > max z ∈ C p ( z ) } , and vote for that y ∈ A which maximizes U : b = arg max y ∈ A U ( y ) with U ( y ) = (cid:80) i ∈ F x u i ( y ). • In Approval Voting, they find y in the same way as in Plurality, only using approval scoresinstead of plurality scores, and then bullet vote for it: b ( y ) = b ( z ) = z (cid:44) y .28 In Range Voting, they find y in the same way as in Plurality, only using Range Votingscores divided by 100 instead of plurality scores, and then bullet vote for it: b ( y ) = b ( z ) = z (cid:44) y . • In IRV, they find the best response truncated ranking ballot by constructing a set A of“candidate” truncated rankings ( x , x , . . . , x (cid:96) ) that cover all possible results they can e ff ectby submitting identical ballots, and then select the member of A that gives the best result. A is constructed iteratively by adding ever longer truncated rankings as follows. Givenall other voters’ ballots, they start by finding the set Y of options y ∈ C for which y survives the elimination process during the tally strictly longer when they rank y A = { ( y ) : y ∈ Y } . Then, for each ranking( x , x , . . . , x (cid:96) ) ∈ A with (cid:96) < k −
1, they find the set Y of options y for which y survives theelimination process during the tally strictly longer when they submit the longer ranking( x , x , . . . , x (cid:96) , y ) than when they submit the shorter ranking ( x , x , . . . , x (cid:96) ). They add allthose ballots ( x , x , . . . , x (cid:96) , y ) to A and iterate until no further ballots are added. One canshow that for each possible truncated ranking ballot, there is a member of A that has thesame e ff ect when used as the unanimous ballot of all faction members, and | A | ≤ k . • In Simple Condorcet, they find the binary opposition values o ( y , z ) resulting from allother voters’ ballots, and put o ( y ) = max z ∈ C o ( y , z ), o = min y ∈ C o ( y ), and A = { y ∈ C : o ( y ) < o + | F x |} . For each y ∈ A , they put A y = { z ∈ A : o ( z ) < o ( y ) } , and check whetherthere is a function g : A y → C such that { ( z , g ( z )) : z ∈ A y } is acyclic and for all z ∈ A y , o ( z , g ( z )) + | F x | > o ( y ). If this is the case, y can be made the winner by ranking the optionsin any way that ranks y first and ranks each z ∈ A y below its g ( z ). Among these y ∈ A ,they find the one with the largest U ( y ) and submit any ranking which ranks y first, rankseach z ∈ A y below its g ( z ), and doesn’t rank any further options. If no such y exists, theysubmit a bullet vote for x . • In Random Ballot, they mark their true favourite since that is a dominant strategy: b = arg max x ∈ C u i ( x ). • In FC, they mark their true favourite as “favourite” and find an optimal option for markingas “consensus” by computing the resulting U for all of the k many possible choices. • In RFC, they submit sincere ratings and find an optimal combination of options for mark-ing as “favourite” and “consensus” by computing the resulting U for all of the k manypossible combinations. • In the Nash Lottery, they try to find a (globally) best response by starting with a commonballot derived by averaging the faction members’ sincere Nash Lottery ballots (see above)and then following a simple steepest ascent optimization algorithm until reaching a (local) See Section 2.3.3 for a proof sketch. U ( x ). Although this local optimum might not be a globally best response, weassume they use the resulting ballot anyway, which can be considered an additional formof bounded rationality. • In MaxParC, they compare the results of all the 2 k many ballots b A of the form b A ( x ) = x ∈ A and b A ( x ) = A ⊆ C . They identify that A whichmaximizes U given all others’ ballots. Note that the corresponding b A is a unanimousbest response since only the resulting approvals matter. For this A , they calculate theapproval scores a y that would result from using ballot b A in MaxParC given all others’ballots. Then they define w ( y ) = (cid:88) i ∈ F x max { , u i ( y ) − u i ( (cid:96) )max z ∈ C u i ( z ) − u i ( (cid:96) ) } / | F x | , (15)which is the average sincere ballot of the faction members, and use the ballot with b ( y ) = max { − a y / N ) , w ( y ) } (16)for y ∈ A and b ( y ) = min { − a y / N ) , w ( y ) } (17)for y (cid:60) A , which leads to the same approvals as b A and is thus also a best unanimousresponse. In other words, they use that best unanimous response which is closest to theaverage sincere ballot. • In polling rounds, they act as in Plurality and Approval Voting.Note that if a steady state emerges, it approximates a pure-strategy Nash equilibrium be-tween the trial-and-error voters as individual players and the factions as aggregate players,which will in general however not be a strong or coalition-proof equilibrium since althoughwe regard factions, we do not regard inter-factional coalitional strategies. Also, the processmay also lead to cyclic or more complex attractors rather than a steady state.
We generated M = , ,
906 many independent group decision problems, drawing their pa-rameters independently from the following probability distributions (where parameter names incode are set in this font ): Number of voters N . We drew odd numbers between 9 and 999 such that log N was approx-imately uniformly distributed in the interval [1 , Number of options k . Uniformly in { , . . . , } . Preference models.
Uniformly in { Unif, BM, GA, QA, LA } .30 M parameters.
For the block model: number of voter blocks
Bmr = r ∼ Unif { , , } , blocksize heterogeneity Bmh = h ∼ Unif { , } , individuality Bmiota = ι ∼ Unif { . , . } . Spatial model parameters.
For GA, QA, and LA: policy space dimension dim = d ∼ Unif { , , } ,voter heterogeneity omega = ω ∼ Unif { , , , } , option broadness heterogeneity rho = ρ ∼ Unif { , / , / , } , where ρ = Risk attitude scenarios.
Uniformly in { all-EUT, all-LCP, all-HCP, mixed } , where in ‘mixed’20% of the voters are EUT, 40% LCP, and 40% HCP. Number of polling rounds R . Uniformly in { , , , , , } . Behavioural type scenarios.
Uniformly in { lazy, middle, strat, all-L, all-S, all-T, all-H, all-F } The following parameters were not varied:
Length of interactive phase.
100 time points.
Trial-and-error frequency.
At each time point, 50% of the trial-and-error voters updated theirballots.
Factional update probability.
At each time point, each faction had a 10% probability to up-date their ballots.For each group decision problem, we constructed a second problem in which a randomlychosen option was replaced by a compromise option y that was constructed from set C ofthe remaining k − ff ect that a specifically designed compromise op-tion would have. Depending on the preference model, voters’ preferences about y were con-structed as follows: In Unif and BM, the compromise got the average utility of the other op-tions, u i ( y ) = (cid:80) x ∈ C u i ( x ) / ( k − ξ y was chosento be a weighted average of the other options’ positions ξ x , with weights w x proportional tofirst-preference support and inversely proportional to options’ platforms’ broadness σ x : ξ y = (cid:88) x ∈ C w x ξ x / (cid:88) x ∈ C w x , (18) w x = |{ i ∈ E : x = arg max z ∈ C u i ( z ) }| /σ x . (19)For each of these 2 M decision problems, we simulated R rounds of polling. Finally, for each ofthe ten voting methods independently, we simulated an initial voting round and an interactivevoting phase based on the same polling results, and determined all options’ resulting winningprobabilities (cid:96) both after the initial voting round and after the interactive phase. For the basically deterministic IRV, we did not calculate tie probabilities since this would have been too costlydue to the iterative nature of the method; instead, we resolved ties in IRV randomly, so that the resulting lottery .3.4 Social welfare metrics To measure the welfare e ff ects of the tested voting methods, we use a set of metrics which arebased on three di ff erent social welfare measures (utilitarian, Gini-Sen, and egalitarian welfare),taken either on an absulute or a relative scale, and either taken before or after the interactivephase of the simulations, giving a total of twelfe di ff erent metrics per problem and method.All these measures are aggregating the voters’ individual utility u i ( (cid:96) ) they get from the re-sulting lottery (cid:96) , as modelled by the various utility models discussed above. In applicationswhere there is only a single decision taken, these measures must hence be interpreted as mea-suring the ‘ex ante’ e ffi ciency of the method, as opposed to the ‘ex post’ e ffi ciency that would bebased on the utilities of the actual options chosen by the resulting lottery. In applications wherewe imagine a sequence of decisions, our e ffi ciency metrics can be interpreted as measuring thelong-run e ffi ciency of the method over the whole sequence of decisions. Utilitarian welfare
The simplest and most popular measure is the one proposed by averageutilitarianism, W util. ( (cid:96) ) = (cid:80) i ∈ E u i ( (cid:96) ) / | E | .Since in all our utility models, lottery utility u i ( (cid:96) ) is a linear combination of option utilities u i ( x ), the lottery that maximizes W util. ( (cid:96) ) is a sure-thing lottery. Gini-Sen welfare As W util. ( (cid:96) ) is insensitive to redistribution of utility across voters, and henceto inequality between voters’ utilities, we also use two inequality-averse metrics, the first ofwhich is the Gini-Sen welfare function W Gini ( (cid:96) ) = (cid:80) i ∈ E (cid:80) j ∈ E \{ i } min { u i ( (cid:96) ) , u j ( (cid:96) ) } / | E | ( | E | − W util. , since we have W Gini ( (cid:96) ) = W util. ( (cid:96) )(1 − I Gini ( (cid:96) )), where I Gini ( (cid:96) ) is the well-known Gini coe ffi cient of inequality in utilities u i ( (cid:96) ) ( ).Note that the lottery (cid:96) ∗ which maximizes W Gini ( (cid:96) ) can be expected to be a proper lotteryrather than a sure-thing lottery. This is because randomization tends to reduce inequality morethan it reduces average utility. Egalitarian welfare
As the most extremely inequality-averse welfare metric, we also con-sider the egalitarian one, W egal. ( (cid:96) ) = min i ∈ E u i ( (cid:96) ). As in the case of Gini-Sen welfare, maximiza-tion of ex-ante egalitarian welfare usually requires randomization. always appeared to be a sure-thing lottery instead of the true tying lottery. For the NL method, the optimizationproblem max S ( (cid:96) ) was solved using Sequential Least Squares Programming (SLSQP); to avoid a convergencefailure due to singular Jacobian matrices because of zero ratings, we added 10 − to all ratings (the maximal ratingalways being 100). In the interactive phase of NL, a faction’s best response ratings optimization problem wassolved using Constrained Optimization By Linear Approximation (COBYLA) since that converged better thanSLSQP. bsolute and relative welfare metrics All three welfare metrics measure welfare on thesame scale as individual utility, hence are hard to compare directly across di ff erent utility modelssince these use quite di ff erent scales. Also, for some models their distribution is quite skewed,having a long lower tail. In addition to the above absolute welfare metrics, we therefore alsocompare the relative metrics relW util. / Gini / egal. ( (cid:96) ) = W util. / Gini / egal. ( (cid:96) ) − min x ∈ C W util. / Gini / egal. ( x )max x ∈ C W util. / Gini / egal. ( x ) − min x ∈ C W util. / Gini / egal. ( x ) ∈ [0 , ∞ ] (20)which rescale the welfare so that the sure-thing (!) lotteries giving the lowest and highestwelfare get scores 0 and 1, respectively. This design allows us to interpret values larger than 1as welfare gains from randomization.Still, as it turned out, the relative versions of Gini-Sen and egalitarian welfare often takevery large values for nondeterministic methods and thus now have a very skewed distributionwith a long upper tail. For this reason, we also study an alternative relative version of all threemetrics, defined as altrelW util. / Gini / egal. ( (cid:96) ) = W util. / Gini / egal. ( (cid:96) ) − min x ∈ C W util. / Gini / egal. ( x ) W util. / Gini / egal. ( (cid:96) ) + max x ∈ C W util. / Gini / egal. ( x ) − x ∈ C W util. / Gini / egal. ( x )(21) = relW util. / Gini / egal. ( (cid:96) )1 + relW util. / Gini / egal. ( (cid:96) ) ∈ [0 , . (22)These are now restricted to the interval [0 , “Cost of fairness” As an alternative to the above relative welfare metrics, one can also com-pare welfare di ff erences between methods with utility di ff erences within the electorate to assessthe influence of method choice on welfare. In analogy to the notion of a “price of anarchy” ( ),we therefore define a “relative cost of fairness”, CF = W RV util − W MPC util W RV util − W RV egal , (23)where the numerator is the absolute di ff erence in average voter utility between the best deter-ministic method RV’s result and the best proportional method MaxParC’s result (which couldbe termed the “absolute cost of fairness”), and the denominator is the di ff erence between theaverage and minimum voter utility under RV (which could be termed the “absolute egalitarianinequality”). 33 .3.5 Randomization metrics To measure the degree of randomization a voting method actually applies, we computed twoestablished entropy measures, Shannon entropy and R´enyi entropy of degree two, and the max-imal probability max x ∈ C (cid:96) x , again applied to the results before and after the interactive phase.This gives six randomization metrics in total per problem and method. As another type of performance indicators, we computed each voter’s “satisfaction level” u i ( (cid:96) ) − min x ∈ C u i ( x )max x ∈ C u i ( x ) − min x ∈ C u i ( x ) ∈ [0 , , (24)which would be zero if i ’s least preferred option won for sure, and unity if i ’s favourite won forsure. Based on these, we report average satisfaction levels in the whole electorate and, to assesspossible advantages of strategic behaviour, by behavioural type. Finally, to get an idea of which methods voters would chose if that choice was itself performedby majority voting, we counted for each decision problem how many voters would prefer thelottery resulting from some method A to that resulting from some method B . A voting method is anonymous i ff it treats all voters alike, i.e., i ff its result isinvariant under permutations of voters. All considered methods have this property. Neutrality.
A voting method is neutral i ff it treats all options alike, i.e., i ff the resulting win-ning probabilities of any two options x , y are swapped when x , y are swapped on all ballots. Allconsidered methods have this property. Pareto-e ffi ciency w.r.t. stated preferences. An option y is Pareto-dominated w.r.t. statedpreferences i ff there is another option x with x P β i y for all i ∈ E . A voting method is Pareto-e ffi cient w.r.t. stated preferences i ff all Pareto-dominated options get zero winning probability.All considered methods except FC and RFC fulfill this. Since in FC, only the fall-back optionis interpreted as stating a preference, a Pareto-dominated y might still be everyone’s proposed34onsensus and win. Similarly, since in RFC only the ratings are interpreted as preferences, y still might be named by someone as fall-back option and thus have positive winning probability.It is more di ffi cult to check whether also an option which is Pareto-dominated w.r.t. true preferences will have zero winning probability, since this depends on whether and how votersbehave strategically. Our numerical simulations at least suggests that all of the consideredmethods, including FC and RFC, fulfill this criterion under normal circumstances. Although there are a number of variants of the ‘monotonicity’ criterion, we here focus ontwo variants of Woodall’s ‘mono-raise’ monotonicity ( ), which di ff er only really for non-deterministic methods, and one properly weaker property related to Woodall’s ‘mono-add-plump’ monotonicity. Strong mono-raise monotonicity.
A voting method is strongly mono-raise monotonic i ff thewinning probability of an option y cannot increase if a di ff erent option x is advanced on oneballot: M ( β ) y ≤ M ( β (cid:48) ) y whenever x (cid:44) y , β i Q Cx β (cid:48) i for some i ∈ E , and β j = β (cid:48) j for all j ∈ E \ { i } . Weak mono-raise monotonicity.
A voting method is weakly mono-raise monotonic i ff thewinning probability of an option x cannot decrease if x is advanced on one ballot: M ( β ) x ≥ M ( β (cid:48) ) x whenever β i Q Cx β (cid:48) i for some i ∈ E and β j = β (cid:48) j for all j ∈ E \ { i } . Weak mono-raise-abstention monotonicity.
We call a voting method weakly mono-raise-abstention monotonic i ff the winning probability of an option x cannot decrease if x is advancedon an abstention ballot: M ( β ) x ≥ M ( β (cid:48) ) x whenever β i Q Cx β (cid:48) i for some i ∈ E , β j = β (cid:48) j for all j ∈ E \ { i } , and β (cid:48) i is an abstention ballot.Obviously, strong mono-raise monotonicity implies weak mono-raise monotonicity, whichin turn implies weak mono-raise-abstention monotonicity.For PV, AV, RV, SC, and RB it is straightforward to prove all three forms of mono-raise mono-tonicity (exercise left to the reader). IRV is known to violate both strong and weak mono-raisemonotonicity ( ) but is easily seen to fulfill weak mono-raise-abstention monotonicity.FC fulfills weak but not strong mono-raise monotonicity since if z is everyone’s proposedconsensus, advancing x on some consensus ballot destroys the consensus so that someone’sfall-back option y (cid:44) x can get gets positive winning probability.RFC violates both weak and strong mono-raise monotonicity. Consider the case of threeoptions x , y , z and two voters who both name z as consensus and rate ( x , y , z ) at (0 , , x and the other y as fall-back, x and y both get winning probability 1 /
2, but if both name x as fall-back, z wins for sure. 35L violates strong mono-raise monotonicity. Again consider three options x , y , z and two voters1 , β ( x , y , z ) = (1 / , , /
6) and β ( x , y , z ) = (0 , , / M ( β ) y =
0. Butif we increase β ( x ) to 1, we get M ( β ) y = / >
0. Numerical simulations suggest that NLfulfills weak mono-raise monotonicity, which we conjecture but were not able to prove yetunfortunately.Regarding weak mono-raise-abstention monotonicity, it was shown in ( ) that NL (therecalled “Nash Max Product”) fulfills a roughly equivalent condition of “Strict Participation”when all ballots are “dichotomous” in the sense that all ratings are either zero or 100 (or someother common, fixed, positive number). Using the Envelope Theorem, we can give an alter-native proof of weak mono-raise-abstention monotonicity for arbitrary ratings. Proof.
Let E , C be fixed, consider some i ∈ E and x ∈ C , assume β : E → B ( C ) is fixed except for its entry β i ( x ), and assume β i ( y ) = (cid:15) for all y ∈ C \ { x } . We will study the change of the NL probabilities p ∗ ( α ) = M ( β ) as a function of the parameter α = β i ( x ) and show that α (cid:48) > α implies p ∗ ( α (cid:48) ) ≥ p ∗ ( α ), which will su ffi ce to prove the claim. p ∗ ( α ) is the solution of the maximization of thecontinuously di ff erentiable function f ( p , α ) = (cid:80) j ∈ E log h j ( p , α ) with h j ( p , α ) = (cid:80) y ∈ C β j ( y ) p y and β i ( x ) = α under the constraint g ( p , α ) = (cid:80) y ∈ C p y =
1. Let V ( α ) = max p , g ( p ,α ) = f ( p , α ) be the cor-responding maximum. Since the constraint is independent of the parameter α , the envelopetheorem implies that V (cid:48) ( α ) = ∂ f ( p , α ) ∂α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = p ∗ ( α ) = p ∗ ( α ) x h i ( p ∗ ( α ) , α ) . (25)Now assume an infinitesimal increase in α from α = α to α = α = α + d α with d α >
0. Thenthe above implies V ( α ) = V ( α ) + V (cid:48) ( α ) d α = f ( p ∗ ( α ) , α ) + p ∗ ( α ) x h i ( p ∗ ( α ) , α ) d α, (26)but also V ( α ) = f ( p ∗ ( α ) , α ) (27) = f ( p ∗ ( α ) , α ) + ∂ f ( p , α ) ∂α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = p ∗ ( α ) ,α = α d α (28) = f ( p ∗ ( α ) , α ) + p ∗ ( α ) x h i ( p ∗ ( α ) , α ) d α. (29)Since optimization means that f ( p ∗ ( α ) , α ) ≥ f ( p ∗ ( α ) , α ), this implies0 ≤ f ( p ∗ ( α ) , α ) − f ( p ∗ ( α ) , α ) d α (30) = p ∗ ( α ) x h i ( p ∗ ( α ) , α ) − p ∗ ( α ) x h i ( p ∗ ( α ) , α ) (31) = p ∗ ( α ) x ( α − (cid:15) ) p ∗ ( α ) x + (cid:15) − p ∗ ( α ) x ( α − (cid:15) ) p ∗ ( α ) x + (cid:15) , (32)36ince (cid:15) >
0, this implies p ∗ ( α ) x ≥ p ∗ ( α ) x . Since this holds for all values α of α , we haveshown that p ∗ ( α ) x is a weakly increasing function of α as claimed. Q.E.D.
MaxParC fulfills all three forms.
Proof.
It su ffi ces to show that if (i) x is advanced by one voter i from β i ( x ) = r to β i ( x ) = r (cid:48) > r , (ii) some voter j is drawn at random, and (iii) some option y (cid:44) x is in the set A (cid:48) j after the change, then y must have been in A (cid:48) j before the change and A (cid:48) j canonly have grown due to the change. It is easy to see that A ( x ) can only have grown and that noother A ( z ) has changed, hence a (cid:48) ( x ) has properly grown but no other a (cid:48) ( z ) has changed. So if y is in A (cid:48) j = arg max z ∈ A j a (cid:48) ( z ), the value of max z ∈ A j a (cid:48) ( z ) has not changed, hence y must have beenin this set before, and the only change in A (cid:48) j can be that now x is also in A (cid:48) j . This means A (cid:48) j canonly have grown and thus y ’s winning probability decreased. Q.E.D.
The idea of this criterion is that the“removal” of a Pareto-dominated option y from all ballots should have no e ff ect on the winningprobabilities. It was first introduced by Steve Eppley on the election-methods emailing list. Since for some methods it is not obvious what is meant by “removal”, we do not study this in aformal way here but rather discuss it verbally.For methods using a ballot type that lets voters rate or rank all options independently (AV,RV, SC, NL and MaxParC), let us assume “removal” means leaving the other options’ ratingsunchanged. For IRV, let us assume “removal” implies decreasing the ranks of the later-rankedoptions by one. Then those six methods all fulfill this criterion, and so do Plurality and RandomBallot whenever no voter has named y as favourite (which rational voters wouldn’t).As this criterion implies Pareto-e ffi ciency, FC and RFC do not fulfill it.We note that in particular many other Condorcet-type methods, which elect a winner of allpairwise comparisons for sure if such an option exists, including the ‘Ranked Pairs’ method byNicolaus Tideman ( ) and the ‘Beatpath’ methods by Markus Schulze ( ), fail this criterion. Independence from losing options.
This criterion demands that the removal of any option y receiving zero winning probability must have no e ff ect on the winning probabilities. Thisis a variation of the famous ‘Independence of Irrelevant Alternatives’ criterion, and which isstronger than Independence from Pareto-dominated alternatives if Pareto-e ffi ciency is given.It is easy to see that again AV, RV, RB, and NL fulfill this and FC and RFC do not. For PV,some voters may have voted for y and now vote for the current runner-up and make it win.IRV and SC also do not, as can be seen from the example of three options x , y , z and threefactions F , , of sizes 4 , , F : x > y > z , F : y > z > x , F : z > x > y . Bothmethods elect x for sure but elect z if y is removed. http://lists.electorama.com/pipermail/election-methods-electorama.com//2003-March/107700.html Proof.
Removal of y does not change who approvesof which other options. If y has zero winning probability, every voter who approves of y alsoapproves of some higher-scoring option. Hence for no voter the set of highest-scoring approvedoptions changes. Thus all other options’ winning probabilities are una ff ected. Q.E.D.
An even stronger variant of ‘Independence of Irrelevant Alternatives’ that can also be in-terpreted as a form of ‘monotonicity’ goes as follows: removing any option y from E mustnot decrease any remaining option x ’s winning probability. NL probably violates this whileMaxParC clearly fulfills it. Independence from cloned options.
Another type of criterion deals with the addition of anoption y , called a ‘clone’, that is very “similar” to some existing option x . Since “similarity”can be defined in di ff erent ways depending on the ballot type, we restrict our interest here tothe special case where y is a unique ‘exact clone’ of x , meaning all voters are truly indi ff erentbetween x and y but not between these two and any further option z . We demand that in thatcase and under plausible assumptions on voter’s voting behaviour, the addition of y shall notchange the winning probability of any other option z (cid:60) { x , y } .Let us assume that after the addition of y , voters will assign y the exact same approval,rating, or ranking (if tied rankings are allowed, otherwise an adjacent ranking) as x , and willname any option z (cid:60) { x , y } as favourite or proposed consensus i ff they named the same optionbefore the addition, only possibly switching from naming x to now naming y . It is then easyto see that AV, RV, IRV, SC, RB, RFC, NL and MaxParC all fulfill this form of ‘exact cloneindependence’, while PV and FC do not.Note that there are other, stronger, forms of clone-independence, including the one discussedin ( ), that some variants of IRV, many Condorcet-type methods, NL, and MaxParC might notfulfill. Revelation of preferences.
Some voting methods have the property that, sometimes depend-ing on the level of strategic behaviour, voters’ filled-in ballots reveal all or part of their prefer-ences. Under RB, for example, whenever a voter has an option she strictly prefers to all otheroptions (a unique favourite), it is a weakly dominant strategy to specify that option. This formof “strategy-proofness” can be interpreted as implying that RB “reveals unique favourites” (butnothing else about voters’ preferences).As another example, it was shown in ( ) that under RFC, whenever a voter has prefer-ences conforming to expected utility theory (see below) with some utility function u , she has nostrategic incentive to specify di ff erent ratings than a properly rescaled version of u , and henceRFC can be said to “reveal von-Neumann–Morgenstern utility functions” (but no preferencesthat do not conform to expected utility theory). Still, RFC is not strategy-proof in the sensethat there always exist weakly dominant strategies, since in its other two ballot components,a rational voter may want to name a proposed consensus option that depends on others’ pref-erences, and may have incentives to name a di ff erent option as “fall-back” than her favourite.38his shows that full preference revelation is related to but neither implied by nor stronger thanstrategy-proofness.FC reveals favourites but its consensus ballot component is strategic. NL and MaxParC alsoreveal favourites in the sense that a voter has no incentive to not rank her favourite first or to rateit below 100, but usually has an incentive to rate all other options strictly below 100. Neither ofthem however reveal much more of a voter’s preferences.AV and RV don’t reveal favourites since typically a rational voter has an incentive to approveof (or rate at 100) some additional options. Still, in the case where voters have no informationabout others’ preferences, AV and RV can be said to reveal something about a voter’s prefer-ences, because in that case a rational expected utility theory voter would approve of (or rate at100) all options she prefers to drawing an option uniformly at random (and would rate all otheroptions at 0), so that one can infer that she strictly prefers each approved to each disapprovedoption.Similarly, under NL, expected utility theory voters who use the zero-information heuristicderived in the end of 2.3.2 also reveal their full preferences, but this heuristic might not bea weakly dominant strategy under zero information, so rational voters may not use it. UnderMaxParC, the linear heuristic derived in 2.3.1 reveals the above-average part of a voter’s utilityfunction but is also typically not weakly dominant under zero information.IRV and SC also do not reveal favourites. For IRV, consider three options and six votersand assume voter 1 has preferences 1 : A > B > C and the others vote A > B , A , B > C , C > A , C . Then if 1 votes sincerely, B is removed and a coin toss between A and C results. But if 1votes B > A > C , A still gets probability 1 / B gets 1 / C gets 1 /
6, which 1 strictlyprefers. For SC, consider three options and three voters and assume voter 1 has preferences1 : A > B > C and the others vote B = C > A , C > A > B . Then 1 would want to vote A = B > C or B > A > C to ensure a coin toss between B and C rather than voting sincerely A > B > C andgetting C for sure. ff ective power This criterion requires that in every situation ( C , E ) and for every option x ∈ C and group ofvoters G ⊆ E , there must be a way of voting β G ∈ B ( C ) G for G so that for all ways of voting β − G ∈ B ( C ) E \ G of the other voters, the winning probability of x is at least as large as G ’s relative size: M ( β G , β − G ) x ≥ | G | / | E | . A related criterion was discussed for the special case of “dichotomouspreferences” under the name “Core Fair Share” in ( ).Since all considered methods are neutral and anonymous, one can summarize the power dis-tribution by drawing the maximal winning probability a group of size | G | / | E | = s can guaranteeany option x of their choice under the various methods, as is done in Fig. 5. For all consid-ered deterministic methods, this “e ff ective decision power” is basically a step function with thevalue zero for s < / s > / s = / G can guarantee x a39 .0 0.2 0.4 0.6 0.8 1.0group size |G|/|E|0.00.20.40.60.81.0 e ff e c t i v e p o w e r majoritarianproportionalfirst to get two Figure 5: E ff ective power of a voter group G as a function of group size, for majoritariandeterministic methods PV, AV, RV, IRV, SC (blue), proportional nondeterministic methods RB,FC, RFC, NL, MaxParC (green), and an example of a nonproportional nondeterministic method(yellow).probability at least | G | / | E | by simply bullet-voting for x , so e ff ective decision power is simplyequal to s (green line), which we call proportional allocation of e ff ective power. Note that of course there are also non-deterministic neutral anonymous methods with dif-ferent allocations of e ff ective power. E.g., one could draw a sequence of plurality ballots atrandom until one option was named twice (“first to get two”), giving a smooth but S-shapednonlinear power curve s (3 − s ).For NL, it is the specific use of the logarithm that gives a linear power curve. Indeed, con-sider a method that puts M ( β ) = arg max (cid:96) S ( (cid:96) ) for S ( (cid:96) ) = (cid:80) i ∈ E f ( r i ( (cid:96) )), some weakly increasingand continuously di ff erentiable function f , and r i ( (cid:96) ) = (cid:80) x ∈ C (cid:96) x β i ( x ). If the power curve is linear,then whenever a group of voters G (cid:60) {∅ , E } bullet-votes for x and the other voters E \ G bullet-vote for y , we must have p : = M ( β ) x = | G | / | E | = : s and q : = M ( β ) y = − | G | / | E | = − s , hencethe first-order condition0 = ( ∂ p − ∂ q ) S | p = s , q = − s = s f (cid:48) (100 s ) + (1 − s ) f (cid:48) (100(1 − s )) (33)implies f (cid:48) (100 s ) ∝ / s for all rational numbers s ∈ (0 ,
1) and thus f ( r ) ∝ log r for all real numbers r ∈ (0 , In this section, we show that our two focus methods NL and MPC support both full and partialconsensus even with strategic voters. To do so, we show that the respective potential consensus40ptions result both from sincere voting (see 2.3.1 for a discussion of sincere voting in MaxParC)and in several forms of strategic equilibrium in archetypial decision situations.
Nash Lottery supports full consensusAssumptions.
We assume two equal-sized factions F , F of m many voters each, andthree options A , B , D . Voters in F have von-Neumann–Morgenstern utility function u ( A , B , D ) = (1 , , u ) and submit ratings r ( A , B , D ) = (1 , , r ), those in F have u ( A , B , D ) = (0 , , v ) and sub-mit r ( A , B , D ) = (0 , , s ) with r , s ∈ (0 ,
1) and 1 / < u , v <
1, so that both factions prefer D to acoin toss between A and B . Resulting lottery and expected utilities. If A , B , D get probabilities p , q , − p − q , theresulting expected utilities are U = p + (1 − p − q ) u , (34) U = q + (1 − p − q ) v , (35)and the Nash sum is f = m log( p + (1 − p − q ) r ) + m log( q + (1 − p − q ) s ) . Because f is concave in both p and q , the unique pair p , q maximizing f can be found as follows.Given q ∈ [0 , f is maximized by that p ∈ [0 , − q ] which is closest to the point p ( q ) of zeroslope, 0 = ∂ p f = − rp + (1 − p − q ) r + − sq + (1 − p − q ) s , (36) p ( q ) = (1 − q )(1 − r ) s + (1 − r ) q − r ) s . (37)Similarly, given p , f is maximized by that q ∈ [0 , − p ] closest to q ( p ) = (1 − p )(1 − s ) r + (1 − s ) p − s ) r . (38)If we introduce the notation [ x ] y = max(0 , min( x , y )), the maximum of f is thus attained where p = (cid:34) (1 − q )(1 − r ) s + (1 − r ) q − r ) s (cid:35) − q , (39) q = (cid:34) (1 − p )(1 − s ) r + (1 − s ) p − s ) r (cid:35) − p . (40)41epending on r , s , the solution ( p , q ) found by the Nash Lottery method and resulting utilities( U , U ) are the following:( p , q , U , U ) = ( , , , ) r + s < , (1 − r , , − r + u , v )2 − r s ≤ , r ≥ − s , (0 , − s , u , − s + v )2 − s r ≤ , s ≥ − r , (0 , , u , v ) s , r ≥ . (41) Outcome with sincere voters.
Sincere voters put r = u > / s = v < / p = q =
0, i.e., the consensus option D wins for sure. Strategic equilibria between factions.
To analyse strategic incentives for the two fac-tions, we treat F , F as the players of a two-player game in which they simultaneously choose r , s ∈ (0 , s , F ’s best responsesare the following: If s < min( , − u ), each r < − s is a best response. If 1 − u ≤ s < , each r ≥ − s is a best response. If s > , only r = − s is a best response. If s = ≥ u , each r ≤ isa best response. Finally, if s = < u , only r = is a best response.Since we assume u , v > , this results in the following sets of NE. Any combination r , s < is a NE giving only U = U = . Any combination ( r , − r ) with r ∈ (1 − v , ) is a NE giving( U , U ) = (1 + u − r , v )2 − r . Any combination (1 − s , s ) with s ∈ (1 − u , ) is a NE giving ( U , U ) = ( u , + v − s )2 − s . Finally, s = r = is the “focal” NE giving ( U , U ) = ( u , v ) and the largest utility sum U + U of all NE. Summary.
The above analysis shows that in this scenario, Nash Lottery supports fullconsensus with both sincere voters (who would put r = u and s = v ) and strategic voters (whowould put s = r = ). We conjecture that similar calculations will show that the same holds withmore and unequally sized factions. Nash Lottery supports partial consensus
Assume that to the above we now add a thirdfaction F of size n − m and a fourth option C , and utilities u ( A , B , D , C ) = (0 , , , u ( C ) = u ( C ) = Strategic equilibria between factions. F has a dominant strategy to bullet-vote for D ,i.e., put r ( A , B , D , C ) = (0 , , , F , F have no reason not to put r ( C ) = r ( C ) =
0. Ifwe parameterize the probabilities of ( A , B , D , C ) as ( p (1 − w ) , q (1 − w ) , (1 − p − q )(1 − w ) , w ), theNash sum becomes f = m log( p + (1 − p − q ) r ) + m log( q + (1 − p − q ) s ) + m log(1 − w ) + ( n − m ) log( w ) , w = − m / n and the same values of ( p , q ) as above. Since F , F ’sutilities are proportional to the case above, U = (1 − w )( p + (1 − p − q ) u ) , (42) U = (1 − w )( q + (1 − p − q ) v ) , (43)the strategic analysis is the same as before, so putting r = s = is again the utility-maximizingand focal equilibrium. Outcome with sincere voters.
Sincere voters in F and F still put r = u > / s = v < / β i ( C ) =
0, and those in F bullet-vote for C . They still get p = q = w = − m / n , i.e., the partial consensus option D now gets probability 1 − w = m / n = ( | F | + | F | ) / | E | ,as required.This shows that in this scenario, Nash Lottery also supports partial consensus. We conjecturethat the same holds with more and unequally sized factions, and with several partial consen-susses between di ff erent sets of factions. MaxParC supports full consensus
Note that under MaxParC, one has never an incentive toapprove of a worst-liked option or to disapprove of one’s favourite. Since the following is notrestricted to voters with expected utility preferences, we don’t use von-Neumann–Morgensternutility functions u i here but rather state a voter’s preferences over lotteries of options (cid:96), (cid:96) (cid:48) bymeans of the binary relations (cid:96) P i (cid:96) (cid:48) (strict preference for (cid:96) over (cid:96) (cid:48) ), (cid:96) R i (cid:96) (cid:48) (weak preference),and (cid:96) E i (cid:96) (cid:48) (indi ff erence), only assuming that R i is a quasi-ordering (not necessarily complete)and that P i and E i are its antisymmetric and symmetric parts. Assumptions.
Assume m ≥ F j with sizes N j ≥ N = (cid:80) mj = N j , with distincttrue favourites x j , and assume voters are indi ff erent about pairs of other factions’ favourites: x j (cid:48) E i x j (cid:48)(cid:48) for all i ∈ F j if j (cid:48) (cid:44) j (cid:44) j (cid:48)(cid:48) . Let (cid:96) b be the benchmark lottery of drawing a randomvoter’s favourite: (cid:96) b ( x j ) = N j / N . Assume there is just one more option y , and this is a potentialfull consensus option: y P i (cid:96) b for all i .Assume the MaxParC ballot profile β has β i ( x j ) = β i ( x j (cid:48) ) =
0, and 0 < β i ( y ) ≤ / N for all j (cid:44) j (cid:48) and i ∈ F j . Then each i ∈ F j approves of x j and y , hence all vote for y and y is thesure winner. Outcome with sincere voters.
As discussed in 2.3.1, there is no unique way to vote “sin-cerely” in MaxParC, hence we rather discuss the results of voters applying one of the heuristicsdiscussed there.A voter applying the conservative satisficing heuristic rates their favourites x j at 100 andthe compromise y at 100(1 − u i ( (cid:96) b ) / u i ( y )) >
0. Also with the informed satisficing heuristic, the43inear heuristic, and the hyperbolic heuristic, voters rate y at >
0. So if all voters apply one ofthese heuristics, y wins for sure. Nash equilibrium.
Since no i ∈ F j can make any i (cid:48) (cid:60) F j vote for x j , the only way i couldonly improve the result would be by making the vote of some i (cid:48) ∈ F j go to x j instead of to y . But this is only possible by lowering β i ( y ) to zero (either certainly or with some positiveprobability), which will make everyone disapprove of y and vote for their favourites, resultingin (cid:96) b . Since no mixture of (cid:96) b with the sure-thing lottery (cid:96) y is an improvement for i , β is a Nashequilibrium in pure strategies.
Likewise, any group of voters G ⊆ F j from the same faction could only improve the resultfor each of them by making the vote of some i (cid:48) ∈ F j go to x j . As above, this is only possible ifat least one i ∈ G lowers β i ( y ) to zero, again resulting in (cid:96) b , which is no improvement. Hence β remains a Nash equilibrium when some group of voters from the same faction is considered toact as one player; in particular, if each faction is considered one player (this could be called a“factional Nash equilibrium”).Still, as with other voting methods, it is easy to see that there are many other Nash equilibria(e.g., the less e ffi cient one where everyone bullet-votes, resulting in (cid:96) b ), so the criterion ofbeing a Nash equilibrium is not su ffi ciently discriminatory and stronger game-theoretic solutionconcepts are called for. Strong Nash equilibrium.
Assume a proper subgroup G ⊂ E intersecting at least twofactions, let’s say it intersects the factions F , . . . , F r . Assume the voters in G change theirballots in some way that improves the result for them all. Assume some i ∈ G stops approvingof y . Then the votes of all i (cid:48) (cid:60) F + · · · + F r will go to their favourites, hence, for j = . . . r , atleast N j + x j for this to be an improvement for all in G , which is impossiblesince there are only N + · · · + N r votes left to distribute. Hence no i ∈ G stops approving of y ,and no x j gets approval by all voters, so y is still the sure winner, and there is no improvementfor G after all.Finally, assume the whole electorate could improve the result for all. Then there would be (cid:96) (cid:48) (cid:44) (cid:96) b with (cid:96) (cid:48) P i y for all i . Hence there would be (cid:96) (cid:48)(cid:48) (cid:44) (cid:96) b with (cid:96) (cid:48)(cid:48) P i y for all i and (cid:96) (cid:48)(cid:48) ( y ) = (cid:96) (cid:48)(cid:48) (cid:44) (cid:96) b implies there is j with (cid:96) (cid:48)(cid:48) ( x j ) < (cid:96) b ( x j ), so all i ∈ F j would have y P i (cid:96) b P i (cid:96) (cid:48)(cid:48) , acontradiction.This shows that β is a strong Nash equilibrium, i.e., no group whether small or large, unan-imous or cross-faction, has an incentive to deviate from β . Other methods.
Under majoritarian methods, in particular PV, AV, RV, IRV, and SC, thereis usually no Nash equilibrium between the factions that would give y positive winning prob-ability, simply because whenever N j > N / j , faction F j will enforce that x j wins.Also, RB does not support full consensus since for all F j it is strictly dominant to vote for x j .FC and RFC however do support full consensus, as shown in ( ).44ith sincere voters, only AV and RV also support full consensus, while PV, IRV and SCwould still elect x j whenever N j > N / MaxParC favours full over partial consensusAssumptions.
As a generalization of the above, assume now that there is an additionaloption z , considered a potential partial consensus by the union H = F + · · · + F h ⊂ E of some ofthe factions, and considered equally bad by all others, so that (cid:96) b / z P i (cid:96) b for all i ∈ H and z E i x j for all i (cid:60) H + F j , where (cid:96) b / z is the result of all i ∈ H voting for z and all others voting for theirfavourites: (cid:96) b / z ( z ) = | H | / N and (cid:96) b / z ( x j ) = | F j | / N for all j > h .Consider an extension of the above ballot profile β with 0 < β i ( z ) ≤ + N − | H | ) / N and β i (cid:48) ( z ) = i ∈ H , i (cid:48) (cid:60) H . Note that then all i ∈ F j approve of x j , all i ∈ H approve of z , andall approve of y , hence again y is the sure winner. Outcome with sincere voters.
A voter applying one of the heuristic in 2.3.1 will rate y at > z at 0 if she is not a member of H . Hence if all voters apply some of theseheuristics, y will be strictly more approved than z and still win for sure. Strong Nash equilibrium.
If all i ∈ H consider full consensus still better than their po-tential partial consensus, i.e., y P i (cid:96) b / z for all i ∈ H , then we will show that β is again a strongNash equilibrium, at least when all i ∈ H have von Neumann–Morgenstern expected utilityfunctions u i ( (cid:96) ) over lotteries. Assume some group G can improve the result to some lottery (cid:96) (cid:48) by modifying their ballots. If no i ∈ G stopped approving of y , y would remain the surewinner, hence some i ∈ G stops approving of y and the votes of all i (cid:48) ∈ H − G go to z , whilethose of i ∈ F j − G for j > h go to x j . As above, for all j with F j ∩ G (cid:44) ∅ , at least N j + x j for this to be an improvement for all i ∈ F j ∩ G , hence less than | H | votes are left that could go to either z or some of x , . . . , x h . Those i ∈ F j ∩ G , j ≤ h ,have Nu i ( (cid:96) (cid:48) ) = v z u i ( z ) + v j u i ( x j ), where v z , v j are the votes going to z or x j , respectively, and v z + (cid:80) hj = v j < | H | . Since Nu i ( y ) > Nu i ( (cid:96) b / z ) = | H | u i ( z ) and Nu i ( y ) > Nu i ( (cid:96) b ) = N j u i ( x j ), we have u i ( y ) < u i ( (cid:96) (cid:48) ) = [ v z u i ( z ) + v j u i ( x j )] / N < [ v z / | H | + v j / N j ] u i ( y ), i.e., v j > N j (1 − v z / | H | ) for all j ≤ h ,thus | H | − v z > (cid:80) hj = v j > | H | (1 − v z / | H | ) = | H | − v z , a contradiction. Note that the same kind ofargument can be made if there are several partial potential consensus options z , z (cid:48) ,. . . , if eachpair of corresponding supporting groups H , H (cid:48) is either disjoint or one contains the other (sothat they form a hierarchy).In other words, no group has an incentive to deviate from electing a good enough full con-sensus, even if a whole hierarchy of narrower and broader partial consensus options is available. MaxParC supports a single partial consensus ssumptions. Assume the same situation as in 2.1.5, but without the full consensus option y , so that only the partial consensus option z remains besides the favourites x j . Then the sameballot profile β , just with y removed, leads to the partial consensus result (cid:96) ( z ) = | H | / N and (cid:96) ( x j ) = | F j | / N for j > h . Outcome with sincere voters.
If the voters from H apply the hyperbolic heuristic, theyrate z at 100(1 − (cid:96) ( x j ) / u i ( z )) which is by assumption larger than 100(1 − | H | / N ), so they all endup voting for z .With the linear heuristic, however, they may rate z too low for getting their votes since100( u i ( z ) − (cid:96) ( x j )) / (1 − (cid:96) ( x j )) might be smaller than 100(1 − | H | / N ). Strong Nash equilibrium.
We can show this β is again a strong Nash equilibrium whenvoters have von Neumann-Morgenstern utilities. Assume some i ∈ G ∩ H stops approving of z .Then, for each j with G ∩ F j (cid:44) ∅ , at least N j + x j for this to be an improvementfor all in G , but for each j with G ∩ F j = ∅ , all N j votes go to x j , a contradiction as above. Sono i ∈ G ∩ H stops approving of z . If G ∩ H (cid:44) ∅ , some i ∈ G − H must vote for z for those votersto profit from the deviation. But then not enough votes are left in G − H to make all i ∈ G − H profit as well. So G ∩ H = ∅ , but since E − H has no potential for even partial consensus, theycannot improve over the benchmark lottery either. This completes the proof. Other methods.
Again, under majoritarian methods, in particular PV, AV, RV, IRV, andSC, there is no Nash equilibrium between the factions that would give z positive winning prob-ability if one of the factions is in a majority. Also, RB does not support partial consensus sincefor all F j it is strictly dominant to vote for x j . FC and RFC also fail to support partial consensus:If z wins because the fallback was not invoked, the voters in E \ H can cause the fallback to beinvoked and have strict incentive to do so; if the fallback is invoked, no voter in any F j ⊂ H willvote for z since they have then a strict incentive to vote for x j instead. MaxParC supports disjoint partial consensuses
If several disjoint groups of factions ex-ist each of which has a potential partial consensus, the situation can get a little trickier, andthe canonical ballot profile might not be a strong Nash equilibrium but only a coalition-proofequilibrium. We treat a simple special case first to demonstrate this.
Example.
Assume N =
6, four factions of sizes N = N = N = N = x . . . x , and two potential partial consensus options z , z (cid:48) , with utilities as in the following table:faction F F F F size 1 2 2 1utility 100 x x x x z z z (cid:48) z (cid:48) β that realizes both partial consensuses is given by this table:faction F F F F willingness β i ( x ) 100 x x x x z z z (cid:48) z (cid:48) F F willingness β (cid:48) i ( x ) 100 x x z z (cid:48) x x F , F approving both x , x so that these options get a higher approval (4)than z , z (cid:48) (having 3), the votes of F , F now go to x , x in equal shares (due to the tiebreaker),and those of F , F still go to z , z (cid:48) .Still, the above deviation by F , F is not coalition-proof since each of these two factionshas an incentive to betray the other by not performing the agreed deviation after all, i.e., bydeviating from the planned deviation. E.g., if F defects in this way, we have the profilefaction F F F F willingness β (cid:48)(cid:48) i ( x ) 100 x x x x z z z (cid:48) z (cid:48) x F ’s and F ’s votes both go to x , profiting F even more and leaving F withstrictly less than under β . Because of this risk of being betrayed by F , F has few incentives toagree with F to perform the original deviation β (cid:48) . Conjecture.
More generally, we conjecture that under quite general conditions, there willbe at least a certain type of coalition-proof equilibrium (similar to ( )) which results in theelection of a broad consensus.More specifically, consider the following type of situation: There are M ≥ B , . . . , B M of voters, each block B k having size N k = | B k | and consisting of m k ≥ F k , . . . , F km k , and we assume their sizes N k j = | F k j | are all at least 2 M . Each faction F k j hasa distinct favourite option x k j , each block B k a potential partial consensus option y k . No otheroptions exist. Let (cid:96) b ( x k j ) = N k j / N define the benchmark lottery and (cid:96) c ( y k ) = N k / N define the par-tial consensus lottery. Each voter i ∈ F k j has a von Neumann–Morgenstern utility function with47 i ( x k j ) = > u i ( y k ) > N k j / N k and u i ( z ) = i ∈ F k j re-sulting from some ballot profile β (cid:48) under MaxParC is thus u i ( β (cid:48) ) = M ( β (cid:48) )( x k j ) + u i ( y k ) M ( β (cid:48) )( y k ),where M ( β (cid:48) ) is the resulting lottery.Now consider the following “canonical” ballot profile β : For i ∈ F k j , β i ( x k j ) = N − N k ) / N < β i y k ≤ + N − N k ) / N , and β i ( z ) = i ∈ F k j approves of x k j and y k , hence ends up voting for y k , so thatthe resulting lottery is (cid:96) c as desired.Also assume that any group G of voters can secretly plan to deviate from β , leading to amodified profile β (cid:48) with β (cid:48) i = β i for all i (cid:60) G , but that no member of G can be sure that the otherswill actually perform the deviation; rather, any subgroup H ⊂ G can secretly plan a furtherdeviation β (cid:48)(cid:48) from β (cid:48) , with β (cid:48)(cid:48) i = β (cid:48) i for all i (cid:60) H .Then we conjecture that with the above strategy profile β , if there is a group G with adeviation β (cid:48) from β that strictly profits all members (i.e., u i ( β (cid:48) ) > u i ( β ) for all i ∈ G ), there is asubgroup H ⊂ G with a further deviation β (cid:48)(cid:48) from β (cid:48) that strictly profits all its members (i.e., u i ( β (cid:48)(cid:48) ) > u i ( β (cid:48) ) for all i ∈ H ) and is strictly worse than β for at least one member of G (i.e., thereis i (cid:48) ∈ G with u i (cid:48) ( β (cid:48)(cid:48) ) < u i (cid:48) ( β )). Summary
We have shown in this section that NL and MaxParC both support full and par-tial consensus in a number of archetypical decision situations, whereas all other eight studiedmethods and all majoritarian methods do not.
We simulated 2 M = moverate ), (ii) the shareof trial-and-error updates that did not lead to a change in the voter’s ballot (metric keeprate ),and (ii) the share of problems in which the ballots after the interactive phase di ff ered from beforethat phase (metric interactivechanged ).Table 2 gives an overview of all metrics’ mean values.In addition to univariate and bivariate statistics for all metrics, we also fitted an OLS gener-alized linear regression model for each metric Y , separately for each preference model U , usingthe following parameters as explanatory variables: dummy variables for the voting method (us-ing RV as reference method); log-transformed numbers of voters ( nvoters ), options ( noptions ),and polling rounds ( npolls ); shares of LCP ( rshare_LCP ), HCP ( rshare_HCP ), strategic( sshare_S ), trial-and-error ( sshare_T ), heuristic ( sshare_H ), and factional ( sshare_F ) vot-ers; a dummy indicating whether the first option was a constructed compromise option ( with_compromise );and the parameters of the preference model ( log(Bmr), Bmh, Bmiota or dim, log(omega),rho ). The regression analysis shows that the case number was large enough to distinguish the48nfluences of all explanatory variables since almost all estimated coe ffi cients were significantlydi ff erent from zero. All six absolute welfare metrics (
Wutil_initial , Wutil_final , Wgini_initial , Wgini_final , Wegal_initial , Wegal_final ) had considerably left-skeweddistributions across problems. When distinguishing by preference model ( umodel ), one can seethat this is due to the spatial preference models, and that their location depends strongly on thepreference model (Fig. 6).In the block ( BM ) and uniform ( unif ) preference models, the majoritarian methods ( PV,AV, RV, IRV, SC ) generated slightly larger utilitarian and slightly smaller egalitarian abso-lute welfare than the proportional methods (
RB, FC, RFC, NL, MPC ), being roughly equiva-lent on the intermediate Gini-Sen welfare metric. In the QA and LA models, the majoritarianmethods also outperformed the proportional ones in the Gini-Sen and egalitarian absolute wel-fare metrics, most significantly in the QA model, less so in the LA model. In the GA model,the di ff erences between methods were still statistically significant (e.g., Tbl. 3) but negligiblein comparison to the overall dispersion of welfare across problems.Throughout the regression models, more voter blocks (larger BMr ), larger policy-space di-mension ( dim ), and larger spatial voter heterogeneity ( omega ) decreased welfare, and so dida larger number of voters except in the
Wgini/GA case. More options and larger block sizeheterogeneity (
BMh ) increased welfare. Larger individuality (
Bmiota ) increased utilitarian butdecreased Gini-Sen and egalitarian welfare. For the spatial option broadness parameter ( rho )and shares of non-EUT voters, there was no clear pattern. More pre-voting polling rounds hada statistically significant but very small negative influence.Larger shares of sincere, trial-and-error, and factional voters and lower shares of lazy voterstended to increase welfare, the share of heuristic voters had no clear influence. Surprisingly,adding a constructed compromise option only increased welfare in the spatial models thoughnot in the block and uniform models.
Relative welfare metrics
Because of the skewed distributions of absolute welfare, we alsoanalysed the two versions of relative welfare metrics, Since the first of these was skewed inthe other direction in case of the nondeterministic methods and inequality-averse metrics, wefocus on the second, alternative version of relative welfare metrics here, which were much morebalanced (Fig. 7).Interestingly, in the uniform preference model, alt_relWgini and alt_relWegal had aparticular trimodal distribution for the proportional methods, which performed similarly to thedeterministic methods in most cases, but much better in a somewhat smaller cluster of casesand much worse in a still smaller cluster of cases.Looking at the fairly inequality-averse Gini-Sen welfare metric in its “middle” version alt_relWgini more closely in a regression analysis, we see that this metric typically in-49igure 6: Distribution of absolute social welfare across decision problems. Top: histograms,kernel density estimators, and boxplots with means for three example metrics / methods. Bottom:distribution of final absolute welfare by preference model and method.50igure 7: Distribution of relative social welfare across decision problems. Top: comparison ofthe two versions of relative welfare with absolute welfare for one combination of metric andmethod. Bottom: distribution of final alternative relative welfare metrics by preference modeland method. 51reased with the no. of options (except for the uniform preference model); it decreased withthe no. of polling rounds, the share of lazy voters; the addition of a constructed compromise op-tion, the risk attitude scenario, and the spatial broadness parameter had no consistent influenceacross preference models. In contrast to their influence on the absolute welfare metric Wgini ,a larger no. of voter blocks
BMr , individuality
BMiota , policy space dimension dim increased alt_relWgini .In addition, the grouped boxplots in Fig. 8 show some specific influences. A larger no.of voters increased alt_relWgini considerably for the proportional, but not for the majori-tarian methods. The behavioural type scenario influenced alt_relWgini under the SimpleCondorcet method much more than the other methods: interestingly, the all-sincere scenarioproduced much less welfare than the all-lazy one. Larger spatial option broadness rho hadopposite e ff ects for the majoritarian methods (increasing alt_relWgini ) and the proportionalmethods (decreasing alt_relWgini ): when options had narrow appeal (or candidates a narrowplatform), proportional methods performed considerably better, with broadly appealing options(or platforms) it was the other way around. Finally, alt_relWutil decreased with increas-ing spatial voter heterogeneity omega more strongly for majoritarian methods, their clearestadvantage mostly restricted to narrow spatial voter distributions. Frequency of best-performing methods
If we focus on the more qualitative question ofwhich methods perform “best” how often, we can study the share of decision problems in whichthe largest welfare was (i) only provided by one or more majoritarian methods (red in Fig. 9),(ii) only provided by one or more proportional methods (green), or (iii) provided by at least onemethod from both groups (yellow). In Fig. 9, this is shown for all three final welfare metrics(
Wutil, Wgini, Wegal ), grouped by parameters that made a significant di ff erence.One can see that across all three welfare metrics, proportional methods were performingbest according to this statistic in the Gaussian allotment and uniform preference models, andwith fewer lazy voters.W.r.t. Gini-Sen and egalitarian welfare, they performed best with larger no. of voter blocks,more individuality, larger spatial voter heterogeneity, and lower spatial option broadness; w.r.t.utilitarian welfare, this was the other way around. “Cost of fairness” As a final indicator of the social welfare e ff ects of using proportionalinstead of majoritarian methods, we show the distribution of the above-defined cost of fairnessmeasure across all simulated decision problems in Fig. 10. It shows that, typically, the decrease(if any) in average voter utility one gets by switching from the best majoritarian method (RangeVoting) to the best proportional (and thus “fair”) method in our study (MaxParC), was about anorder of magnitude smaller than the di ff erence between the utilitarian and egalitarian absolutesocial welfare resulting from using Range Voting (which can be seen as a natural measure ofabsolute inequality in voters’ utility). The 2%-trimmed mean of this “relative cost of fairness”was 0.08. 52igure 8: Distribution of relative social welfare across decision problems. Specific influencesof no. of voters, behavioural type scenario, spatial option broadness, and spatial voter hetero-geneity. 53 modelBMrBMiotaomegarhoriskmodelscenario Wutil_final Wgini_final Wegal_final Figure 9: Frequency of majoritarian and proportional methods being best according to finalutilitarian (left column), Gini-Sen (middle), and egalitarian (right) social welfare, by variousparameters (rows). 54igure 10: Distribution of the “relative cost of fairness” across decision problems.
Summary
Overall, one can conclude that proportional methods can well compete with ma-joritarian ones regarding social welfare e ff ects and that the welfare assessment depends stronglyon the choice of welfare metric used: the more inequality-averse the welfare metric, the moreit favours proportional methods. In addition, welfare e ff ects of method choice depend stronglyon the distribution of voters’ preferences, which suggests that proportional methods may par-ticularly well perform in situations with heterogeneous voters and when many options have acomparatively narrow appeal.Since proportional methods achieve this by randomization, and hence only over sequencesof several decisions (which is why we used ‘ex-ante’ / long-run welfare metrics here), we needto look next at the amount of randomization actually used. In “deterministic” methods, randomization is only used to resolve the odd tie, hence Shannonentropy (and similarly R´enyi entropy) is mostly zero, sometimes log 2, and rarely larger, andmaximal option probability is mostly one, sometimes 1 /
2, and rarely smaller. Fig. 12 comparesthis to the level of randomization in the nondeterministic methods used in this study. Under RB,entropy is distributed in a left-skewed distribution with a peak at log noptions , leading to amixture distribution with mean around 1.3. Under FC, this distribution is further mixed with apeak at 0 representing the cases where the full compromise was found. RFC was similar to FCregarding randomization.Under NL, this probability of finding a full compromise was almost twice as large; while itstill shows a mixture of left-skewed distributions with peaks at log k for some integer k , these k are now generally smaller than for RB since the optimization of the Nash sum typically leavessome options with zero probability. The mean entropy for NL is thus smaller, 1.05. For MPC,this is even more pronounced, with two clear peaks at 0 and log 2 and mean entropy initially 0.9.For MPC, the interactive phase changed entropy the most, bringing it down to 0.8 on average.For RB and FC, the 25% quantile [and mean] of the largest option probability were only ataround 0.3 [0.43], for NL is was at around 0.35 [0.53], and for MPC initially around 0.45 [0.6]55nd finally (after the interactive phase) around 0.5 [0.65].Regression analysis reveals that Shannon entropy increases (and max. probability decreases)with the no. of options, no. of voters (slightly), no. of voter blocks, and share of lazy voters (con-siderably). Adding a constructed compromise option decreased entropy (and increased max.probability) even if it had no clear welfare e ff ects. Surprisingly, the share of non-expected-utility, risk- (and hence randomization-) averse voters had no clear e ff ect on the level of ran-domization, and neither had the no. of pre-voting polls.Fig. 13 shows some further parameter influences on individual methods’ level of random-ization. With more options and more trial-and-error voters, MPC’s advantage becomes morepronounced, while NL randomizes less if all voters are sincere or are factionally strategic, thelatter reflecting the fact that the factional strategy in NL was implemented as an optimizationproblem in our simulations which however might be hard to solve in reality.In the next subsection we will see whether strategizing pays o ff and whether it gives arelative advantage. In order to see whether and when di ff erent behavioural types of voters have advantages, westudy the distribution of the average satisfaction of all voters of a certain type across our sim-ulations. Fig. 14 shows that typical shape of these distributions depends very much on themethod, but is almost identical for the two polar behavioural types of lazy voters and faction-ally strategic voters (and also for the other three types). In other words, voting strategically doesnot so much give a comparative advantage to the strategic voters over the lazy ones, but ratherincreases overall welfare (as seen above).As can be seen in Fig. 11, the majoritarian methods fare somewhat better regarding averagevoter satisfaction with mean values around 0.66 compared to MPC’s mean of around 0.61,again quite much depending on the preference model. As can be expected, for the proportionalmethods the risk-averse non-EUT voters were less satisfied. If voters were to decide between the ten voting methods and would use for this “meta-decision”a pairwise comparison method such as Simple Condorcet or any other majoritarian method,which voting method would win?If one assumes that voters are purely consequentialist and judge a method only by its gener-ated utility, the surprising answer seems to be that they would then end up with Instant-Runo ff Voting. As table 2 shows, IRV is the Condorcet Winner of this meta-decision since it would wina pairwise decision against all other nine methods. Which method would win if the meta-decision was made using a proportional method, we can only speculatehere since our current results do not provide us a way to predict this.
Our experiments indicate that assessments via agent-based simulations involving individualpreferences will usually depend very much on the particular assumptions about voters’ prefer-ence distributions, whether from a spatial or other model (or “culture”), and on the particularfunctional forms (e.g., linear, quadratic, or Gaussian) and parameter values used in these mod-els. They also seem to suggest that using an interactive phase only rarely has any considerablee ff ect on the most important metrics, with the decrease in randomization under MaxParC beingan exception. This might however be due to our very restricted assumptions on what agents cando during the interactive phase. Future work, whether empirical or numerical, should thereforeconsider the possibilities of information gathering, communication and other forms of socialdynamics during the interactive phase.To this end, we are currently developing a social app that o ff ers an interactive version ofMaxParC for making everyday group decisions, which we plan to use in empirical studies toassess the real-world performance and social dynamics of nondeterministic proportional con-sensus decision making methods. Since the MaxParC ballot explicitly asks for a quantity (the level “willingness” to approve of anoption) whose meaning ultimately depends on ones beliefs about the other voters’ preferences,voting under MaxParC always incorporates some form of “strategic” thinking in some sense, soit is in a way pointless to ask what “the” sincere way of filling in a MaxParC ballot is. Rather,one may apply any of a number of di ff erent heuristics that all lead to a sincere ballot in the sensethat more-preferred options are assigned higher willingness values. Conservative satisficing heuristic.
This sincere voting heuristic is based on the idea to assignto any option y a willingness b ( y ) just small enough to guarantee that if I end up approving of y and all who don’t approve of y approve of their favourite only, the resulting lottery will not beworse than the benchmark lottery (cid:96) that would result if all approve of their favourite only.To find this willingness value b ( y ), a EU-type voter would proceed as follows. Assume (cid:96) ( x ) > x ∈ C (otherwise ignore those x for which (cid:96) ( x ) = x , x , . . . by descending utility, so that u i ( x ) > u i ( x ) > . . . > u i ( x k ) and x = f i is the favourite of i . For all a = . . . k , let F a = (cid:80) kc = a + (cid:96) ( x c ) and U a = (cid:80) kc = a + (cid:96) ( x c ) u i ( x c ),58oting that 1 − (cid:96) ( f i ) = F > . . . > F k = U = u i ( (cid:96) ) and U k =
0. For each β ∈ [0 , − (cid:96) ( f i )], let a ( β ) the smallest a with F a ≤ β , noting that a (0) = k and a (1 − (cid:96) ( f i )) =
1. For each y ∈ C \ { f i } with u i ( y ) ≥ u i ( (cid:96) ), let V y ( β ) = (1 − β ) u i ( y ) + U a ( β ) + ( β − F a ( β ) ) u i ( x a ( β ) ) , (44)noting that V y (0) = u i ( y ) ≥ u i ( (cid:96) ) and V y (1 − (cid:96) ( f i )) = u i ( (cid:96) ) − (cid:96) ( f i )( u i ( f i ) − u i ( y )) < u i ( (cid:96) ). V y ( β ) isthe evaluation of the lottery that results if the β N voters whose favourites I like least (this is the“conservative” aspect of the heuristic) approve of their favourite only while the rest (includingme) approve also of y so that y will win with probability 1 − β while the rest of the winningprobability goes to options I rather don’t like. The “satisficing” aspect of the heuristic is tobe satisfied if this evaluation is not worse than that of the benchmark lottery. Hence one lets b ( y ) = β for the largest β with V y ( β ) ≥ u i ( (cid:96) ), so that 0 ≤ b ( y ) < − (cid:96) ( f i )), and completesthe ballot by putting b ( f i ) =
100 and b ( y ) = y ∈ C with u i ( y ) < u i ( (cid:96) ).LCP-type voters may use the same formula based on the probability weights w j instead ofthe actual probabilities (cid:96) ( x j ).Also for HCP-type voters, one can easily derive a similar formula.Note that this heuristic indeed produces a sincere ballot since u i ( y (cid:48) ) > u i ( y ) will imply b ( y ) > b ( y (cid:48) ). If one has to expect that a fraction α ∈ [0 ,
1] of all voters is lazy, one would adjust b ( y ) to b ( y ) = α + (1 − α ) β ) if β > Informed satisficing heuristic.
If more information about the other voters’ preferences isavailable or can be estimated, one may rather want to apply this heuristic in which i assumesthat if a fraction of voters j will eventually approve of some option y , it will be those j with thelargest relative utility ρ j ( y ) : = u j ( y ) − u j ( (cid:96) ) u j ( f j ) − u j ( (cid:96) ) . Hence let us assume i has beliefs regarding the distri-bution of ρ j ( y ) inside each faction F x and hence can sort the voters into an ordering j , . . . , j N by descending ρ j ( y ), so that ρ j ( y ) > . . . > ρ j N ( y ) and so that she knows f j a for all a . If the first n ≤ N voters in this ordering rather assign their winning probability to y than to f j a , i ’s utilitybecomes V y ( n ) = ( nu i ( y ) + N (cid:88) a = n + u i ( f j a )) / N . (45)Now if i is satisfied if this is no smaller than u i ( (cid:96) ), she would seek the smallest n with ρ j n ( y ) ≤ ρ i ( y ) and V y ( n (cid:48) ) ≥ u i ( (cid:96) ) for all n (cid:48) > n and put b ( y ) = − n / N ), or, if there is no such n , put b ( y ) =
0. Note that also this heuristic has b ( y ) > ff u i ( y ) ≥ u i ( (cid:96) ) (since then n < N ).However, this heuristic may produce insincere ballots in which b ( y (cid:48) ) < b ( y ) despite u i ( y (cid:48) ) > u i ( y ) since the voter ordering used for b ( y (cid:48) ) may be completely di ff erent than the one used for b ( y ). Still, one can argue that in many situations, the ballot will be approximately sincere.This is because often (i) ρ j n ( y ) ≤ ρ i ( y ) will imply V y ( n (cid:48) ) ≥ u i ( (cid:96) ) for all n (cid:48) > n , and hence b ( y ) ≈ − |{ j : ρ j ( y ) ≥ ρ i ( y ) }| ), and (ii) the distribution of ρ j ( y ) in the electorate will be similar forall relevant options y , so that b ( y ) is approximately monotonic in ρ i ( y ). In a spatial model with59oncave utilities u i ( y ) = f ( || η i − ξ y || ) (such as the LH and QH models) and smoothly and widelydistributed voter and option positions, i will indeed prefer y to a lottery of favourites of thosevoters j with ρ j ( y ) ≥ ρ for any ρ since those voters are distributed approximately uniformly andsymmetrically around η y , so the average distance from η i to their favourites is at least || η i − ξ y || ,translating into an expected utility from the lottery that is below u i ( y ) since f is concave. Moreparticularly, both in the 1-dimensional LH and the 2-dimensional QH model, the number ofvoters j with ρ j ( y ) ≥ ρ i ( y ) scales roughly linearly with 1 − ρ i ( y ), hence b ( y ) will scale roughlylinearly with ρ i ( y ), whereas in a higher-dimensional model, b ( y ) will become a concave functionof ρ i ( y ).Our next two heuristics mimic this linear or concave behaviour to some extent with muchsimpler formulae. Linear heuristic.
A much simpler heuristic is the one we assume in our simulations, where b ( y ) = (cid:16) α + (1 − α ) u i ( y ) − u i ( (cid:96) )max x ∈ C u i ( x ) − u i ( (cid:96) ) (cid:17) for all y ∈ C with u i ( y ) ≥ u i ( (cid:96) ), i.e., one assigns a willing-ness of 0 to options worse than the benchmark, 100 to one’s favourite, and interpolates linearlybetween 100 α and 100 based on the options’ utilities, where α ∈ [0 ,
1] is the expected share oflazy voters in the electorate.One motivation for this heuristic is that under certain assumptions, it can be interpreted as anapproximation of the conservative satisficing heuristic. Assume the number of options is large,their utilities u i ( y ) for i are distributed uniformly, say (without loss of generality) between 0 and u i ( f i ) =
1, and their benchmark winning probabilities (cid:96) ( y ) are not correlated to i ’s evaluations u i ( y ). Then u i ( (cid:96) ) ≈ / F a and a ( β ) decrease approximately linearly in a or β , respectively, U a ( β ) ≈ β /
2, and β − F a ( β ) is small. Hence V y ( β ) ≈ (1 − β ) u i ( y ) + β /
2, which equals u i ( (cid:96) ) for β ≈ u i ( y ) −
1, hence b ( y ) ≈ (cid:16) α + (1 − α ) u i ( y ) − u i ( (cid:96) )max x ∈ C u i ( x ) − u i ( (cid:96) ) (cid:17) for all y with u i ( y ) ≥ u i ( (cid:96) ). The samederivation can be made under the weaker assumption that only those options y with u i ( y ) ≥ u i ( (cid:96) )are numerous, have uniformly distributed u i ( y ) and have (cid:96) ( y ) uncorrelated to u i ( y ). These as-sumptions are, e.g., approximately fulfilled if k is large and utility follows the LH model. If,instead, utility depends more concavely on distance, as in the QH model, the linear heuristic willtend to produce larger willingness values than the conservative satisficing heuristic, hence willproduce more compromise outcomes which however may sometimes be worse than the bench-mark lottery for some voters. On the contrary, if utility depends more convexly on distance,as in the tails of the GH model, the linear heuristic will tend to produce smaller willingnessvalues than the conservative satisficing heuristic, hence may sometimes not produce a partialconsensus when there is a potential one. Hyperbolic heuristic.
A little less simple is the heuristic that puts b ( f i ) = b ( y ) = y with u i ( y ) < u i ( (cid:96) ), and b ( y ) = − u i ( (cid:96) ) − min j u i ( f j ) u i ( y ) − min j u i ( f j ) ) for all other y , which has a hyperbolicalrather than a linear dependency on u i ( y ), growing fast for u i ( y ) slightly above u i ( (cid:96) ) and muchslower for u i ( y ) approaching u i ( f i ).Also this formula can be derived as an approximation of the conservative satisficing heuris-60ic, under di ff erent assumptions on the distribution of utility. Assume that i considers all otheroptions than f i that occur as favourites of any voter as approximately equally bad, so that wecan assume u i ( f i ) = u i ( f j ) ≈ j with f j (cid:44) f i . Then u i ( (cid:96) ) ≈ (cid:96) ( f i ), V y ( β ) ≈ (1 − β ) u i ( y ),hence b ( y ) ≈ − u i ( (cid:96) ) u i ( y ) ) ≈ − u i ( (cid:96) ) − min j u i ( f j ) u i ( y ) − min j u i ( f j ) ). Since these assumptions on utility are evenmore extremely “convex” than in the GA model, the hyperbolic heuristic may be a better choicein Gaussian utility situations than the linear heuristic. Assume N (cid:29) C = { , . . . , k } , put m = k − e = (1 , . . . , ∈ R m , p = ( (cid:96) , . . . , (cid:96) m ), v = u k , w = ( u − v , . . . , u m − v ). We focus on voter 1’s choice of ratings r ∗ = β i and consider s = r k ≥ t = ( r − s , . . . , r m − s ) with t x ≥ − s the control variables, all vectors being column vectors. Thenthe Nash sum ( = log of Nash lottery target function) is f ( p | s , t ) = g ( p | s , t ) + h ( p ) (46)with g ( p | s , t ) = log( s + p (cid:62) t ) , h ( p ) = (cid:88) i log( r ik + (cid:88) x p x r ix ) , (47)where summation over i means i = . . . N (likewise for j ) and summation over x means x = . . . m (likewise for y , z ).Since N (cid:29)
1, we can approximate f ( p | s , t ) = g ( q | s , t ) + d (cid:62) G ( s , t ) + h ( q ) + d (cid:62) Hd / , (48)where q = arg max p h ( p ) , (49) d = p − q , (50) G ( s , t ) x = ∂ p x g ( p | s , t ) | p = q = ts + q (cid:62) t = γ ( s , t ) t , (51) H xy = ∂ p x ∂ p y h ( p ) | p = q = (cid:88) i ∂ p x r iy r ik + (cid:80) z p z r iz = − (cid:88) i r ix r iy ( r ik + (cid:80) z p z r iz ) (52)with γ ( s , t ) = s + q (cid:62) t > , (53) ∂ s γ ( s , t ) = − s + q (cid:62) t ) = − γ ( s , t ) < , (54) ∇ t γ ( s , t ) = − q ( s + q (cid:62) t ) = − γ ( s , t ) q . (55)61ssume that H is nonsingular with I = H − , (56)and note that H , I are symmetric and negative semidefinite. Assume that q x > x and t (cid:44)
0, so that s + q (cid:62) t > γ ( s , t ) < ∞ .The Nash lottery p ∗ ( s , t ) is that p ∈ [0 , m with e (cid:62) p ≤ f ( p | s , t ). Assumethis is an interior solution (e.g., since there is at least one bullet voter for each option), then thefirst-order condition is0 = ∇ p f ( p | s , t ) = ∇ d ( d (cid:62) G ( s , t ) + d (cid:62) Hd / = G ( s , t ) + Hd , (57)hence d ∗ ( s , t ) = − IG ( s , t ) = − γ ( s , t ) It , (58) ∂ s d ∗ ( s , t ) = γ ( s , t ) It , (59) ∇ t d ∗ ( s , t ) (cid:62) = γ ( s , t ) qt (cid:62) I − γ ( s , t ) I . (60)Voter 1’s expected utility is then U ( s , t ) = v + p ∗ ( s , t ) (cid:62) w = v + q (cid:62) w + d ∗ ( s , t ) (cid:62) w = v + q (cid:62) w − γ ( s , t ) t (cid:62) Iw . (61)If she considers abstaining (which is equivalent to putting t ≡ s >
0, w.l.o.g. s =
1) and wonders what small change in ratings ∆ r would improve her utility most, she wouldcalculate ∂ r x U ( s , t ) | t ≡ = − ∂ t x ( γ ( s , t ) t (cid:62) Iw ) | t ≡ = − ( ∂ t x t (cid:62) | t ≡ ) Iw = − ( Iw ) x , (62) ∂ r k U ( s , t ) | t ≡ = − ( ∂ s − (cid:88) x ∂ t x )( γ ( s , t ) t (cid:62) Iw ) | t ≡ = (cid:88) x ( Iw ) x . (63)Not knowing H and hence I , voter 1 might use the following heuristic to estimate an ap-proximate H from the latest favourite polling data, simply assuming every other voter j > r jx = x , putting all others to r jy = f p ( x ) > x , H xy ≈ N (cid:16) / f p ( k ) + δ xy / f p ( x ) (cid:17) (64)where δ xy = ff x = y , else δ xy =
0. Hence H is a matrix filled with equal positive entries plussome positive diagonal. Its inverse then has I xx ≈ ζ f p ( x ) (cid:0) N − f p ( x ) (cid:1) (65) I xy ≈ − ζ f p ( x ) f p ( y ) (66)62or some ζ > x (cid:44) y . This would imply that voter 1’s utility grows fastest in the direction ∆ r with ∆ r x = − ( Iw ) x ≈ ζ f p ( x ) Nw x − (cid:88) y f p ( y ) w y = ζ N f p ( x )( u x − υ ) , (67) ∆ r k = (cid:88) x ( Iw ) x ≈ ζ N (cid:88) x f p ( x ) ( υ − u x ) = ζ N f p ( k )( u k − υ ) , (68)where υ = (cid:80) kx = u x f p ( x ) / N is voter 1’s expected utility of the benchmark lottery based on thelatest favourite polling data. A natural heuristic is then that voter 1 moves her ratings from r x ≡ r k = r x = + ρ f p ( x )( u x − υ ) ≥ x = . . . k , where ρ = / f p ( y )( υ − u y ) > , (70) y = arg k min x = f p ( x )( u x − υ ) , (71)so that r y =
0. In the special case where f p ( x ) ≡ N / k (e.g. before the first poll), we get y = arg min kx = u x , ρ = k / N ( υ − u y ), and r x = u x − min kz = u z υ − min kz = u z ∝ u x − min kz = u z , i.e., voter 1 wouldthen vote sincerely. If, however, some options appear to have much higher chances than others,she would exaggerate her stated preferences regarding those options that appear to have higherchances (high f p ( x )) while playing down her stated preferences regarding those options thatappear to have lower chances, which can result in rating some promising well-liked compromiseoption higher than her favourite if the latter has low chances, or rating some lurking less-likedcompromise option lower than a very improbable least-liked option. We show that w.l.o.g., one can restrict the analysis on the described set A of ballots. First,assume some ballot ranks some option y which however gets eliminated before all higher-rankedoptions are eliminated. Then submitting a shorter ballot with y left out instead leads to the exactsame tally process. Second, assume y is ranked but gets eliminated at the same point as whensubmitting the shorter ballot with y left out. Then submitting the shorter ballot also leads to theexact same tally process. Hence we can restrict our focus on ballots ranking only options thatsurvive the elimination process strictly longer than when not ranked, and don’t get eliminatedbefore any higher-ranked option. For any ballot b = ( x , x , . . . , x (cid:96) ) ∈ A , let a ( b ) ∈ { , , . . . , k − } be the number of options eliminated strictly before x (cid:96) when submitting b , and assume that also b (cid:48) = ( x , x , . . . , x (cid:96) , y ) ∈ A . Note that if submitting b , y is eliminated after at least a ( b ) + b (cid:48) (cid:60) A ), hence there are at most k − a ( b ) − ff erent y ∈ C such that ( x , x , . . . , x (cid:96) , y ) ∈ A . Thus the number a (cid:48) ( b , y ) of options eliminatedstrictly before y when submitting b (not b (cid:48) !) is one of the numbers in { a ( b ) + , . . . , k − } and isdi ff erent for all y for which ( x , x , . . . , x (cid:96) , y ) ∈ A . If submitting b (cid:48) , y must survive longer thanwhen submitting b , hence a ( b (cid:48) ) > a (cid:48) ( b , y ) ≥ a ( b ) + > a (cid:48) (( x , x , . . . , x (cid:96) − ) , x (cid:96) ) + , (72)i.e., a ( b (cid:48) ) ≥ a ( b ) +
2. This implies that any ballot b = ( x , x , . . . , x (cid:96) ) ∈ A can be uniquely encodedvia a sequence of integers ( a (cid:48) ( ∅ , x ), a (cid:48) (( x ) , x ), a (cid:48) (( x , x ) , x ), . . . , a (cid:48) (( x , x , . . . , x (cid:96) − ) , x (cid:96) )) thatfulfils a (cid:48) (( x , . . . ) , x i ) + < a (cid:48) (( x , . . . ) , x i + (73)for all i . There are less than 2 k such sequences in 0 , . . . , k −
1, hence | A | ≤ k . References
1. D. Lewis,
Direct democracy and minority rights: A critical assessment of the tyranny ofthe majority in the American states (Routledge, 2013).2. P. Collier,
Oxford Economic Papers , 563 (2004).3. L. Cederman, A. Wimmer, B. Min, World Politics (2010).4. F. S. Cohen, Comparative Political Studies , 607 (1997).5. F. Zakaria, Foreign a ff airs , 22 (1997).6. N. McCarty, K. T. Poole, H. Rosenthal, Polarized America: The dance of ideology andunequal riches (MIT Press, 2016).7. P. Dubey, L. S. Shapley,
Mathematics of Operations Research , 99 (1979).8. J. H. Davis, Organizational Behavior and Human Decision Processes , 3 (1992).9. K. O. May, Econometrica: Journal of the Econometric Society pp. 680–684 (1952).10. J. Heitzig, F. W. Simmons,
Social Choice and Welfare , 43 (2012).11. R. Carnap, The Journal of Philosophy , 141 (1947).12. J. G. Cross, The Quarterly Journal of Economics , 239 (1973).13. D. P. Kingma, J. Ba, arXiv preprint arXiv:1412.6980 (2014).14. J. C. Harsanyi, International journal of game theory , 1 (1973).645. P. Troyan, Games and Economic Behavior , 936 (2012).16. H. Aziz, A. Bogomolnaia, H. Moulin, ACM EC 2019 - Proceedings of the 2019 ACMConference on Economics and Computation pp. 753–781 (2019).17. J.-F. Laslier, M. R. Sanver,
Handbook on approval voting (Springer Science & BusinessMedia, 2010).18. M. S. Granovetter, Threshold Models of Collective Behavior (1978).19. M. Wiedermann, E. K. Smith, J. Heitzig, J. F. Donges,
Scientific Reports (in press) (2020).20. C. Duddy,
Mathematical Social Sciences , 1 (2015).21. R. Lanphier et al., https: // electorama.com / em, last accessed 14 April 2020 (1996).22. H. Moulin, Fair division and collective welfare (MIT Press, 2004).23. J.-F. Laslier,
Handbook on Approval Voting (2010), pp. 311–335.24. R. Carroll, J. B. Lewis, J. Lo, K. T. Poole, H. Rosenthal,
American Journal of PoliticalScience , 1008 (2013).25. A. Bruhin, H. Fehr-Duda, T. Epper, Econometrica , 1375 (2010).26. J. Behnke, S. Hergert, F. Bader, Stimmensplitting Kalkuliertes Wahlverhalten unter denBedingungen der Ignoranz (2004).27. J. Sommer, Wer w¨ahlt strategisch und warum? Eine Analyse strategischen Wahlverhaltensbei der Bundestagswahl 2013, Phd dissertation, Heinrich-Heine-Universit¨at D¨usseldorf(2015).28. G. Gigerenzer, W. Gaissmaier, Annual review of psychology , 451 (2011).29. K. van der Straeten, J. F. Laslier, N. Sauger, A. Blais, Social Choice and Welfare , 435(2010).30. J. S. Bower-Bir, N. J. D’Amico, A Tool for All People, but Not All Occasions: How VotingHeuristics Interact with Political Knowledge and Environment (2013).31. J. Laslier, Journal of Theoretical Politics , 113 (2009).32. R. Myerson, R. Weber, American Political Science Review , 102 (1993).33. A. Dellis, Handbook on Approval Voting , J.-F. Laslier, M. R. Sanver, eds. (Springer, 2010),chap. 18, pp. 431–454. 65 ehavioural types scenariotype lazy middle strategic all-L all-S all-T all-H all-F
L (lazy) 1 / / /
20 1 0 0 0 0S (sincere) 1 / / /
20 0 1 0 0 0T (trial-and-error) 1 / / / / / / / / / ff erent behavioural type scenarios34. H. Moulin, Econometrica , 1337 (1979).35. P. K. Bag, H. Sabourian, E. Winter, Journal of Economic Theory , 1278 (2009).36. A. Sen,
Journal of Public Economics , 387 (1974).37. E. Koutsoupias, C. Papadimitriou, Lecture Notes in Computer Science (including subseriesLecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Springer Ver-lag, 1999), vol. 1563, pp. 404–413.38. D. R. Woodall,
Discrete Applied Mathematics , 81 (1997).39. T. N. Tideman, Social Choice and Welfare , 185 (1987).40. M. Schulze, Social Choice and Welfare , 267 (2011).41. B. D. Bernheim, B. Peleg, M. D. Whinston, Journal of Economic Theory , 1 (1987).66 V AV RV IRV SC RB FC RFC NL MPC anonymous yes yes yes yes yes yes yes yes yes yesneutral yes yes yes yes yes yes yes yes yes yesPareto-e ffi cient w.r.t. stated preferences yes yes yes yes yes yes no no yes yesstrongly mono-raise monotonic yes yes yes no yes yes no no no yesweakly mono-raise monotonic yes yes yes no yes yes yes no yes ? yesweakly mono-raise-abstention mon. yes yes yes yes yes yes yes yes yes yesindependent from Pareto-dominated alternatives partial full full full full partial no no full fullindependent from losing options no full full no no partial no no full fullindependent from exact clones no yes yes yes yes yes no yes yes yesstronger forms of clone-proofness no yes yes no yes yes no ? ? ?strategy-freeness no no no no no yes no no no noreveals preferences no some some no no fav. fav. utility fav. fav.allocates power proportionally no no no no no yes yes yes yes yessupports full consensus with sincere voters no yes yes no no no yes yes yes yessupports full consensus with strategic voters no no no no no no yes yes yes yessupports partial consensus with strategic voters no no no no no no no no yes yes PV AV RV IRV SC RB FC RFC NL MPC moverate keeprate interactivechanged
Eshannon_initial
Eshannon_final
Erenyi2_initial
Erenyi2_final maxprob_initial maxprob_final pcompromise_initial pcompromise_final
Wutil_initial -2.3 -2.36 -2.27 -2.25 -4.06 -4.17 -3.8 -3.7 -4.04 -2.97
Wutil_final -2.29 -2.28 -2.26 -2.24 -3.22 -4.17 -3.78 -3.62 -3.58 -2.93
Wgini_initial -3.18 -3.26 -3.13 -3.09 -5.76 -5.86 -5.3 -5.15 -5.66 -4.07
Wgini_final -3.17 -3.14 -3.11 -3.09 -4.51 -5.86 -5.27 -5.03 -4.98 -4.02
Wegal_initial -8.43 -8.77 -8.32 -8.15 -16.8 -18.6 -16.5 -15.9 -17.1 -11.8
Wegal_final -8.4 -8.33 -8.24 -8.13 -12.7 -18.6 -16.3 -15.4 -14.8 -11.6 relWutil_initial relWutil_final relWgini_initial +
20 2.1e +
20 2.1e +
20 2.1e +
20 2.1e + relWgini_final +
20 2.1e +
20 2.1e +
20 2.1e +
20 2.1e + relWegal_initial +
20 3.6e +
26 1.91e +
20 1.91e +
20 1.91e +
20 2.2e +
67 2.2e +
67 2.2e +
67 2.2e +
67 2.21e + relWegal_final +
20 1.35e +
59 1.91e +
20 1.91e +
20 1.31e +
21 2.2e +
67 2.2e +
67 2.2e +
67 2.2e +
67 2.21e + alt_relWutil_initial alt_relWutil_final alt_relWgini_initial alt_relWgini_final alt_relWegal_initial alt_relWegal_final avgsatisfaction_initial_F avgsatisfaction_final_F avgsatisfaction_initial_H avgsatisfaction_final_H avgsatisfaction_initial_L avgsatisfaction_final_L avgsatisfaction_initial_S avgsatisfaction_final_S avgsatisfaction_initial_T avgsatisfaction_final_T pctprefer_PV_over — 18 15.4 13.3 34.3 65.4 63.4 63.1 61.2 58.7 pctprefer_AV_over pctprefer_RV_over pctprefer_IRV_over pctprefer_SC_over pctprefer_RB_over pctprefer_FC_over pctprefer_RFC_over pctprefer_NL_over pctprefer_MPC_over PV AV RV IRV SC RB FC RFC NL MPC
Table 2: Level of compliance with voting method consistency criteria, and average performancemetrics from agent-based simulations. 67
LS Regression Results==============================================================================Dep. Variable: Wgini_final R-squared: 0.661Model: OLS Adj. R-squared: 0.661Method: Least Squares F-statistic: 2.663e+05Date: Sat, 04 Apr 2020 Prob (F-statistic): 0.00Time: 15:34:58 Log-Likelihood: 1.0244e+07No. Observations: 5124153 AIC: -2.049e+07Df Residuals: 5124130 BIC: -2.049e+07Df Model: 22Covariance Type: HC1===================================================================================coef std err z P>|z| [0.025 0.975]-----------------------------------------------------------------------------------Intercept 0.1581 0.000 1223.248 0.000 0.158 0.158PV -0.0018 6.48e-05 -27.023 0.000 -0.002 -0.002AV 5.993e-05 6.55e-05 0.916 0.360 -6.84e-05 0.000IRV -0.0002 6.53e-05 -3.193 0.001 -0.000 -8.05e-05SC -0.0030 6.58e-05 -44.931 0.000 -0.003 -0.003RB 0.0019 6.47e-05 29.673 0.000 0.002 0.002FC 0.0018 6.47e-05 28.243 0.000 0.002 0.002RFC 0.0018 6.47e-05 28.405 0.000 0.002 0.002NL 0.0017 6.5e-05 26.489 0.000 0.002 0.002MPC 0.0014 6.52e-05 22.143 0.000 0.001 0.002log(nvoters) 0.0003 1.06e-05 25.800 0.000 0.000 0.000log(noptions) 0.0046 4.01e-05 115.265 0.000 0.005 0.005with_compromise 0.0004 2.9e-05 12.443 0.000 0.000 0.000rshare_LCP 0.0004 4.05e-05 10.453 0.000 0.000 0.001rshare_HCP 0.0009 4.08e-05 22.671 0.000 0.001 0.001log(npolls) 6.738e-05 1.87e-05 3.607 0.000 3.08e-05 0.000sshare_S 0.0005 5.79e-05 9.036 0.000 0.000 0.001sshare_T 0.0007 5.69e-05 13.150 0.000 0.001 0.001sshare_H 0.0012 5.68e-05 20.744 0.000 0.001 0.001sshare_F 0.0013 5.42e-05 23.297 0.000 0.001 0.001dim -0.0465 1.95e-05 -2384.367 0.000 -0.047 -0.046log(omega) -0.0447 3.11e-05 -1437.194 0.000 -0.045 -0.045rho -0.0018 3.89e-05 -46.644 0.000 -0.002 -0.002==============================================================================Omnibus: 909575.095 Durbin-Watson: 1.060Prob(Omnibus): 0.000 Jarque-Bera (JB): 1549242.812Skew: 1.174 Prob(JB): 0.00Kurtosis: 4.319 Cond. No. 63.3==============================================================================