NNegative votes to depolarize politics
Karthik H. Shankar ∗ Center for Memory and Brain, Boston University
The controversies around the 2020 US presidential elections certainly casts serious concerns on theefficiency of the current voting system in representing the people’s will. Is the naive Plurality votingsuitable in an extremely polarized political environment? Alternate voting schemes are graduallygaining public support, wherein the voters rank their choices instead of just voting for their firstpreference. However they do not capture certain crucial aspects of voter preferences like disapprovalsand negativities against candidates. I argue that these unexpressed negativities are the predominantsource of polarization in politics. I propose a voting scheme with an explicit expression of thesenegative preferences, so that we can simultaneously decipher the popularity as well as the polarityof each candidate. The winner is picked by an optimal tradeoff between the most popular and theleast polarizing candidate. By penalizing the candidates for their polarization, we can discouragethe divisive campaign rhetorics and pave way for potential third party candidates.
Social choice theorists have pondered over alternatevoting systems for more than two centuries since Nicolasde Condorcet [1]. The ranked voting systems, whereinthe voters rank their choices instead of just voting fortheir first preference, have been thoroughly explored andtheir deficiencies have been mathematically nailed down.Arrow’s Impossibility theorem [2] proves that it is impos-sible to always pick a winner in a ranked voting systemthat satisfies certain basic intuitive criteria. Further, theGibbard-Satterthwaite (GS) theorem [3, 4] proves that inany ranked voting system, some voters can always strate-gically misrepresent their votes to alter the results, whichimplies an impossibility of a strategy-proof voting system.So, it appears that there cannot exist a voting systemthat fairly represents the voter preferences, promptingthe notion that a perfect democracy is mathematicallyimpossible. It may very well be true that we cannot design a perfectvoting system, nevertheless we can certainly do muchbetter than the naive plurality voting system wherein thevoters only vote for their first preference. Any alternatevoting method is of course more complex, but the adventof electronic voting vastly simplifies its implementation.So there is no valid excuse to not facilitate smarter andfairer elections.The major problem with the plurality voting systemis that when two strong candidates emerge with certaincritical mass of support prior to the election, then rest ofthe support will automatically coagulate around them,simply because many voters don’t want to waste theirvotes on a third candidate. This is very deceptive be-cause many of those votes are not direct support for therespective candidates, rather they are anti-votes againstthe opposing candidate. To make matters worse, the can-didates understand this phenomenon and indulge in some ∗ Email: [email protected] The Stanford Encyclopedia of Philosophy provides agood nontechnical introduction to various voting systems.https://plato.stanford.edu/entries/voting-methods/ divisive campaign rhetorics to collect more of their oppo-nent’s anti-votes. Many voters would thus strategicallyrefrain from voting for their real preference (the thirdparty candidate) because their priority is to defeat oneof the top two contenders by voting for the other. Con-sequently, the third party candidates are stifled out ofcompetition.Ranked voting systems alleviate this issue significantlybecause the voters have the opportunity to elaboratetheir preferences in a more detailed fashion. The vot-ers can express their dislike for a candidate by rankingthem the last, however this is not the same as explicitlycasting a negative vote. The only way to prevent theanti-votes for a candidate from being masked into votesfor another candidate is to explicitly express them on theballot as negative votes.Moreover, it is well established that people’s choicevery much depends on how the decision problem in frontof them is framed [5] . By not allowing the votersto freely express their negative preferences, we are es-sentially ill-framing the decision problem and needlesslytampering with voter’s psychology. Here I shall proposea voting system which explicitly incorporates negativevotes. Normed Negative Voting :
Consider a votingsystem where each voter assigns a positive or negativenumber to each candidate such that the magnitudes ofall the numbers sum up to 10.Example : In a three candidate election amongst { A, B, C } , voter-1 could vote as { +7 , − , − } whilevoter-2 could vote as { +3 , +2 , − } . Both the votersexpress the same rank ordering of preferences, namely A > B > C , however their ballots clearly contain farricher information than just the relative ranks. Requir-ing the sum of magnitudes of the vote to be a constant(10) is mathematically termed as
Normed ; it serves toensure the rule of equality one-person-one-vote . For example, people make completely different choices when aproblem is framed as an increase in monetary gain as opposed toavoiding monetary loss. a r X i v : . [ ec on . T H ] D ec Vote aggregation :
For each candidate, aggregatethe positive votes from all voters as P, and aggregatethe negative votes from all voters as N. Then define foreach candidate
Popularity ≡ P-N, and
Polarity ≡ N/P.An absolutely noncontroversial candidate who does notacquire any negative votes will have the lowest polarityof zero. On the other hand, a candidate with almostequally large positive and negative votes is by definitionextremely polarizing with a polarity of 1. For obviousreasons, candidates with polarity larger than 1 should bedisqualified.
Winning metric :
To determine the winner, weshall construct a metric W that rewards popularity andpenalizes polarity. It has to be a monotonically increas-ing function of popularity and a monotonically decreas-ing function of polarity. The candidate with the highestvalue of W is the winner. If two candidates have equalpopularity, the one with lower polarity will have to win.Consider the following metric parametrized by two posi-tive constants ( c, b ). W cb ( P, N ) ≡ P − cN b N/P (1)Let us first examine this function with c = b = 1.Consider an election with three candidates and two votersas shown below. Candidate voter-1 voter-2 P N P-N N/P W A 10 -5 10 5 5 0.5 3.33B 0 4 4 0 4 0 4C 0 1 1 0 1 0 1
Although A has the highest popularity, B still winsover A because of lower polarity. This seems to be a fairresult for this election. However, we should note thatthis voting scheme is severely prone to strategic-voting.Voter-1 has naively expressed a clear preference for A andno dislike for B. On the other hand, voter-2 might sim-ply prefer B over A (and no real dislike for A), howeverby misrepresenting the preference and casting a negativevote to increase the polarity of A, voter-2 can enhancethe chances of B’s victory. Game-theoretically speaking,voter-1 would foresee this and in turn misrepresent hispreference to include a negative vote for B. Avoid over-penalization of Negative Votes :
Strategically misrepresenting the negative votes as de-scribed above should be discouraged. It cannot be com-pletely eliminated, but it can be suppressed by appropri-ately constraining the winning metric. We should ensure When there are more than two candidates it is difficult to strate-gically cast negative votes without hampering the positive votesfor the preferred candidate, which could inhibit the winningchances of the preferred candidate. that the winning metric does not over-penalize the neg-ative votes because that would give the voters an oppor-tunity to exploit it. In order to quantify what exactly wemean by “over-penalize”, let us first note that pluralityvoting method is indeed the optimal voting scheme in atwo-candidate election. So we shall demand that a per-fect preference for a candidate by one voter cannot beoverridden by another voter’s negative vote. To be morespecific, • In a 2-candidate election with just two voters, ifvoter-1 gives a perfect preference for one candidate,then that candidate cannot loose the election regard-less of how voter-2 votes.
To work out this constraint, let voter-1 cast +10 votesfor A and voter-2 cast a negative vote -X for A as shownbelow.
Candidate voter-1 voter-2 P-cN N/P W cb A 10 -X 10-cX X/
10 10 (cid:16) − cX bX (cid:17) B 0 10-X 10-X 0 10-X
For any value of X between 0 and 10, we demand thatcandidate B should not be able to win. (cid:18) − cX b X/ (cid:19) ≥ (10 − X ) (2) ⇒ c + b − b ≤ X/
10 (3) ⇒ c + b ≤ X = 0, so that A and B are tied up.We can now generalize the constraint for m candidatesand a large number of voters. Again, let voter-1 assign+10 votes to A, and voter-2 assign -X votes to A. Thereminder of 10-X votes of voter-2 can be assigned as pos-itive votes for any of the other ( m −
1) candidates. Herevoter-2 should be viewed as a statistical representativeof all voters who voted -X for A; and all these voters areassumed to be independent minds uninfluenced by eachother. On average, each of the other candidates wouldhave received (10 − X ) / ( m −
1) votes from voter-2. So,the above constraint can be reframed at a statistical levelas (cid:18) − cX b X/ (cid:19) ≥ (10 − X ) m − ⇒ m − ≥ [( m − c + b − X/ − b ( X/ (5)In eq.3 the r.h.s attains a minimum at X=0, which iswhere the equality should be implemented. But that isnot true in eq.5 when m >
2. The acceptable range ofparameters (c,b) for which eq.5 holds for all values of Xis plotted in fig.1 for m = 2 , , ,
5; the region under thecurves corresponding to each m contain the admissible b c m = = = = FIG. 1. Maximal-Penalty metrics values for the parameters. First we note the obvious,that c cannot be greater than 1 because negative votesshouldn’t weigh more than positive votes. For m = 2,note that c = 1 is admissible only with b = 0. Butfor m > c = 1 is admissible for any b ≤ m −
2. Inparticular, note that ( c = 1 , b = 1) is acceptable for all m >
2. By choosing the parameters on the curves of fig.1,we are hitting the limit of over-penalization of negativevotes; so these curves are essentially the maximal-penalty metrics.Here we have only analyzed the winning metric func-tion of the form eq. 1. But it is fairly straightforward toimagine other functional forms that introduce nonlinear-ities to penalize the negative votes more adversely, like( P − N ) e N/P , ( P − N ) ( P + N ) , ( P − N ) − N/P . (6)The first two are acceptable, but the third is not ac-ceptable because the metric function must overall be lin-ear in popularity so that its properties do not dependon the size of the electorate. These functions can beanalyzed in a procedure similar to that discussed abovewith introduction of some free parameters (analogous to c, b ) to ensure that negative votes are not over-penalized.The maximal-penalty metrics thus obtained show simi-lar qualitative behavior as shown in fig.1, so there is notmuch utility to further discuss these alternate functionalforms of metric. MAXIMIZING VOTER SATISFACTION
Let’s denote the voters by latin indices i, j, k... and thecandidates by greek indices µ, ν, α... . Let’s denote thepositive votes from voter- i to candidate- µ as p iµ , and thenegative votes as n iµ . Only one of the two, p iµ or n iµ ,will be nonzero for a specific i and µ . Summing over allthe voters will give the net positive votes P µ = (cid:80) i p iµ and net negative votes N µ = (cid:80) i n iµ gathered by eachcandidate- µ . If candidate- α is declared the winner of the election,we define can define the voter-satisfaction to be s iα = p iα − n iα + µ (cid:54) = α (cid:88) µ n iµ S α = (cid:88) i s iα = P α − N α + µ (cid:54) = α (cid:88) µ N µ (7)¯ S α = S α − (cid:34) µ (cid:54) = α (cid:88) µ P µ (cid:35) (8)The first two terms in the r.h.s of eq. 7 correspondto the satisfaction/dissatisfaction explicitly triggered bythe winning candidate, while the third term representsthe satisfaction triggered by all the loosing candidates.Negative votes are directly responsible for a candidate toloose the election, and so those negative votes assigned toloosing candidates can be deemed to have satisfactorilyperformed their job, and hence contributes to the votersatisfaction as the third term in the r.h.s of eq. 7.One might be tempted to include the dissatisfactiondue to the positive votes accrued by the loosing candi-dates as shown in eq. 8. But notice that it is a measureof inaction of certain positive votes that failed to de-liver victory. It should be treated simply as a lack ofsatisfaction that was potentially attainable, rather thana negative quantity to be subtracted from voter satisfac-tion. While defining the voter satisfaction, it is critical toacknowledge the intrinsic asymmetry between the posi-tive and negative votes in the voter’s psychology– positivevotes succeed only when they are cast to the winner, butnegative votes succeed when they are cast to any loser. Ideally we would like the election outcome to max-imize the satisfaction of all voters, i.e. S α should bemaximum for the winning candidate- α . Notice that if weonly considered the first two terms in the r.h.s of eq.7,then we simply have to maximize the popularity givenby ( P α − N α ), which is exactly what the winning metric W would implement. It is however not obvious whethersome other winning metric will always yield the winnerwho maximizes voter satisfaction. Consider an exampleof a 4-candidate election with aggregated positive andnegative votes as shown below. Election-0
Candidate P N
S W W W W . . A 11 5 10 6.0 4.12 3.14 6.92B 7 3 10 4.0 2.8 2.15 4.53C 6 1 13 5.0 4.28 3.75 5.08D 3 0 12 3.0 3.0 3.0 3.0
Here, candidate-C has the maximum voter satisfaction,and also wins under the metrics W and W . But thismay not always happen; in some elections the winningmetrics may not yield the candidate who maximizes votersatisfaction. To understand how reliably these metricscorrelate to maximal voter satisfaction, we shall simulatea large number of m -candidate elections with randomlydistributed ( P µ , N µ ) with at least one candidate qualifiedto win. Then we calculate the probability that a winningmetric yields a winner who maximizes the voter satisfac-tion. Table I shows the results for four different metricsand various values of m . W W W W . . m = 3 87% 91% 86% 80% m = 4 84% 89% 87% 76% m = 5 82% 88% 88% 73% m = 8 79% 88% 91% 70% m = 20 79% 91% 93% 70% TABLE I: Winning Metrics Correlation to MaximalVoter SatisfactionInterestingly, the metrics W and W perform betterin aligning with voter satisfaction for larger values of m ,while this effect is not seen in W . This suggests that theeffect of polarity in the functional form of the metric (de-nominator of eq. 1) truly captures the voter satisfactionin a very non-obvious way. This is indeed nontrivial be-cause the metric is a function of only the votes obtainedby any particular candidate, but voter satisfaction is afunction of votes obtained by all candidates. It is alsovery clear from the performance of W . . that a metricwith c < m ≥ W .If we altered our definition of voter satisfaction to be¯ S α in eq. 8, we find that the winner picked by W almostalways maximizes voter satisfaction. So, whether or notthe metric function should penalize polarity (eq.1) verymuch depends on our definition of voter satisfaction.More generally, we could ask the following question–if the aim is to pick the candidate who would maximizethe voter satisfaction, then why do we need to pick thewinner in a round-about manner using a winning metric,rather than directly picking the candidate maximizingthe voter satisfaction? This is because the prescriptionof a winning metric a priori informs the voters how muchtheir negative votes weigh against their positive votes.Without a winning metric we wouldn’t have a tool todiscourage the voters from strategic voting. COMPARISON WITH RANKED VOTING
It is straightforward to convert the normed negativevotes into ranked votes, as long as voters cast distinctvotes to various candidates. For example, consider the4-candidate election with three voters as shown below. Candidate-B is the winner according to the metrics W , W and the voter satisfaction. Election-1
Candidate voter-1 voter-2 voter-3 Borda W W SA 5 [1] -5 [4] 3 [2] 3+0+2 3.0 1.84 6B 2 [2] 0 [3] 4 [1] 2+1+3 6.0 6.0 14C 1 [3] 1 [2] -2 [4] 1+2+0 0.0 0.0 6D -1 [4] 4 [1] 1 [3] 0+3+1 4.0 3.33 11
The rank corresponding to each vote is expressed insquare brackets next to the vote. Let us now bypass thevote itself and only consider the ranks. There are dif-ferent ways to pick a winner in a ranked voting system.In
Condorcet method [6], we split the m -candidate raceinto a bunch of 2-candidate head-to-head competitions.The winner is the candidate who wins every head-to-head competition. In the above example, voter-1 ranksA higher than B while voter-2 and voter-3 rank B higherthan A; hence B defeats A. Similarly B defeats C andD in head-to-head competitions.Thus candidate-B is theCondorcet winner in this example. However, there aremany situations where this procedure does not yield aclear winner, and we end up in a rock-paper-scissor con-figuration of cyclic loop of winners.In Instant Runoff , votes are counted in multiple stages.In each stage, the candidates with the least number ofrank-1 votes get eliminated, and subsequently the ranksof all other candidates are boosted up by one in thoseballots that had the eliminated candidate in the rank-1position. So, the voter’s ballot is not eliminated afterthe first preference is eliminated, instead the subsequentpreferences are considered in order. In the above exam-ple, candidate-C gets eliminated in the first round withzero rank-1 votes, and A, B & D end up in a tie. Sucha tie situation is unlikely when there are many voters,and the counting will proceed to next stage of elimina-tion. For instance if there were two voters who vote likevoter-3, then A and D get eliminated in the second stagewith just one rank-1 vote each, leaving candidate-B thewinner.A major issue with Instant Runoff is that it does notsatisfy the basic criterion of monotonicity . That is, awinning candidate can become a looser by getting rankedhigher by a voter, which is intuitively strange and unac-ceptable. It happens because a stronger candidate sur-vives an earlier stage of elimination due to the rank mod-ifications. This gives a lot of leeway for the voters tostrategically attempt to eliminate strong opposition can-didates at earlier stages of elimination, rather than voteaccording to their innate preferences. It is of course verydifficult to strategize for multistage elimination with alarge number of voters, but there is nothing stoppingthe voters from attempting to strategize. Any procedurewhich calls for multistage elimination of candidates isprone to this issue. The GS theorem [3] shows that it isnot possible to totally prevent strategic voting, but wecan curtail it significantly by satisfying the monotonicitycriterion.
Borda proposed a metric that aggregates a weightedsum of the ranks accrued by each candidate [7]. In an m -candidate race, rank-1 gets a weight ( m − m −
2) and so on, with rank-m ending at weightzero. In the above example, candidate A has accruedone rank-1, one rank-4 and one rank-2, yielding a netBorda count of 3+0+2 =5; while B attains a Borda countof 6 and wins the election. Borda metric satisfies themonotonicity criterion, but it violates another intuitivecriterion,
Independence of Irrelevant Alternatives (IIA).The IIA criterion requires that if the voters were al-lowed to modify their votes without changing the relativepreference between a winning candidate and a specificlooser, then that looser shouldn’t be able to win due tothe vote modifications. Suppose candidate- α is the win-ner and is ranked higher than a loosing candidate- µ bysome voter, then no matter how the voter changes theranks while holding α higher than µ , there should be nochance for candidate- µ to win. The Condorcet method ofcounting the head-to-head competitions only cares aboutthe relative rankings, hence it will satisfy IIA criterion.Any other method that weighs the absolute rank posi-tions, like the Borda metric, will violate this criterion.The IIA criterion seems overrated for its intuitiveness,primarily because it is a necessary condition for Arrow’simpossibility theorem which proves that it is not alwayspossible to find a winner in a ranked voting election whencertain intuitive conditions are met [2, 8, 9].The normed negative voting (NNV) method in thisarticle would not satisfy the IIA criterion, because thesevotes are cardinal in nature, which represents much richerinformation than relative ranking. Hence it is not subjectto implications of Arrow’s theorem [2]. Furthermore, theNNV also violates the monotonicity criterion. If a voterincreases the positive votes for the winner, then obviouslythe winning metric value as calculated by eq. 1 would fur-ther increase. But any increase in positive votes for thewinning candidate must be compensated by a decreasein votes to other candidates, as per the normed votingrules. Let’s suppose that the negative votes to anothercandidate is reduced. It is now possible for this can-didate to emerge as the new winner, if c + b >
1. Thisviolates the monotonicity criterion. However, notice thatthis is precisely what we prevented at a statistical level,by restricting the metric to stay under maximal penalty(eq. 5). Although the monotonicity criterion does nothold at individual instances, it holds on average withlarge number of voters. That is why it is not possibleto strategically exploit the negative votes without largescale coordinated effort among voters.In
Approval voting method, the voters can approve ofany number of candidates and disapprove of the others,and every approved candidate gets +1 vote. In a way,this violates the basic principle of one-person-one-vote . But we can rectify the method by assigning -1 vote to ev-ery disapproved candidate, thereby revising it as normed-Approval voting. We can view this as a special case ofthe NNV procedure with a uniform magnitude of positiveor negative vote for every candidate, which is clearly anunnecessary restriction on voter expression.The NNV method gathers the voter preferences in avery rich format. To emphasize this, let’s modify thevotes of voter-2 in the previous example of
Election-1 without affecting the relative ranks, as shown below.
Election-2
Candidate voter-1 voter-2 voter-3 Borda W W SA 5 [1] -1 [4] 3 [2] 3+0+2 7.0 6.2 10B 2 [2] 0 [3] 4 [1] 2+1+3 6.0 6.0 10C 1 [3] 1 [2] -2 [4] 1+2+0 0.0 0.0 2D -1 [4] 8 [1] 1 [3] 0+3+1 8.0 7.2 11
Candidate-B is still the winner under all the rankedvoting methods discussed above. But candidate-B loosesto both A and C under NNV metrics W , W , and thevoter satisfaction. Election-1 is identical to
Election-2 under ranked voting methods, but very different underNNV. This clearly illustrates the effect of superficiallyconsidering the ranks while ignoring the deeper negativepreferences of voters. It is also not difficult to imagine thefrustration that would develop among the voters if theirvotes aligned with
Election-2 , but ranked voting methodresults in B’s victory, which is aligned with
Election-1 . CONCLUSION
I have discussed the importance of negative votes incapturing the voter preferences in a richer format. Toprevent voters from exploiting the negative votes forstrategic voting, we constrained the winning metric toavoid over-penalization of the negative votes. Since thenegative votes for the disliked candidates come at theexpense of positive votes for their preferred candidates,the voters are incentivized to vote according to their truepreference, suppressing the intent for strategic voting.This would also have a serious impact on how elec-tion campaigns are conducted. For the fear of accumu-lating negative votes, the candidates would refrain fromdivisive rhetorics and stay focused on constructive issues.The major parties cannot afford to nominate a polarizingcandidate because they understand that it would directlypave way for a third party victory. The whole politicalarena can thus be depolarized.The candidates will stay well-behaved by conductingdecent campaigns and the voters will stay well-behavedby voting their true preferences. We can then hope fora future where no candidate receives any negative vote.–
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