Evaluating the Properties of a First Choice Weighted Approval Voting System
aa r X i v : . [ ec on . T H ] M a y Evaluating the Properties of a First Choice WeightedApproval Voting System
Peter Butler and Jerry LinJune 2, 2020
Abstract
Plurality and approval voting are two well-known voting systems with different strengths andweaknesses. In this paper we consider a new voting system we call beta(k) which allows votersto select a single first-choice candidate and approve of any other number of candidates, where k denotes the relative weight given to a first choice; this system is essentially a hybrid of pluralityand approval. Our primary goal is to characterize the behavior of beta(k) for any value of k . Undercertain reasonable assumptions, beta(k) can be made to mimic plurality or approval voting in theevent of a single winner while potentially breaking ties otherwise. Under the assumption thatvoters are honest, we show that it is possible to find the values of k for which a given candidatewill win the election if the respective approval and plurality votes are known. Finally, we showhow some of the commonly used voting system criteria are satisfied by beta(k). Introduction
Plurality is a popular voting system where each voter votes for exactly one candidate; closeto a third of all countries use plurality for government elections (Ace Project, n.d.). One majordrawback of plurality is that it tends to force a two-party system as a result of Duverger’s Law(Riker 1982, 753). This makes third-party candidates practically noncontenders in plurality sys-tems. Another issue with plurality arises in elections with more than two candidates. In such cases,a candidate who does not have a majority of the votes may still be elected, as may a candidatewho is less preferred overall than other candidates (Brams and Fishburn, 2007, 1-3). In particular,plurality can at times elect more extremist candidates in elections when there are more than twocandidates since a candidate does not need a simple majority of the votes to win such an election.This is seen as a problem with plurality by critics. Approval voting is an alternative system wherevoters may approve of any number of candidates. It has been adopted by several professionalgroups including the American Mathematical Society and the American Statistical Association1Karlin and Peres, 2017, 223). One clear advantage of approval voting is that minority candidatesactually stand a chance to win (Brams and Fishburn, 2007, 7). While approval does not sufferfrom Duverger’s Law as does plurality, approval is not a Pareto-efficient system; a candidate who isunanimously preferred to another candidate is not guaranteed to have a higher approval score (weshow this in sub-section 4.4). This is mainly due to the fact that, under approval, voters cannotexpress the degree to which they prefer one candidate over another. Although there is a scarcityof empirical evidence on the efficacy of approval voting, some critics believe that it promotes morecentrist candidates than plurality (Brams and Fishburn, 2007, 10; Cox, 1985, 118).We have developed a new voting system, called beta(k), that we believe can function as areasonable hybrid of plurality and approval. Under beta(k), a voter may approve of candidateswhile also denoting their top preference. In this system, approvals have a weight of while first-choice votes are weighted by k ≥ . We believe that this system maintains the benefits of approvalvoting and mitigates some of the potential drawbacks. Introducing the additional weight forfirst-choice preferences gives voters more options for expressing their levels of preference and coulddiscourage candidates from simply taking moderate stances on all issues. One of our goals is to findhow the value of k affects the outcome of a beta(k) election. We make the reasonable assumptionsthat any voter who would approve of a candidate in an approval election would approve of thatcandidate in a beta(k) election, and that any voter who would vote for a candidate in a pluralityelection would denote that candidate as their first-choice in a beta(k) election. We find values of k that guarantee a beta(k) winner to agree with a plurality winner or an approval winner, assuminga single winner exists, while potentially breaking ties otherwise. We also show that by knowingthe approval votes and plurality votes corresponding to a set of voters, we can find the subset ofcandidates who could potentially win in a beta(k) election, and we can find the ranges of k forwhich each of these candidates will win. Lastly, we compare the three systems by checking theircompliance with several common voting criteria.Much research has been done on the behavior of different voting systems and many criteriahave been developed to determine which systems function "better." In general, while the conceptof better and worse voting systems depends on criteria being checked, we believe that some of themost desirable properties are non-dictatorship, monotonicity, unanimity, and Pareto-efficiency (allof which we formally define in section 4). 2 Summary of relevant research
Brams and Fishburn have speculated that some elections would have had quite different resultshad approval been used instead of plurality (Brams and Fishburn, 2007, 59-69). Karlin and Pereshave shown previously that approval and plurality are monotonic (Karlin and Peres, 2017, 221-224). One issue frequently discussed by theorists is the possibility of voting strategically. TheGibbard-Satterthwaite theorem shows that no system (except dictatorship) is strategy-proof whenthere are more than two candidates, a famous result from the 1970s that is proven elegantlyin (Karlin and Peres, 2017, 226-228). There is theoretical and experimental research on votingstrategies for plurality and approval. In a plurality system where voters act strategically, boththe Condorcet Winner and Condorcet Loser Criteria are violated (Niou, 2001, 225). However, arecent experimental study with students at New York University suggests that plurality is lessmanipulable and more socially efficient than approval (Bassi, 2015, 77).
Definition 2.1
Let X = { C , C , . . . , C c } be a finite, ordered set with cardinality c ∈ N such that c ≥ . We refer to the elements of X as candidates . Definition 2.2
We assume that there exists a preference relation for each voter on the set ofcandidates. We denote this relation by ≻ , i.e. A ≻ B means that the voter prefers A over B. Weassume that this relation is complete (for all candidates A and B, either A ≻ B or B ≻ A ) and transitive (if A ≻ B and B ≻ C , then A ≻ C ). Furthermore, we denote the i th voter’spreference relation by ≻ i , and the collection of preference relations for all voters is a preferenceprofile ( ≻ , ≻ , . . . , ≻ n ) . Note: Definition 2.2 is based on definitions in (Karlin and Peres 2017, 218) . Definition 2.3 A vote over X is a non-zero real-valued vector in R c such that the j th elementin the vector corresponds to candidate C j in X . Definition 2.4
Let k ∈ R such that k ≥ . A beta(k) vote over X is a type of vote in R c suchthat each element in the vector is in { , , k } and exactly one element is equal to k. Definition 2.5 An approval vote is a type of vote in R c such that each element in the vector isin { , } and at least one element is equal to 1. Definition 2.6 A plurality vote is a type of vote in R c such that each element in the vector isin { , } and exactly one element is equal to 1. 3 efinition 2.7 A vote matrix is a n × c matrix of real numbers such that every row within thematrix is a vote of the same type. A beta(k) matrix is a vote matrix in which all the votes arebeta(k) votes and every beta(k) vote uses the same k value. An approval matrix is a votematrix in which all the votes are approval votes. A plurality matrix is a vote matrix in whichall the votes are plurality votes. Definition 2.8
Let K ( P, A ) denote a function that takes in a n × c plurality matrix P and a n × c approval matrix A and outputs a n × c beta(k) matrix B such that: B = K ( P, A ) = P · ( k −
1) + A. Definition 2.9
Let E ( X , Z ) denote a function that takes in a set of candidates X withcardinality c and a n × c vote matrix Z as inputs and outputs a × c matrix equal to the sum ofthe row vectors of Z . We call this × c matrix the score for the vote matrix Z . The type of thematrix used for the score is also used to describe the score (e.g. a plurality score is a score for aplurality matrix). We call any candidate in X whose corresponding element in a score is equal tothe maximum value in the score a winner for that score (there can be multiple winners ifmultiple candidates are tied for the highest score). Any candidate who is not a winner for thatsame score is a loser for that score. Any time there is more than one winner for a given score, wecall the score a tie . Winners, losers, scores, and ties can be described by the type of score used todetermine them (e.g. a plurality winner is the candidate whose corresponding element in aplurality score is equal to the maximum value in the plurality score). Definition 2.10 A voting system (we will often refer to this as simply a system for short) is amethod for selecting a single candidate from a set of candidates. In particular, an approvalsystem , plurality system , or beta(k) system is a voting system that randomly selects acandidate from the set of approval winners, plurality winners, or beta(k) winners (respectively). From now on, n i will denote the i th row of a vote matrix, n will denote the number of votes, c will denote the number of candidates, and X = { C , C , . . . , C c } will denote the set of candidates. P will denote a n × c plurality matrix, A will denote a n × c approval matrix, and B will denote a n × c beta(k) matrix; p j will denote the j th element in E ( X , P ) , a j will denote the j th element in E ( X , A ) , and b j will denote the j th element in E ( X , B ) . Lemma 3.1.
For any type of score, the set of winners is non-empty. roof. By definition, the set of candidates has cardinality c ≥ . Within the × c score, theremust be a maximum element. Therefore, the set of winners is non-empty. (cid:4) Theorem 3.2. If • B i,j = k if and only if P i,j = 1 , and • k > n ,then the set of beta(k) winners is a subset of the set of plurality winners.Proof. Suppose, for the sake of contradiction, there exists a candidate C l who is a beta(k) winnerand not a plurality winner. Let C w be any candidate in the set of plurality winners. Since k > n ,we shall say k = n + ǫ where ǫ > .Because C w is a plurality winner and C l is not, p w ≥ p l + 1 . Because C l is a beta(k) winner, b l ≥ b w and b l ≤ p l · k + ( n − p l ) .Then b l ≤ ( p l )( n + ǫ ) + ( n − p l ) < ( p l + 1)( n + ǫ ) ≤ p w ( n + ǫ ) ≤ b w . This means C l is not a beta(k) winner. Therefore, we have arrived at a contradiction. (cid:4) Corollary 3.2.1. If • B i,j = k if and only if P i,j = 1 , • there is exactly one plurality winner, and • k > n ,then C w is a plurality winner if and only if C w is a beta(k) winner.Proof. If C w is a beta(k) winner, then C w must also be the plurality winner as the set of beta(k)winners is a subset of the set of plurality winners.If C w is the plurality winner, then C w must also be a beta(k) winner because the set of beta(k)winners is a subset of the set of plurality winners and the set of beta(k) winners is non-empty. (cid:4) Remark. If • B i,j = k if and only if P i,j = 1 , and • k > n ,then the set of plurality winners may not be a subset of the set of the beta(k) winners. roof. Suppose, for the sake of contradiction, the set of plurality winners is always a subset ofthe set of the beta(k) winners. The following counterexample disproves this: B = k k , P = (cid:4) Lemma 3.3. If • A i,j = 1 if P i,j = 1 , • B = K ( P, A ) , and • C w is both a plurality winner and an approval winner,then C w is a beta(k) winner for any k ≥ .Proof. Without loss of generality, let C l be any other candidate in X . p w ≥ p l and a w ≥ a l .Thus, A’s score under beta(k) is kp w + ( a w − p w ) , while B’s score under beta(k) is kp l + ( a l − p l ) .A wins beta(k) if kp w + ( a w − p w ) ≥ kp l + ( a l − p l ) . This is equivalent to the condition k ≥ a l − a w + p w − p l p w − p l = 1 + a l − a w p w − p l . Because a l − a w is negative and p w − p l is positive, W wins for any k ≥ . (cid:4) Corollary 3.3.1.
Under the same assumptions, suppose X = { C , C } , candidate C is aplurality winner, and C is an approval winner. C is the beta(k) winner if k > a − a p − p .C is the beta(k) winner if k < a − a p − p .C and C are both beta(k) winners if k = 1 + a − a p − p . orollary 3.3.2. For any set of candidates X = { C , C , . . . , C c } , if C w has a higher score than C l under both approval and plurality votes, then C l cannot have the highest score under beta(k)for any k ≥ .Proof. Assume a candidate C w has a higher score under plurality and approval votes than C l . Asshown in the lemma, this implies that C w has a higher beta(k) score than C l for all k > , so C l cannot be a beta(k) winner. (cid:4) Theorem 3.4.
Given a set of candidates { C , C , . . . , C c } , let Y denote the subset of candidates { C , C , . . . , C r } (WLOG) that can win a beta(k) election given a value of k ≥ where n is thenumber of voters. Suppose (again, WLOG) p < p < . . . < p r . Then a r < a r − < . . . < a .Conversely, if (again, WLOG) a < a < . . . < a r , then p r < p r − < . . . < p .Proof. Suppose there exists a different ordering of the approval scores. Then for some pair C i , C j ( i = j ) , p i > p j and a i > a j or p i < p j and a i < a j . Then, C i or C j would not be apotential beta(k) winner, hence not in Y .To clarify, let p < p < . . . < p r . If the approval ordering is not the exact reverse of the pluralityordering i.e. there exists C i , C j such that p i < p j and a i < a j . Then C j is not a potential winnerby Corollary 3.3.2.By a similar argument, it is not hard to prove the other direction. (cid:4) Lemma 3.5.
Suppose as in Theorem 3.4 that Y is the subset { C , . . . , C r } that can win abeta(k) election (i.e. for each of those candidates there exists some k ≥ such that the givencandidate is in the set of winning candidates) and suppose that r > . Suppose that p < p < . . . < p r (hence a > . . . > a r ). Then the following series of inequalities holds: a − a p − p ≤ a − a p − p ≤ . . . ≤ a r − − a r p r − p r − . Proof.
Suppose that Y is the subset { C , ..., C r } that can win a beta(k) election and r > . Wewill proceed to prove this series of inequalities by induction.Consider the base case. Since C is a potential winner, that implies that there exists a k suchthat b ≥ b and b ≥ b . Thus we get the system of inequalities k · p + a − p ≥ k · p + a − p k · p + a − p ≥ k · p + a − p , a − a p − p ≤ k ≤ a − a p − p . Thus, we must have a − a p − p ≤ a − a p − p . Otherwise there would not exist any k for which candidate C could win, contradicting the initialassumption.Now choose w such that w < r and suppose that a − a p − p ≤ . . . ≤ a w − − a w p w − p w − . Since we assume that C w is a potential winner, there exists a k such that b w > b w − and b w > b w +1 . This gives us the system of inequalities k · p w + a w − p w ≥ k · p w − + a w − − p w − k · p w + a w − p w ≥ k · p w +1 + a w +1 − p w +1 , which is equivalent to a w − − a w p w − p w − ≤ k ≤ a w − a w +1 p w +1 − p w . Thus, a k such that C w is a potential winner exists only if a w − − a w p w − p w − ≤ a w − a w +1 p w +1 − p w . This completes the proof by induction. (cid:4)
Theorem 3.6.
Suppose as in the above Lemma we have potential candidates { C , . . . , C r } with r > . Then candidate C w is a beta(k) winner if and only if a w − − a w p w − p w − ≤ k ≤ a w − a w +1 p w +1 − p w . If w = 1 or w = r , then the left or the right inequality (respectively) should be omitted. roof. ( ⇒ )For candidate C w to be a beta(k) winner, the equation k · p w + a w − p w ≥ k · p j + a j − p j must be satisfied for all j = w .To establish the lower bound for k , consider all integers l such that ≤ l < w . We must have k · p w + a w − p w ≥ k · p l + a l − p l . Equivalently, k ≥ max { a l − a w p w − p l | ≤ l < w } . Notice however, that by the previous Lemma, a l − a w p w − p l ≤ a w − − a w p w − p w − for all such l , thus k ≥ a w − − a w p w − p w − . To establish the upper bound, we use a similar calculation but for g where w < g ≤ r . For C w towin, we need k so that k ≤ min { a w − a g p g − p w | w < g ≤ r } . Again, the previous Lemma implies that a g − a w p w − p g ≥ a w +1 − a w p w − p w +1 for all such g , thus k ≤ a w − a w +1 p w +1 − p w . Therefore, C w is a beta(k) winner whenever a w − − a w p w − p w − ≤ k ≤ a w − a w +1 p w +1 − p w . Of course, if w = 1 then the lower bound for k is just and if w = r then there is no upperbound for k .( ⇐ )Suppose, for the sake of contradiction, that C w is a beta(k) winner and there exist l, w ∈ N suchthat ≤ l < w and a w − a l p l − p w > k .If a w − a l p l − p w > k , then k · p w − p w + a w < k · p l − p l + a l .However, this contradicts C w being a beta(k) winner.9imilarly, suppose that C w is a beta(k) winner and there exist w, g ∈ N such that w < g ≤ r and k > a g − a w p w − p g .If k > a g − a w p w − p g , then k · p w − p w + a w < k · p g − p g + a g .Again, this contradicts C w being a beta(k) winner. (cid:4) Theorem 3.7. If • A i,j = 1 if and only if B i,j = 0 , and • ≤ k < n ,then the set of beta(k) winners is a subset of the set of approval winners.Proof. Suppose, for the sake of contradiction, there exists a candidate C l who is a beta(k) winnerand not an approval winner. If C w is any approval winner, then a w ≥ a l + 1 and b w ≥ a w , but b w ≤ b l ≤ a l · ( k ) < a l (1 + n ) ≤ a w . This is a contradiction. (cid:4) Corollary 3.7.1. If • A i,j = 1 if and only if B i,j = 0 , • there exists exactly one approval winner, and • ≤ k < n ,then C w is a beta(k) winner if and only if C w is an approval winner.Proof. If C w is a beta(k) winner, then C w must also be the approval winner as the set of beta(k)winners is a subset of the set of approval winners.If C w is the approval winner, then C w must also be a beta(k) winner because the set of beta(k)winners is a subset of the set of approval winners and the set of beta(k) winners is non-empty. (cid:4) Remark. If • A i,j = 1 if and only if B i,j = 0 , and • ≤ k < n ,then the set of approval winners may not be a subset of the set of beta(k) winners. roof. Suppose, for the sake of contradiction, the set of approval winners is always a subset ofthe set of the beta(k) winners. The following counterexample disproves this: B = k k , A = (cid:4) Definition 4.1
A voting system is a dictatorship if there exists a vote whose maximum valuealways corresponds to a winning candidate under that system.
Observation: beta(k) with at least 3 votes, is a non-dictatorship.Proof.
For the sake of contradiction, assume this is a dictatorship. This means there exists n j such that the maximum value in this vector always corresponds to the beta(k) winner. WLOG,suppose the maximum value in n j corresponds to the w th candidate. Suppose for all n p = n j , k points are assigned to the l th candidate and points are assigned to the w th candidate.Then b l > b w and we have arrived at a contradiction. (cid:4) Note: Since approval is equivalent to beta(1), this shows that approval is also a non-dictatorship.Under this construction, multiple winners cannot be selected for a fixed set of plurality votes.Furthermore, since plurality winners agree with beta(k) winners whenever k > n , this shows thatplurality is also a non-dictatorship (since n > and all but one of the voters vote for n j not n i ,the plurality winner is unique). Definition 4.2
A voting system is monotonic if increasing a value in a vote corresponding to awinning candidate cannot make that candidate lose. Similarly, in a monotonic system, decreasingthe value in a vote for a losing candidate cannot make that candidate win.
Observation: beta(k) is a monotonic voting system.Proof.
WLOG, assume that beta(k) selects the w th candidate. Then b w > b l for all l = w .Suppose that the n j vote vector assigns either or to its w th element. If this element isincreased, then b neww > b w > b l ≥ b newl for all l = w where b newi denotes the new beta(k) total for11andidate i after the vote was increased. Similarly, if candidate l does not win, then there exists w such that b w > b l . Suppose that the n j vote vector assigns either or k to its l th element.Then, if this element is decreased, we have b newl < b l < b w ≤ b neww , so candidate l still does notwin the election. (cid:4) Note: Again, since the result above holds for all k, it holds for approval since beta(1) is equivalentto approval. This also holds for plurality since plurality winners agree with beta(k) winnerswhenever k > n (assuming that there is not a plurality tie). However, it is trivial to show thateven under a tie, the plurality system is still monotonic.
Definition 4.3
If there exists an element in X that corresponds to the maximum value in everyvote, this element is called a unanimous winner . Observation: beta(k) selects a unanimous winner, if such a candidate exists.Proof.
Assume the w th candidate in the beta(k) votes is the unanimous winner. Then b w ≥ b j forall j = w , and the beta(k) system selects the w th candidate. (cid:4) Note: Again, since the result above holds for all k, it holds for approval since beta(1) is equivalentto approval and this property holds trivially for plurality elections.
Definition 4.4
For a set of n votes, a system is Pareto efficient if, for any of the system’spossible winners C w and any l = w , there is some voter who prefers C w to C l (i.e. there existsvoter i such that C w ≻ i C l ). In English, a Pareto winner is a candidate such that there does notexist a different candidate who is unanimously preferred and a Pareto efficient system is one thatalways elects a Pareto winner. Proposition: beta(k) is pareto efficient when k > c − .Proof. If a candidate receives at least one k-vote, then at least one voter prefers that candidateto any other candidate. Thus, if we can define bounds for k such that any beta(k) winner isguaranteed to have at least one k-vote, the result follows. We will consider two cases: Case 1:
Every candidate receives at least one k-vote. In this case, any winner will be Pareto.12 ase 2:
There is at least one candidate who receives no k-votes, so there are at most c − candidates that do receive a k-vote. Notice that the candidate(s) who does not receive a k-votecan have no more than n total points under beta(k). Among the other candidates, at least onemust have received at least n/ ( c − k-votes. Thus, by setting k > c − , at least one of thecandidates with a k-vote will earn more than n points under beta(k). Therefore any winnerunder such a system must have at least one k-vote.Taking into consideration the cases above, it is clear that for k > c − , any beta(k) winner mustbe Pareto. (cid:4) Note: It is trivial to see that any plurality winner is Pareto. Such a winner must have at leastone plurality vote, implying that at least one voter prefers that candidate to every other candidate.Notice, however, that with the bound established above, it can be shown that approval does notalways select a Pareto winner. Consider a case where there is a tie between two candidates underapproval and both candidates are approved by the same voters, but one candidate is unanimouslypreferred over the other. Then both of these candidates are approval winners, but only one isPareto.
In this paper we derived some interesting properties for beta(k) and saw how its performancecompares to plurality and approval. In particular, for certain k , beta(k) is no "worse" thanplurality or approval from the results of 3.2 and 3.7. Furthermore, beta(k) has an advantage overapproval by being fully Pareto-efficient for k > c − . This result is not hard to extend to showthat if a certain condition is satisfied by either plurality or approval but not the other, then thatcondition is satisfied under beta(k) for a certain range of k . In particular, if the given criterion issatisfied by plurality but not by approval, then the criterion is satisfied under beta(k) for k greater than some lower bound; if the criterion is satisfied by approval but not plurality, then thecriterion is satisfied under beta(k) for k less than some upper bound. We conjecture that if acriterion is satisfied by both plurality and approval, then it is satisfied by beta(k) for any k > .Beta(k) could be a good alternative to approval or plurality in elections where the number ofvoters is large and the number of candidates is small. In such cases, one could choose a k largeenough to guarantee Pareto results and this k would still be small enough that the beta(k) wouldbe similar to approval.There are further ways to determine the viability of beta(k). This would include checking whether13t satisfies other common voting criteria (i.e. Condorcet-winner, consistency, et cetera) and seeinghow it performs in Monte Carlo simulations. Additionally, it would be interesting to see whatpossible voting strategies exist for beta(k) and how they affect the outcome of beta(k) elections.A method for determining rational voter strategies may be found in (Myerson and Weber, 1993). Ace Project: The Electoral Knowledge Network. (Accessed January 2019)
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