aa r X i v : . [ ec on . T H ] M a y ExploringWeak Strategy-Proofnessin Voting Theory
Anne Elizabeth Carlstein
Advisor: Eric Maskin
A thesis submitted in partial fulfillment of therequirements for the degree of Bachelor of Arts withHonors in the Mathematics DepartmentHarvard University
March 29, 2020
Contents
Introduction 2Step 1 8Step 2 33Step 3 45Acknowledgements 70References 71
Introduction
In social choice theory, the intent is to choose a winning alternative, given pref-erences of voters. For the purposes of this thesis, we are only considering a choicebetween three alternatives { x, y, z } . To understand how this is accomplished, wemust first determine how voters and their preferences are expressed. Here, we de-fine a voter (denoted i ) as a member of a continuum , drawn from the unit interval(i.e., i ∈ [0 , x ≻ y means that option x is preferred to option y ).A domain (denoted by U ) is a collection of preference rankings, each of whichwe express as a column vector. For example, consider the following domain: U = ( x y yy x zz z x ) The third column vector yzx represents the preference ranking y ≻ z ≻ x . Witha specific profile (denoted by u ), we can represent the relative proportion of votersand their preferences about the alternatives. For example, consider the followingprofile: u = p q 1 - p - qx y yy x zz z xIn this profile, a proportion q of the voters have preference ranking yxz . We cansimilarly identify the other preference rankings and their weights in the profile.Given a preference ranking of this form, how would we determine the winningoutcome? This can be accomplished by a voting rule. For the purposes of thispaper, given a profile of voter preferences u , and a collection of alternatives X , a voting rule (denoted F ) selects a winner (if one exists). More formally, given aset of alternatives X ⊆ { x, y, z } , a voting rule F maps each profile u on domain U to an alternative in X . This is denoted F ( u, X ). Given this formulation of a voting rule, we still have not specified how the votingrule actually selects the result. There are numerous methods to choose a winner.We would not be happy with a voting rule that says “given options { x, y, z } , x is always the winner, regardless of voter preferences.” (For example, a Russianelection where Vladimir Putin is on the ballot). So, we want voting rules to satisfycertain desirable axioms. For example, consider the profile u that we defined above.Intuitively, if a voting rule selected z as the winner, we might be dissatisfied withthat result, because clearly, everyone prefers option y over option z . (This is thePareto property, as is defined below). This will ensure that the chances of a tie are negligible, i.e., we say that ties are nongeneric. This definition only requires the voting rule to map to a winning alternative on generic profiles.On a nongeneric profile, a voting rule can evaluate to ∅ . Certain axioms are extremely natural:
Pareto (denoted P) - As we saw before, if one alternative ( y ) is preferred byeveryone to another alternative ( z ), then z should not be the alternative that isselected by the voting rule. It also follows from this axiom that if everyone ranksalternative y above every other option, then y should be the winner. Anonymity (denoted A) - This tells us that if the names of voters are relabeled,then the outcome of the voting rule should not change. To express this moreformally, we take some measure preserving permutation π : [0 , → [0 ,
1] (i.e., forany C ⊆ [0 , , µ ( C ) = µ ( π ( C ))). Apply permutation π to all voters i in someprofile u , so that each voter i has voter π ( i )’s preference rankings, and denotethe fully permuted profile as u π . To satisfy this property, it must be true that F ( u, X ) = F ( u π , X ). Neutrality (denoted N) - If we permute the names of the candidates on theballot (and accordingly adjust the voter preferences associated to them), then theadjusted winner should be the permuted winning candidate.For example, consider the following permutation on profile u : σ ( x ) = y ; σ ( y ) = z ; σ ( z ) = xu σ = p q 1 - p - qy z zz y xx x ySuppose that the winner according to some voting rule F ( u, X ) = y . So then,to satisfy Neutrality, it should be true that F ( u σ , X ) = σ ( y ) = z .The above axioms are clearly ones that would be attractive for a voting rule tofulfill. However, some other axioms in the voting literature are less obvious. Inparticular, the next axiom is particularly controversial: Independence of Irrelevant Alternatives (denoted IIA) - Let X be a collec-tion of alternatives, and let X ′ be a collection of alternatives where x ∈ X ′ ⊂ X . Ifalternative x is the winner F ( u . , X ), then to satisfy IIA, x must also be the winner F ( u . , X ′ ). In other words, dropping non-winning candidates from considerationshould not change who the winner is.Surprisingly, these axioms are not easy to fulfill. This is apparent in the mostcelebrated result of social choice theory, Arrow’s Impossibility Theorem (Arrow1951). Theorem (Arrow’s Impossibility Theorem, as stated in Dasgupta andMaskin 2019)
Assume the domain U consists of all possible preference rankingsover { x, y, z } . There is no voting rule that satisfies P, A, N, and IIA. This result might seem disheartening - however, beyond this result, there is stilla wide variety of directions that voting theory can take. We can consider how torestrict the domain of preferences. We can reconsider which axioms are included.With these modifications, we can analyze how these restrictions limit the votingrules that are possible. And that is what we will do!First, we introduce some of the common voting rules: This theorem applies to all sets of 3 or more alternatives, but for the purposes of this thesis,we are focusing on just the case of 3 alternatives.
Rank-Order Voting a.k.a. Borda Count (Borda 1781) - The alternativesare ranked by each voter (in the order of their preferences), and the alternatives arescored according to the following rule: an alternative gets k points for each voterthat ranks it above k other alternatives. The highest scoring alternative wins. Thisvoting method dates back to 1781, and was proposed by Jean-Charles Borda. Majority Rule a.k.a. Condorcet’s Method (Condorcet 1785) - Again,the alternatives are ranked by each voter. Each pair of alternatives are comparedin a head-to-head contest (aggregating over all voters), and the winning option isthe alternative x where a majority of voters prefer x over any other opponent y .This method also dates back to the 18 th century (1785), and was championed by theMarquis de Condorcet. (Borda and Condorcet were contemporaries and intellectualfoes - they disagreed on which voting rule was superior.) Plurality Rule - Given the preference rankings of each voter, the alternativethat is ranked in the first spot by the most voters is declared the winner (even ifthey are short of a majority) . This method is used in U.S. elections (e.g., to electmembers of Congress, to determine the winner of each state’s electoral votes, formany local elections).Of course, there are many voting rules besides the methods described above.However, in the spirit of Arrow’s Impossibility Theorem, Dasgupta and Maskin(2019) show in their Theorem 6 that the Borda Count and Condorcet’s Methodare the only possible voting rules when certain intuitive axioms are imposed. Thisthesis generalizes Dasgupta and Maskin’s
Theorem 6 by relaxing a key axiom.In
Theorem 6 , attention is restricted to rich domains:
A domain U is a rich domain if for each x ∈ X there exists some y, z ∈ X andsome preference ranking in the domain such that y ≻ x ≻ z . In other words, eachalternative must appear in a non-extremal place in some preference ranking in thedomain.For example, consider the following domains: U = ( x x y zy z z yz y x x ) U = ( x x y yy z z xz y x z ) (Clearly, U is not rich, while U is).Additionally, their Theorem 6 replaces the common IIA axiom with
Strategy-Proofness . Strategy-Proofness (denoted SP) - Consider a generic profile u on the domain U , and alternatives X , and suppose that F ( u, X ) = x . Let the notation x ≻ i y denote that voter i prefers alternative x over alternative y . Consider all coalitions C ⊆ [0 , u ′ where for all i / ∈ C , ≻ i = ≻ ′ i . Then, for y = F ( u ′ , X ),there exists some i ∈ C such that x ≻ i y .Suppose that u represents the true preferences of the voters. For any strategicprofile u ′ , where members of the coalition C misreport their preferences, there issome voter in the misreporting coalition who winds up with a worse result thanwhat they would have received had they reported their true preferences. Or, inother words, if a coalition of voters can misreport and improve their outcome, then This definition of plurality rule makes sense given the preference rankings of voters - however,in practice, plurality rule does not actually require voters to rank all alternatives (they only needto report their one top choice). the voting rule is not Strategy Proof. Clearly, this is an appealing axiom, as wewant voters to be incentivized to report their true preference rankings, and to avoidattempting to (potentially misguidedly) game the system for a better result.Before stating Dasgupta and Maskin’s
Theorem 6 , we formally define the twovoting rules, Condorcet’s Method (denoted F C ), and Borda Count (denoted F B ): Majority Rule (a.k.a. Condorcet’s Method) - F C ( u, X ) = { x ∈ X | µ [ i | x ≻ i y ] ≥ ∀ y = x, y ∈ X } In other words, when considering the head-to-head comparisons of all pairs,the Condorcet winner x is the one for which a majority prefer x to every otheralternative y .Recall profile u that we defined above: u = p q 1 - p - qx y yy x zz z xFor example, apply F C to profile u . We see that in the pairing of ( x, y ), x ispreferred to y with proportion p . And, in the comparison ( x, z ), x is preferred to z with proportion p + q . Finally, in the comparison ( y, z ), y is always preferred. Thistells us that z can never be the Condorcet winner. We also can see that if p > ,then x is the Condorcet winner.However, note that Condorcet’s Method does not always select a winner. Con-sider the following profile: u c = 1/3 1/3 1/3x y zy z xz x yWe apply F C to profile u c . We can see that a proportion prefer x to y ; aproportion prefer y to z , and a proportion prefer z to x . This is a CondorcetCycle. According the head-to-head contests, we have that the overall electorateprefers x ≻ y , y ≻ z , and z ≻ x . This means that no one can be the Condorcetwinner - giving us the Condorcet paradox (as noted by the Marquis de Condorcet). Ranked-Choice Voting (a.k.a. Borda Count) - F B ( u, X ) = { x ∈ X | Z r i ( x ) dµ ( i ) ≥ Z r i ( y ) dµ ( i ) ∀ y ∈ X } (Here, the point score r i ( x ) = |{ y ∈ X | x (cid:23) i y }| )The score for each alternative is determined by integrating over the point scores r i for all voters i . The point score r i ( x ) is measured as the number of candidatesthat are ranked no higher than x by voter i . The Borda winner is the alternativewhose total score is higher than the score of any other candidate.As an example, we can apply the Borda Count to profile u :The score for alternative x is: 2 p + q The score for alternative y is: p + 2 q + 2(1 − p − q ) = 2 − p .The score for alternative z is: 1 − p − q . Note that z can never be the winner under the Borda Count because this wouldrequire that 1 − p − q > − p , which would imply that q < −
1, which is impossible.Also note that p > is not enough to guarantee that x is the Borda winner, eventhough p > guaranteed that x would be the Condorcet winner.So, with this setup, we can state the theorem: Theorem 6 (Dasgupta and Maskin) If F satisfies P, A, N, and SP on U,and U is rich, then F = F B or F = F C . The proof of this theorem proceeds in three steps. In the first step of the argu-ment, it is shown that the Borda Count is the only voting rule that satisfies theaxioms on the
Condorcet Cycle domain.A Condorcet Cycle domain (on 3 alternatives) is: U CC = ( x y zy z xz x y ) Next, for the second step, it is proven that no voting rule satisfies the axiomson any expansion of the Condorcet Cycle domain. The final step shows that Con-dorcet’s Method is the only voting rule that satisfies the axioms on any remainingrich domain.Richness is an important component of this result. Without this domain restric-tion, there are non-rich domains where a voting rule other than F C and F B satisfythe given axioms. Additionally, in the proof, richness also allows for permuting thealternatives (applying axioms A and N), while still remaining in the same domain.Returning to the definition of Strategy-Proofness, we notice that there is no limiton the size of the coalition that deviates. However, manipulation on a large scale isunrealistic in practice, so instead, we focus only on manipulations by coalitions ofa small size ( ε ). This allows us to formulate a less restrictive axiom, namely WeakStrategy-Proofness (as suggested by Shengwu Li, and as defined by Dasguptaand Maskin 2019).A voting rule is manipulable on U if for all ε >
0, there exist some coalition C where | C | < ε , profiles u and u ′ (where for all i / ∈ C , ≻ i = ≻ ′ i ) and x, y ∈ X , where x = F ( u, X ) , y = F ( u ′ , X ), and y ≻ i x for all i ∈ C . Weak Strategy-Proofness (denoted WSP) - A voting rule F satisfies WeakStrategy-Proofness on U if F is not manipulable on U .In other words, if a coalition of arbitrarily small size can misreport their prefer-ences and achieve an improved result, then the voting rule is manipulable.Because Weak Strategy-Proofness allows for a larger class of voting rules to beconsidered, when taken as an axiom, this leads to a generalization of Dasgupta andMaskin’s Theorem 6 .So, for my senior thesis, I prove the following result:
Theorem If F satisfies P, A, N, and WSP on U, and U is rich, then F = F B or F = F C . This proof proceeds in three steps, as in the proof of
Theorem 6 . However,this proof requires many more subcases, to check for the possibility of small groupmisrepresentations. This utilizes a partitioning argument (based on ε ), which is ageneralization of the argument used in the proof of Theorem 5 (Dasgupta andMaskin). The proof also introduces iterative and inductive machinery to relate theevaluation of profiles that differ by more than ε . Without further ado, we begin the proof.
Step 1
First, we want to show that the Borda Count is the unique voting rule thatsatisfies P, A, N, and WSP on the Condorcet Domain: U CC = ( x y zy z xz x y ) Because WSP is less restrictive than SP, we know from Barbie et al. 2006 that F B also satisfies WSP on the Condorcet Domain. (By properties of the BordaCount, F B always satisfies P, A, N).We consider the following profile: u . = a b 1 - a - bx y zy z xz x ySince U CC is symmetric, for this entire step, we can assume without loss ofgenerality that x is the winner according to the Borda Count. This means that:2 a + (1 − a − b ) > a + 2 b a + (1 − a − b ) > b + 2(1 − a − b )Assume that F is some voting rule that satisfies P, A, N, and WSP, and that F = F B . Fix ε > Step 1, Case I:
Suppose that according to the Borda Count, y beats z .So, we have the following inequalities: a + 1 − b > a + 2 b > − a − ba + b > a > > b Assume that F has the following evaluation: F ( u. ) = y ( A σ on u. , and the resulting profile, where σ ( x ) = z, σ ( y ) = x, σ ( z ) = y : u = a b 1 - a - bz x yx y zy z x = b 1 - a - b ax y zy z xz x yBy A and N, we have: F ( u ) = x ( A : Suppose that 1 − a − b ≥ b Case 1.I.1.1.1 ≤ a − b < ε In this case, voters in profile u . with preferences xyz will want to improvetheir outcome. A weight of a − (1 − a − b ) will misreport as zxy , and a weightof (1 − a − b ) − b will misreport as yzx . This will have a total coalition size of2 a − b + 1 − a − b = a − b < ε . It will induce profile u and evaluation ( A x . This would contradict WSP, so Case 1.I.1.1.1 can nothold.
Case 1.I.1.1.2 ε ≤ a − b < ε Consider the following profile: u = b + (k + m) 1 - a - b - k a - mx y zy z xz x yWhere: k = − a − b ; m = a + b − Suppose that F ( u ) = x . But then, note that voters in profile u with preferences xyz would form a coalition of size k + m and misreport as yzx with a weight of k , and would misreport as zxy with a weight of m . This would induce profile u ,and would result in improved result x . This would contradict WSP, implying that F ( u ) = x .But then, in profile u . , voters with preferences xyz would want to induce profile u .Note that the following weights are equivalent: a − ( k + m ) = b + ( k + m ) b + k = 1 − a − b − k − a − b + m = a − m So then, from profile u . , voters with true preferences xyz would misreport as yzx with a weight of k , and would misreport as zxy with a weight of m (for a total coalition size of k + m < ε ). This would form profile u and improved outcome x . This would contradict WSP, so Case 1.I.1.1.2 does not hold.
Case 1.I.1.1.n+1 nε ≤ a − b < ( n + 1) ε Again, as assumptions for this step, we have that: F ( u . ) = y ( A F ( u ) = x ( A u = a b 1 - a - bz x yx y zy z x = b 1 - a - b ax y zy z xz x yConsider the following profile: u ( j ) = b + ( jk ′ + jm ′ ) 1 - a - b - jk ′ a - jm ′ x y zy z xz x y(Where k ′ = − a − bn +1 ; m ′ = a + b − n +1 , and j iterates from 1 to n ).Suppose that F ( u (1) ) = x . But then, voters in profile u (1) with preferences xyz would misreport as yzx with a weight of k ′ , and would misreport as zxy with aweight of m ′ (for a total coalition size of k ′ + m ′ < ε ). This would induce profile u ,and improved result x . And, this would contradict WSP, meaning that F ( u (1) ) = x .Now, assume that F ( u ( j ) ) = x for some general 1 ≤ j < n . We can show that F ( u ( j +1) ) = x . Suppose instead that F ( u ( j +1) ) = x . But then, note that votersin profile u ( j +1) with preferences xyz would misreport as yzx with a weight of k ′ , and would misreport as zxy with a weight of m ′ (for a total coalition size of k ′ + m ′ < ε . This would induce profile u ( j ) , and improved result x . And, this wouldcontradict WSP. Instead, it must be true that F ( u ( j +1) ) = x .This means that for the following profile: u ( n ) = b + ( nk ′ + nm ′ ) 1 - a - b - nk ′ a - nm ′ x y zy z xz x y F ( u ( n ) ) = x Then, note that: b + ( nk ′ + nm ′ ) = b + ( n ( a − b ) n + 1 ) = ( n + 1) b + na − nbn + 1 = na + bn + 1= ( n + 1) a − ( a − b ) n + 1 = a − ( k ′ + m ′ )Similarly, it is also true that:1 − a − b − nk ′ = b + k ′ a − nm ′ = 1 − a − b + m ′ So, profile u ( n ) is exactly equivalent to the following: u ( n ) = a - ( k ′ + m ′ ) b + k ′ m ′ x y zy z xz x yBut then, those in profile u . with preferences xyz will form a coalition of size k ′ + m ′ , and misreport as yzx with weight k ′ , and misreport as zxy with weight m ′ . This will induce profile u ( n ) and improved result x , contradicting WSP. Thismeans that this case in general fails to satisfy the properties, and we have that cannot hold. : Suppose instead that 1 − a − b < b .For this case, we consider the following profile: u = 1 - 2b b bx y zy z xz x y Suppose that F ( u ) = x ( A F ( u . ) = y ( A Case 1.I.1.2.1.1 ≤ b + a − < ε If F ( u ) = x , then voters in profile u . with preferences xyz will form a coalitionof size 2 b + a −
1, and misreport as zxy . This will induce profile u and improvedresult x , which would contradict WSP. So, in this case, F ( u ) = x . Case 1.I.1.2.1.2 ε ≤ b + a − < ε Again, for this case, we have: F ( u . ) = y ( A F ( u ) = x ( A u = a - k b 1 - a - b + kx y zy z xz x y(Here, k = b + a − . Note that the definition of k is local to this case.)Also: a − k = 1 − b + k − a − b + k = b − k So: u = 1 - 2b + k b b - kx y zy z xz x ySuppose that F ( u ) = x . But then, voters in u with preferences xyz willform a coalition of size k , and misreport as zxy . This will induce profile u andimproved result x , which would contradict WSP. So, F ( u ) = x .But then, note that voters in profile u . with preferences xyz will misreport as zxy with a weight of k . This will induce profile u , and achieve improved result x , contradicting WSP. So, this case fails to hold. Case 1.I.1.2.1.n+1 nε ≤ b + a − < ( n + 1) ε For this case, we have: F ( u . ) = y ( A F ( u ) = x ( A u ( j ) = 1 - 2b + j k ′ b b - j k ′ x y zy z xz x y(Here, j iterates from 1 to n , and k ′ = b + a − n +1 . Also, note that this definition of k ′ and u ( j ) is local to this case.)We consider: u (1) = 1 - 2b + k ′ b b - k ′ x y zy z xz x ySuppose that F ( u (1) ) = x . But then, voters in u (1) with preferences xyz willform a coalition of size k ′ , and misreport as zxy . This will induce profile u ,and achieve improved outcome x , contradicting WSP. So, it must be the case that F ( u (1) ) = x .Now we assume that for some general j that F ( u ( j ) ) = x . We can show that F ( u ( j +1) ) = x . Suppose instead that F ( u ( j +1) ) = x . But then, voters in u ( j +1) with preferences xyz will form a coalition of size k ′ , and misreport as zxy . Thiswill induce profile u ( j ) , and will achieve improved outcome x . This would contradictWSP, so, F ( u ( j +1) ) = x . This is true for any j .In particular, we have: u ( n ) = 1 - 2b + nk ′ b b - nk ′ x y zy z xz x y F ( u ( n ) ) = x ( A − b + nk ′ = 1 − b + n (2 b + a − n + 1 == na − b + 1 n + 1 = a − b + a − n + 1 = a − k ′ Likewise: b − nk ′ = 1 − a − b + k ′ So, profile u ( n ) is exactly equivalent to the following profile (with relabeledweights): u ( n ) = a - k ′ b 1 - a - b + k ′ x y zy z xz x yIn profile u . , voters with preferences xyz will form a coalition of size k ′ , and mis-report as zxy . This will induce profile u ( n ) and evaluation ( A x over y . This would contradict WSP. This means that our supposition that F ( u ) = x is not possible. So, this case also cannot hold, and concludes . Now suppose that F ( u ) = y .Recall that we are still in the case where 1 − a − b < b . Apply permutation σ to u (where σ ( x ) = z, σ ( y ) = x, σ ( z ) = y ). From A and N, this results in thefollowing profile and evaluation: u = b b 1 - 2bx y zy z xz x y F ( u ) = x ( A F ( u . ) = y ( A Case 1.I.1.2.2.1 ≤ a − b < ε But then, voters in u . with preferences xyz will form a coalition of size a − b < ε ,and misreport as zxy . This will induce profile u and evaluation ( A x over y . This would contradict WSP, so Case 1.I.1.2.2.1 cannothold. Case 1.I.1.2.2.n+1 nε ≤ a − b < ( n + 1) ε Again, we have: F ( u . ) = y ( A F ( u ) = x ( A u ( j ) = b + jk ′ b 1 - 2b - jk ′ x y zy z xz x y(Here, j iterates from 1 to n , and k ′ = a − bn +1 . Again, note that the definitions for k ′ , n, u ( j ) are local to this section).We consider the profile: u (1) = b + k ′ b 1 - 2b - k ′ x y zy z xz x ySuppose that F ( u (1) ) = x . But then, voters in profile u (1) with preferences xyz will form a coalition of size k ′ < ε , and misreport as zxy . This will induce profile u and evaluation ( A x over y . This would contradictWSP, meaning that F ( u (1) ) = x .Now, for a general j , assume that F ( u ( j ) ) = x . We can show that F ( u ( j +1) ) = x . Suppose instead that F ( u ( j +1) ) = x . But then, voters in profile u ( j +1) withpreferences xyz will form a coalition of size k ′ < ε , and misreport as zxy , whichwould induce profile u ( j ) and improved result x . This would contradict WSP. So,in general, F ( u ( j ) ) = x .In particular, we have the following profile and evaluation: u ( n ) = b + nk ′ b 1 - 2b - nk ′ x y zy z xz x y F ( u ( n ) ) = x ( A b + nk ′ = b + n ( a − b ) n + 1 = b + nan + 1 = ( n + 1) a − a + bn + 1 = a − k ′ Similarly, it is true that:1 − b − nk ′ = 1 − a − b + k ′ So, profile u ( n ) is equivalent to the following relabeling of weights: u ( n ) = a - k ′ b 1 - a - b + k ′ x y zy z xz x yWe have that F ( u ( n ) ) = x . But then, in profile u . , voters with preferences xyz will form a coalition of size k ′ , and misreport as zxy . This will induce profile u ( n ) and evaluation ( A x over y . This would contradict WSP.So, this case also cannot hold, meaning that for F ( u ) = y . Finally, suppose that F ( u ) = z .Again, we consider profile u : u = b 1 - a - b ax y zy z xz x yRecall that: F ( u ) = x ( A b > − a − b . Case 1.I.1.2.3.1 ≤ a − b < ε Here, voters in profile u with preferences zxy will misreport as xyz with a weight of 1 − b , and will misreport as yzx witha weight of 2 b + a − a − b < ε ). This would induceprofile u , achieving improved outcome z . This would contradict WSP. Case 1.I.1.2.3.n+1 nε ≤ a − b < ( n + 1) ε Consider the following profile: u ( j ) = 1 - 2b - jk b - jm b + j(k + m)x y zy z xz x y(Here, k = − bn +1 , m = b + a − n +1 , and j iterates from 1 to n . Also note that thisdefinition of u ( j ) is local to this case).Suppose that F ( u (1) ) = z . But then, voters in profile u (1) with preferences zxy will misreport as xyz with a weight of k , and will misreport as yzx with a weightof m (for a total coalition size of k + m < ε ). This would induce profile u , andimproved result z . This would contradict WSP, meaning that F ( u (1) ) = z . Now suppose for a general j that F ( u ( j ) ) = z . We can show that F ( u ( j +1) ) = z as well. Suppose instead that F ( u ( j +1) ) = z . But then, voters in profile u ( j +1) withpreferences zxy will misreport as xyz with a weight of k , and will misreportas yzx with a weight of m (for a total coalition size of k + m < ε ). This wouldinduce profile u ( j ) , and would achieve improved result z , contradicting WSP. So, F ( u ( j +1) ) = z . This is true for any j .In particular, we have this for: u ( n ) = 1 - 2b - nk b - nm b + n(k + m)x y zy z xz x yNote that: 1 − b − nk = b + kb − nm = 1 − a − b + mb + n ( k + m ) = a − ( k + m )So: u ( n ) = b + k 1 - a - b + m a - (k + m)x y zy z xz x yBut then, voters in profile u with preferences zxy will misreport as xyz witha weight of k , and will misreport as yzx with a weight of m (for a total coalitionsize of k + m < ε ). This would induce profile u ( n ) , and improved result z . Thiswould contradict WSP, meaning that this case also fails.So, in general, we see that is not possible - when y beats z according tothe Borda count, it cannot be true that F ( u . ) = y . Now we assume that F has the following evaluation on u . : F ( u . ) = z ( A u . written out): F a b 1 - a - bx y zy z xz x y = z ( A σ to this profile, where σ ( x ) = z, σ ( y ) = x, σ ( z ) = y :This gives us profile u , and the following evaluation: u = a b 1 - a - bz x yx y zy z x = b 1 - a - b ax y zy z xz x y F ( u ) = y ( A Suppose that 1 − a − b ≥ b . Case 1.I.2.1.1 ≤ a − b < ε Note that voters in profile u . with preferences xyz will want to improve theiroutcome. A weight of a − (1 − a − b ) will misreport as zxy , and a weight of(1 − a − b ) − b will misreport as yzx . This will induce profile u and evaluation( A y . This will have a total coalition size of2 a − b + 1 − a − b = a − b < ε . This would contradict WSP, meaning that Case 1.I.2.1.1 cannot hold.
Case 1.I.2.1.n+1 nε ≤ a − b < ( n + 1) ε Consider the following profile: u ( j ) = a - j(k + m) b + jk 1 - a - b + jmx y zy z xz x y(Here, k = − a − bn +1 ; m = a + b − n +1 . Also note that these definitions of u ( j ) andthe weights k and m are local to this case).First consider F ( u (1) ). Suppose that F ( u (1) ) = z . But then, note that voters inprofile u . with preferences xyz will want to improve their outcome. A weight of k will misreport as yzx , and a weight of m will misreport as zxy . This will induceprofile u (1) , giving an improved outcome over z . This deviation will have a totalcoalition size of k + m = a − bn +1 < ε . This would contradict WSP, so F ( u (1) ) = z .Now, we assume that F ( u ( j ) ) = z for some general j . We want to show that F ( u ( j +1) ) = z . Suppose that F ( u ( j +1) ) = z . But then, in profile u ( j ) , voters with preferences xyz will misrepresent: a weight of k will misreport as yzx , and aweight of m will misreport as zxy . This will induce profile u ( j +1) , and result inthe improved outcome over z . This would contradict WSP, so, in general, we knowthat F ( u ( j ) ) = z for all j .This tells us the evaluation for the following profile: u ( n ) = a - n(k + m) b + nk 1 - a - b + nmx y zy z xz x y F ( u ( n ) ) = z Note that the following weights are equivalent: a − n ( k + m ) = ( n + 1) a − na + nbn + 1 = a + nbn + 1 = b + ( k + m ) b + nk = ( n + 1) b + n − na − nbn + 1 = b − nb + n − nan + 1 = 1 − a − b − k − a − b + nm = a − m So, equivalently for u ( n ) , we have: u ( n ) = b + (k + m) 1 - a - b - k a - mx y zy z xz x yWe recall that F ( u ) = y . Voters in profile u ( n ) will misreport their preferencesas yzx with a weight of k , and a weight of m will misreport as zxy . This willinduce profile u and evaluation ( A y . So, in general,this case cannot hold, and this concludes . Now we suppose that b > − a − b .We apply permutation ν to profile u . (where ν ( x ) = y, ν ( y ) = z, ν ( z ) = x ). ByA, N, and ( A u = 1 - a - b a bx y zy z xz x y F ( u ) = x ( A Case 1.I.2.2.1 ≤ a + b − < ε Voters in profile u . with preferences xyz will misreport: a weight of ( a − b )will misreport as yzx , and a weight of (2 b + a −
1) will misreport as zxy . Theresulting coalition size is 2 a + b − < ε . This will induce profile u and improvedoutcome x , which would contradict WSP. So, Case 1.I.2.2.1 cannot hold.
Case 1.I.2.2.n+1 nε ≤ a + b − < ( n + 1) ε Now, consider the following profile: u ( j ) = a - j(k + m) b + jk 1 - a - b + jmx y zy z xz x y(Here, k = a − bn +1 , m = b + a − n +1 , and j iterates from 1 to n . Note that the definitionof u ( j ) is local to this case).Consider F ( u (1) ). Suppose that F ( u (1) ) = z . But then, voters in profile u . withpreferences xyz will misreport: a weight of k will misreport as yzx , and a weightof m will misreport as zxy . The resulting coalition size is k + m = a + b − n +1 < ε .This would induce profile u (1) , and give an improved outcome over z . This wouldcontradict WSP, so, we know that F ( u (1) ) = z .Suppose for some j that F ( u ( j ) ) = z . We can show that F ( u ( j +1) ) = z . Supposeinstead that F ( u ( j +1) = z . But then, voters in F ( u ( j ) ) with preferences xyz willmisreport: a weight of k will misreport as yzx , and a weight of m will misreportas zxy . Again, the total coalition size is k + m < ε . This would induce profile u ( j +1) and an improved outcome over z for those who deviated. So, this wouldcontradict WSP, meaning that F ( u ( j +1) ) = z .This means that: u ( n ) = a - n(k + m) b + nk 1 - a - b + nmx y zy z xz x y F ( u ( n ) ) = z Note that: a − n ( k + m ) = 1 − a − b + ( k + m ) b + nk = b ( n + 1) + n ( a − b ) n + 1 = b + nan + 1 = ( n + 1) a − ( a − b ) n + 1 = a − k − a − b + nm = n + 1 − na − a − nb − b + 2 nb + na − nn + 1 == 1 − a − b + b + nbn + 1 = ( n + 1) b − (2 b + a − n + 1 = b − m So: u ( n ) = 1 - a - b + (k + m) a - k b - mx y zy z xz x ySo then, voters in profile u ( n ) with preferences xyz will misreport: a weightof k will misreport as yzx , and a weight of m will misreport as zxy . This willinduce profile u , and improved result x . This contradicts WSP, meaning that Case 1.I.2.2.n+1 cannot hold.This concludes , and means that
Case I of Step 1 is impossible.So, we move on to
Case II of Step 1. Step 1, Case II:
Now suppose that by the Borda Count, z beats y .So, we have the following inequalities: a + 1 − b > − a − b > a + 2 ba + b < a > > b Again, we are considering the following profile: u . = a b 1 - a - bx y zy z xz x y Suppose that F has the following evaluation on u . : F ( u . ) = z ( A σ to u . (where σ ( x ) = z, σ ( y ) = x, σ ( z ) = y ), resulting inprofile u from before, and the following evaluation: u = a b 1 - a - bz x yx y zy z x = b 1 - a - b ax y zy z xz x y F ( u ) = y ( A Suppose that a ≥ − a − b .And, note that 1 − a − b > b . Case 1.II.1.1.1 ≤ a − b < ε Voters in profile u . with preferences xyz misreport to improve their outcome,where a weight of 1 − a − b will misreport as yzx , and a weight of 2 a − b will misreport as zxy . This deviation has a coalition size of a − b < ε . This willinduce profile u and evaluation ( A y . Case 1.II.1.1.n+1 nε ≤ a − b < ( n + 1) ε Again, by assumption, we have that: F ( u . ) = z ( A F ( u ) = y ( A u ( j ) = a - (jk + jm) b + jk 1 - a - b + jmx y zy z xz x y(Here, k = − a − bn +1 ; m = a + b − n +1 , and j iterates from 1 to n . Also note that thisdefinition of u ( j ) is local to this case).Suppose that F ( u (1) ) = z . But then, voters in profile u . with preferences xyz would misreport as yzx with a weight of k , and would misreport as zxy witha weight of m (for a total coalition size of k + m < ε ). This would induce profile u (1) , and an improved result over z . And, this would contradict WSP, meaningthat F ( u (1) ) = z .Now, assume that F ( u ( j ) ) = z for some general 1 ≤ j < n . We can show that F ( u ( j +1) ) = z . Suppose instead that F ( u ( j +1) ) = z . But then, note that votersin profile u ( j ) with preferences xyz would misreport as yzx with a weight of k , and would misreport as zxy with a weight of m (for a total coalition size of k + m < ε ). This would induce profile u ( j +1) , and an improved result over z , whichwould contradict WSP. Instead, it must be true that F ( u ( j +1) ) = z .This means that for the following profile: u ( n ) = a - (nk + nm) b + nk 1 - a - b + nmx y zy z xz x y F ( u ( n ) ) = z Then, note that: a − ( nk + nm ) = b + ( k + m )Similarly, it is also true that: b + nk = 1 − a − b − k − a − b + nm = a − m So, profile u ( n ) is exactly equivalent to the following: u ( n ) = b + (k + m) 1 - a - b - k a - mx y zy z xz x y But then, those in profile u ( n ) with preferences xyz will form a coalition ofsize k + m , and misreport as yzx with weight k , and misreport as zxy withweight m . This will induce profile u and evaluation ( A y , contradicting WSP. This means that this case in general fails to satisfythe properties, and we have that cannot hold. Now assume instead that a < − a − b .Consider the following profile: u = a 1 - 2a ax y zy z xz x y Suppose that F ( u ) = x ( A F ( u . ) = z ( A Case 1.II.1.2.1.1 ≤ − a − b < ε Note that voters in profile u with preferences yzx will misreport their pref-erences as zxy with a weight of 1 − a − b < ε . This will induce profile u . , andachieve improved outcome z . This would contradict WSP, so Case 1.II.1.2.1.1 fails.
Case 1.II.1.2.1.n+1 nε ≤ − a − b < ( n + 1) ε Now, consider the following profile: u ( j ) = a 1 - 2a - jk a + jkx y zy z xz x y(Where k = − a − bn +1 , and j iterates from 1 to n . Also, note that this definitionof u ( j ) is local to this case).Suppose that F ( u (1) ) = x . But then, voters in profile u with preferences yzx will misreport their preferences as zxy with a weight of k < ε . This would induceprofile u (1) and an improved result over x , contradicting WSP. So, F ( u (1) ) = x .Now suppose that F ( u ( j ) ) = x for some general j . We can show that F ( u ( j +1) ) = x . Suppose instead that F ( u ( j +1) ) = x . But then, voters in profile u ( j ) with preferences yzx will misreport their preferences as zxy with a weight of k < ε .This would induce profile u ( j +1) and an improved result over x , contradicting WSP.This means that F ( u ( j +1) ) = x for any general j .Note that in particular, we have this for: u ( n ) = a 1 - 2a - nk a + nkx y zy z xz x y = a b + k 1 - a - b - kx y zy z xz x yBut then, voters in profile u ( n ) with preferences yzx will misreport their pref-erences as zxy with a weight of k < ε . This will induce profile u . , and achieveimproved outcome z over x . This contradicts WSP, so in general, Case 1.II.1.2.1 cannot hold.
Now suppose that F ( u ) = y ( A F ( u . ) = z ( A σ to u , where σ ( x ) = z, σ ( y ) = x, σ ( z ) = y . By propertiesA and N, this gives us: u = 1 - 2a a ax y zy z xz x y F ( u ) = x ( A Case 1.II.1.2.2.1 ≤ a − < ε Note that voters in profile u with preferences xyz will misreport their pref-erences as yzx with a coalition size of 3 a − < ε . This would induce profile u ,and would achieve improved result x , contradicting WSP. So, Case 1.II.1.2.2.1 cannot hold.
Case 1.II.1.2.2.n+1 nε ≤ a − < ( n + 1) ε Now, consider the following profile: u ( j ) = 1 - 2a + jk a - jk ax y zy z xz x y (Here, k = a − n +1 , and j iterates from 1 to n . Again, we note that this definitionof u ( j ) is local to this case).Suppose that F ( u (1) ) = x . But then, voters in profile u (1) with preferences xyz will misreport their preferences as yzx with a coalition size of k < ε . This wouldinduce profile u , and would achieve improved result x , contradicting WSP. So, F ( u (1) ) = x .Now, we assume that F ( u ( j ) ) = x for some general j . We can show that F ( u ( j +1) ) = x as well. Suppose instead that F ( u ( j +1) ) = x . But then, notethat voters in profile u ( j +1) with preferences xyz will misreport their preferencesas yzx with a coalition size of k < ε . This misrepresentation would induce profile u ( j ) , and would lead to improved result x . This would contradict WSP, meaningthat F ( u ( j +1) ) = x for any j .In particular, this is true for the following profile: u ( n ) = 1 - 2a + nk a - nk ax y zy z xz x y = a - k 1 - 2a + k ax y zy z xz x yBut then, voters in profile u with preferences xyz will misreport their pref-erences as yzx with a coalition size of k < ε , inducing profile u ( n ) and achievingimproved result x over y . This would contradict WSP. So, also cannothold. Now we suppose that F ( u ) = z ( A , we have: F ( u . ) = z ( A σ to u .This gives us profile u and the following evaluation (from properties A and N): u = 1 - 2a a ax y zy z xz x y F ( u ) = y ( A Case 1.II.1.2.3.1 ≤ a − < ε Note that voters in profile u with preferences xyz will misreport their prefer-ences as yzx with a coalition size of 3 a − < ε . This would induce profile u , andwould achieve improved result y over z contradicting WSP. So, Case 1.II.1.2.3.1 cannot hold.
Case 1.II.1.2.3.n+1 nε ≤ a − < ( n + 1) ε Now, consider the following profile: u ( j ) = a - jk 1 - 2a + jk ax y zy z xz x y(Here, k = a − n +1 , and j iterates from 1 to n . Again, we note that this definitionof u ( j ) is local to this case).Suppose that F ( u (1) ) = z . But then, voters in profile u with preferences xyz will misreport their preferences as yzx with a coalition size of k < ε . This wouldinduce profile u (1) , and would achieve an improved result over z contradicting WSP.So, F ( u (1) ) = z .Now, we suppose for a general j that F ( u ( j ) ) = z . We can then show that F ( u ( j +1) ) = z . Suppose instead that F ( u ( j +1) ) = z . But then, voters in profile u ( j ) with preferences xyz will misreport their preferences as yzx with a coalition sizeof k < ε . This would induce profile u ( j +1) , and would achieve an improved outcomeover z . This would contradict WSP, so, for any general j , F ( u ( j +1) ) = z .In particular, note that this is true for the following profile: u ( n ) = a - nk 1 - 2a + nk ax y zy z xz x y = 1 - 2a + k a - k ax y zy z xz x yBut then, note that voters in profile u ( n ) with preferences xyz will misreporttheir preferences as yzx with a coalition size of k < ε . This will induce profile u and evaluation ( A y over z . This contradicts WSP.So, these cases are exhaustive for , and we conclude that when z beats y according to the Borda count, then F ( u . ) = z . We move on to Now suppose that F has the following evaluation on u . : u . = a b 1 - a - bx y zy z xz x y F ( u . ) = y ( A σ to profile u . and evaluation ( A
17) where σ ( x ) = z, σ ( y ) = x, σ ( z ) = y . This gives us: u = a b 1 - a - bz x yx y zy z x = b 1 - a - b ax y zy z xz x yBy A and N, we have: F ( u ) = x ( A Suppose that a ≥ − a − b Again, note that 1 − a − b > b . Case 1.II.2.1.1 ≤ a − b < ε In this case, voters in profile u . with preferences xyz will misreport as zxy with a weight of a − (1 − a − b ) and will misreport as yzx with a weight of 1 − a − b ,for a total coalition size of a − b < ε . This will induce profile u and improved result x . This would contradict WSP, meaning that this case cannot hold. Case 1.II.2.1.n+1 nε ≤ a − b < ( n + 1) ε Now, consider the following profile: u ( j ) = b + jk + jm 1 - a - b - jk a - jmx y zy z xz x y(Where k = − a − bn +1 ; m = a + b − n +1 , and j iterates from 1 to n . Also, note thatthe definition of u ( j ) is local to this case).Suppose that F ( u (1) ) = x . But then, voters in profile u (1) with preferences xyz will misreport as yzx with a weight of k and will misreport as zxy with a weightof m . This would induce profile u and evaluation ( A x .This would contradict WSP, so F ( u (1) ) = x .Now assume for some general j that F ( u ( j ) ) = x . We can show that F ( u ( j +1) ) = x . Suppose instead that F ( u ( j +1) ) = x . But then, voters in profile u ( j +1) with preferences xyz will misreport as yzx with a weight of k and will misreport as zxy with a weight of m . This would induce profile u ( j ) and an improved outcome x . This would contradict WSP, meaning that for any general j , F ( u ( j +1) ) = x .In particular, we have this for: u ( n ) = b + nk + nm 1 - a - b - nk a - nmx y zy z xz x yNote that: b + n ( k + m ) = a − ( k + m )1 − a − b − nk = b + ka − nm = 1 − a − b + m So: u ( n ) = a - (k + m) b + k 1 - a - b + mx y zy z xz x yBut then, note that voters in profile u . with preferences xyz will misreport as yzx with a weight of k and will misreport as zxy with a weight of m , for atotal coalition size of k + m < ε . This would induce profile u ( n ) and an improvedoutcome x . This would contradict WSP. So, in general, we see that failsto hold. Suppose that a < − a − b .Consider the following profile: u = a 1 - 2a ax y zy z xz x yNote that we are still assuming: u . = a b 1 - a - bx y zy z xz x y F ( u . ) = y ( A Suppose that: F ( u ) = y ( A Case 1.II.2.2.1.1 ≤ − a − b < ε But then, voters in profile u with preferences yzx will misreport as zxy witha coalition size of 1 − a − b . This will induce profile u . and evaluation ( A y . This would contradict WSP, so this case fails to hold. Case 1.II.2.2.1.n+1 nε ≤ − a − b < ( n + 1) ε Consider the following profile: u ( j ) = a 1 - 2a - jk a + jkx y zy z xz x y(Here, k = − a − bn +1 , and j iterates from 1 to n . Again, we note that this definitionof u ( j ) is local to this case).Suppose that F ( u (1) ) = y . But then, voters in profile u with preferences yzx will misreport as zxy with a coalition size of k < ε . This will induce profile u (1) ,for improved outcome y . This would contradict WSP, meaning that F ( u (1) ) = y .Now, we assume that F ( u ( j ) ) = y for some general j . We can show that F ( u ( j +1) ) = y as well. Suppose instead that F ( u ( j +1) ) = y . But then, votersin profile u ( j ) with preferences yzx will misreport as zxy with a coalition sizeof k < ε . This will induce profile u ( j +1) , for improved outcome y . This wouldcontradict WSP, meaning that F ( u ( j +1) ) = y .This is true for the following profile: u ( n ) = a 1 - 2a - nk a + nkx y zy z xz x y = a b + k 1 - a - b - kx y zy z xz x yBut then, note that voters in profile u ( n ) with preferences yzx will misreportas zxy with a coalition size of k < ε . This would induce profile u . and evaluation ( A y . This would contradict WSP, meaning that this casecannot hold. So instead, it must be true that: F ( u ) = y ( A σ to profile u (where σ ( x ) = z, σ ( y ) = x, σ ( z ) = y ), which,by N and A, gives us: u = 1 - 2a a ax y zy z xz x y F ( u ) = x ( A Case 1.II.2.2.2.1 ≤ a − < ε Note that voters in profile u with preferences xyz will misreport as yzx witha coalition weight of 3 a − < ε . This will induce profile u , and achieve improvedoutcome x . This would contradict WSP, meaning that Case 1.II.2.2.2.1 cannothold.
Case 1.II.2.2.2.n+1 nε ≤ a − < ( n + 1) ε Now, consider the following profile: u ( j ) = 1 - 2a + jk a - jk ax y zy z xz x y(Here, k = a − n +1 , and j iterates from 1 to n . Also note that the definition of u ( j ) is local to this case).Suppose that F ( u (1) ) = x . But then, voters in profile u (1) with preferences xyz will misreport as yzx with a coalition weight of k < ε . This will induce profile u , and achieve improved outcome x . This would contradict WSP, meaning that F ( u (1) ) = x .Now, assume that for some general j that F ( u ( j ) ) = x . We can show that F ( u ( j +1) ) = x . Suppose instead that F ( u ( j +1) ) = x . But then, voters in profile u ( j +1) with preferences xyz would misreport as yzx with a coalition weight of k < ε . This would induce profile u ( j ) , and result in improved outcome x , whichwould contradict WSP. So, F ( u ( j +1) ) = x .In particular, this is true for the following profile: u ( n ) = 1 - 2a + nk a - nk ax y zy z xz x y = a - k 1 - 2a + k ax y zy z xz x yBut then, voters in profile u with preferences xyz will misreport as yzx witha coalition weight of k < ε . This will induce profile u ( n ) , giving improved result x ,and would violate WSP.But this means that Case II is impossible. We have shown that there is novoting rule that differs from the Borda Count which satisfies the properties P, A,N, and WSP on the Condorcet Cycle domain. This concludes
Step 1 . Step 2
Now, we want to show that there is no voting rule that satisfies the axioms P,A, N, and WSP on any expansion of the Condorcet cycle domain; by symmetry, itsuffices to show that there is no voting rule on: U ∗ = ( x y z xy z x zz x y y ) Proceed by contradiction. Assume that F is a voting rule that satisfies theproperties. Without loss of generality, we assume > ε > . We know that on the Condorcet domain, the only voting rule that satisfies P,A, N, and WSP is the Borda Count. So, this tells us how F must behave on thefollowing profile: u = (1/3) - d (1/3) + 2d (1/3) - dx y zy z xz x y F ( u ) = y ( E < d < . We define d based on ε : d = ε <
224 = 112For this step, we will consider the following profile: u = (1/3) - d (1/3) + 2d (1/3) - dx y xz z zy x yWe will show that F ( u ) cannot be defined in a way that satisfies the propertiesP, A, N, and WSP. Suppose that the profile u has the following evaluation. F ( u ) = x ( E d and ε , indexing by the interval nε ≤ − d < ( n + 1) ε for some integer n ≥
0. Note that this exists for any choiceof > ε > n ∈ Z ≥ such that: nε ≤ − d < ( n + 1) ε We have set d = ε , so the inequality nε ≤ − d < ( n + 1) ε is equivalent to: nε ≤ − ε < ( n + 1) ε Set n = (cid:4) ε − (cid:5) . Since ε > n will always be finite. Case 2.I.1 ≤ − d < ε Then, voters in profile u with preferences xyz will misreport their preferencesas xzy , and voters in profile u with preferences zxy will also misreport theirpreferences as xzy , inducing u and evaluation ( E y to x . This is a coalition size of 2( − d ) < ε . This would contradict WSP,meaning that the profile of u could not evaluate to x . Case 2.I.2 ε ≤ − d < ε Now, for convenience, denote − d as q . Recall that: F ( u ) = y And, we are supposing that: F ( u ) = x Consider the following profile: u ′ = q (1/3) + 2d qx y xy z zz x yWhat can F ( u ′ ) evaluate to? Suppose that F ( u ′ ) = y . Then, in profile u , thosewith preferences zxy would misreport as xzy with coalition size q < ε , whichwould improve their outcome from y . This would contradict WSP, so we concludethat F ( u ′ ) = y .But then (as an assumption for ), we are assuming that F ( u ) evaluates to x .Given this, in u ′ , people with preferences xyz would misreport their preferencesas xzy (for a coalition of size q ), inducing improved outcome x , and contradictingWSP. In this case, it cannot be true that F ( u ) evaluates to x . Case 2.I.n+1 nε ≤ q < ( n + 1) ε (for n ≥ F ( u ) = y We are supposing that: F ( u ) = x Let r = (cid:6) n +12 (cid:7) . Note that r ≥ n +12 . Consider the set of profiles: u ( j ) = q (1/3) + 2d (r - j)(q/r) j(q/r)x y z xy z x zz x y y(where integer j iterates from 0 to r − j = 0, we have: u (0) = q (1/3) + 2d qx y zy z xz x yThis profile is exactly equivalent to u , meaning that F ( u (0) ) = y .Then, consider u (1) = q (1/3) + 2d (r - 1)(q/r) (q/r)x y z xy z x zz x y yIf F ( u (1) ) = y , then in profile u (0) , those with preferences zxy would form acoalition of size qr (as qr ≤ − d ) n +1 < ε ) and misreport as xzy , creating profile u (1) ,and inducing an outcome that is better than y for them. This would contradict W SP , so F ( u (1) ) = y .Now, assume that for some general 0 < j < r − F ( u ( j ) ) = y . We want to showthat F ( u ( j +1) ) = y .So, we have: u ( j ) = q (1/3) + 2d (r - j)(q/r) j(q/r)x y z xy z x zz x y y u ( j +1) = q (1/3) + 2d (r - j - 1)(q/r) (j + 1)(q/r)x y z xy z x zz x y ySuppose that F ( u ( j +1) ) = y . But then, voters in u ( j ) with preferences zxy would form a coalition of size qr and misreport as xzy , creating profile u ( j +1) , andimproving their outcome. This means that F ( u ( j +1) ) = y . A similar logic applies for all j , so we have that F ( u ( j ) ) = y . In particular, wehave that: F ( u ( r − ) = F q (1/3) + 2d (q/r) (r - 1)(q/r)x y z xy z x zz x y y = y ( E u ( k ) = k(q/r) (1/3) + 2d (2r - k)(q/r)x y xy z zz x y(where integer k iterates from 0 to r ).Note that when k = 0, we have the following profile: u (0) = (1/3) + 2d 2qy xz zx yBy the property A, this is equivalent to the profile u . So, we have that F ( u (0) ) = x . Consider the next profile: u (1) = (q/r) (1/3) + 2d (2r - 1)(q/r)x y xy z zz x ySuppose that F ( u (1) ) = x . But then, those in profile u (1) with preferences xyz would misreport as xzy (coalition size qr ), inducing profile u (0) and improvedoutcome x . So, F ( u (1) ) = x .Again, this logic will apply for all k . Assume for some general k < r that F ( u ( k ) ) = x . So then, we want to show that F ( u ( k +1) ) = x . u ( k ) = k(q/r) (1/3) + 2d (2r - k)(q/r)x y xy z zz x y u ( k +1) = (k + 1)(q/r) (1/3) + 2d (2r - k - 1)(q/r)x y xy z zz x y Suppose that F ( u ( k +1) ) = x . But then, those in profile u ( k +1) with preferences xyz would misreport as xzy (coalition size qr ), inducing profile u ( k ) and improvedoutcome x . So, F ( u ( k +1) ) = x .This means that in particular, we have: F ( u ( r ) ) = F q (1/3) + 2d qx y xy z zz x y = x Note that for the evaluation F ( u ( r − ) in ( E
3) above, those with preferences zxy would misreport their preferences as xzy (with coalition size qr ). This wouldinduce profile u ( r ) and outcome x , contradicting WSP. So, this case also fails.This means that for any ε , for F to satisfy the properties P, A, N, and WSP, itis impossible for F to evaluate as x on profile u . Suppose that the profile u has the following evaluation. F (1/3) - d (1/3) + 2d (1/3) - dx y xz z zy x y = y By the property A , the profile u is equivalent to the following: u ′ = (2/3) - 2d (1/3) + 2dx yz zy xThis means that the following evaluation is also true: F ( u ′ ) = y ( E σ ∗ , where σ ∗ ( x ) = y, σ ∗ ( y ) = x, σ ∗ ( z ) = z . By A and N , we create the following profile and evaluation: u = (2/3) - 2d (1/3) + 2dy xz zx y F ( u ) = x ( E d < . This is still thecase, as we have set d = ε . Therefore, (2 / − d > > (1 /
3) + 2 d .Additionally, there exists m ∈ Z ≥ such that mε ≤ − d < ( m + 1) ε . Because ε < , we can set m = (cid:4) ε − (cid:5) . Since 0 < ε < , then m will always be greaterthan or equal to zero and finite.For convenience, in the following steps, denote − d as D , and + 2 d as 1 − D .Also, note that < D < , and 2 D − − d . Case 2.II.1 : 0 ≤ − d < ε In profile u , voters with preferences yzx would form a coalition of size (2 / − d − ((1 /
3) + 2 d ) = − d = 2 D −
1, and misrepresent their preferences as xzy ,inducing ( E y . This would violate WSP. Forthis case, to satisfy the properties the we want, it cannot be true that F ( u ) = y . Case 2.II.2 : ε ≤ − d < ε (i.e., ε ≤ D − < ε )Again, recall that we are assuming: F ( u ′ ) = y ( E E
4) and the properties A, N, it follows that: F ( u ) = x ( E u = D - k (1 - D) + kx yz zy x(where k = D − , so k < ε . Also, note that in this case, D − k = , and(1 − D ) + k = ).We want to consider what F ( u ) can evaluate to. Suppose that F ( u ) = y .But then, in profile u ′ , voters with preferences xzy would form a coalition of size k < ε , and misreport as yzx , inducing u and an improved outcome over y . Thiswould contradict WSP, so, F ( u ) = y .But then, voters in profile u with preferences yzx would form a coalition ofsize k < ε , and misreport as xzy , inducing profile u and improved outcome y .This would contradict WSP. So, again, for this case, to satisfy the properties thewe want, it cannot be true that F ( u ) = y . Case 2.II.m+1 mε ≤ − d < ( m + 1) ε (for m ≥ F ( u ′ ) = y ( E E
4) and the properties A, N, it follows that: F ( u ) = x ( E u ( j ) = D - j k ′ (1 - D) + j k ′ x yz zy x(where j iterates from 1 to m , and k ′ = D − m +1 . And, note that this definition of u ( j ) is local to Case 2.II.m+1 ).We first consider the evaluation of u (1) . Suppose that F ( u (1) ) = y . But then,in profile u ′ , voters with preferences xzy would form a coalition of size k ′ , andmisreport as yzx , inducing profile u (1) and an improved outcome over y . Thiswould contradict WSP, so, F ( u (1) ) = y .For a general j , assume that F ( u ( j ) ) = y . By a similar logic to before, we canshow that F ( u ( j +1) ) = y . Suppose that F ( u ( j +1) ) = y . Then, in profile u ( j ) , voterswith preferences xzy would form a coalition of size k ′ , and misreport as yzx ,inducing profile u ( j +1) and an improved outcome over y . This would contradictWSP, so, F ( u ( j +1) ) = y , for any j .We have the following profile and evaluation: u ( m ) = D - m k ′ (1 - D) + m k ′ x yz zy x F ( u ( m ) ) = y But then, note that in profile u , voters who have preferences yzx would forma coalition of size k ′ , and misreport as xzy , inducing profile u ( m ) and improvedoutcome y .Clearly, D − (1 − D + mk ′ ) = 2 D − − mk ′ = ( m +1)(2 D − − m (2 D − m +1 = k ′ < ε .This would contradict WSP. So, for a general − d , then F will not satisfy theproperties. Thus, it is impossible for F to evaluate as y on u . Finally, suppose that the profile u has the following evaluation. F (1/3) - d (1/3) + 2d (1/3) - dx y xz z zy x y = z ( E u ′ = (2/3) - 2d (1/3) + 2dx yz zy x F ( u ′ ) = z Consider the profile: u = (2/3) - 2d (1/3) + 2dx zz xy yWe can show that it must be true that: F ( u ) = z ( E (set up) - showing that F ( u ) = z We will use an iterative approach to show that F ( u ) must equal z for any d and ε . The cases will be based on the following inequality: hε ≤
13 + 2 d < ( h + 1) ε Note that there exists some integer h ≥ + 2 d fulfills theinequality. We have set d = ε , so we can let h = (cid:4) ε + (cid:5) . Case 2.III.0.1 : 0 ≤ + 2 d < ε So, suppose that F ( u ) = z . Then, in profile u , voters with preferences zxy would form a coalition of size + 2 d and misreport as yzx , inducing profile u ′ and improved outcome z . This would contradict WSP, so F ( u ) = z . Case 2.III.0.2 : ε ≤ + 2 d < ε Again, we proceed by contradiction, and suppose that F ( u ) = z . Consider thefollowing profile: u ∗ = (2/3) - 2d c (1/3) + 2d - cx z yz x zy y x(Here, we let c = +2 d )So, what can F ( u ∗ ) evaluate to? Suppose that F ( u ∗ ) = z . But then, votersin u ∗ with preferences zxy would form a coalition of size c < ε , and misreporttheir preferences as yzx , inducing profile u ′ and improved outcome z . This wouldcontradict WSP, so F ( u ∗ ) = z . Suppose that F ( u ) = z . But then, voters in u with preferences zxy wouldform a coalition of size c < ε , and misreport their preferences as yzx , inducingprofile u ∗ and improved outcome z . This would contradict WSP, so F ( u ) = z . Case 2.III.0.h+1 : hε ≤ + 2 d < ( h + 1) ε (for h ≥ u ( j )5 = (2/3) - 2d j c ′ (1/3) + 2d - j c ′ x z yz x zy y x(Here, c ′ = +2 dh +1 , and j iterates from 0 to h .)When j = 0, then this profile is just equivalent to u ′ . So, we know that F ( u (0)5 ) = z . Then, we want to know what the evaluation F ( u (1)5 ) is. Suppose that F ( u (1)5 ) = z . But then, voters in u (1)5 with preferences zxy would form a coalition of size c ′ < ε , and misreport their preferences as yzx , inducing profile u (0)5 and improvedoutcome z . This would contradict WSP, so F ( u (1)5 ) = z . A similar logic appliesfor all j , so this means that F ( u ( h )5 ) = z . Assume that F ( u ( j )5 ) = z for somegeneral j . Then, we want to know what the evaluation F ( u ( j +1)5 ) is. Suppose that F ( u ( j +1)5 ) = z . But then, voters in u ( j +1)5 with preferences zxy would form acoalition of size c ′ < ε , and misreport their preferences as yzx , inducing profile u ( j )5 and improved outcome z . This would contradict WSP, so F ( u ( j +1)5 ) = z .But then this means that F ( u ) must equal z : otherwise, if F ( u ) = z , thenvoters in profile u with preferences zxy would deviate with a coalition size of c ′ , and misreport their preferences as yzx , inducing profile u ( h )5 and outcome z .Therefore, F ( u ) = z .We have that F ( u ) = z , and we see that is true for a general ε . This concludes Step 2.III.0 .Apply the following permutation: σ ∗∗ ( x ) = z, σ ∗∗ ( y ) = y , σ ∗∗ ( z ) = x to u .With N and A, this gives us the following profile and evaluation: u = (2/3) - 2d (1/3) + 2dz xx zy y F ( u ) = x ( E , denote D = − d , and set m as it was defined before. Case 2.III.1 ≤ D − < ε In evaluation ( E
7) of profile u , voters with preferences xzy would form acoalition of size 2 D −
1, and misrepresent their preferences as zxy , inducing profile u and evaluation ( E x . This would violateWSP, meaning that for this case, F ( u ) = z . Case 2.III.2 ε ≤ D − < ε Again, recall that we are assuming: F ( u ′ ) = z We also have: F ( u ) = z ( E F ( u ) = x ( E u = D - k (1 - D) + kx zz xy y(where k = D − , so k < ε . Again, note that in this case, D − k = , and(1 − D ) + k = ).We want to consider the evaluation F ( u ). Suppose that F ( u ) = x (also notethat y is Pareto dominated on this profile, and cannot be the result). But then,in profile u , voters with preferences xzy would form a coalition of size k , andmisreport as zxy , inducing u and an improved outcome over z . This wouldcontradict WSP, so, F ( u ) = z . But then, voters in profile u with preferences zxy would form a coalition ofsize k , and misreport as xzy , inducing profile u and improved outcome z . Thiswould contradict WSP. This means that F ( u ) cannot equal z . But, this violates theinitial hypothesis that F ( u ) = z . So, again, for this case, to satisfy the propertiesthat we want, it cannot be true that F ( u ) = z . Case 2.III.m+1 mε ≤ D − < ( m + 1) ε (for m ≥ F ( u ′ ) = z And, we have that: F ( u ) = z ( E F ( u ) = x ( E u ( j ) = D - j k ′ (1 - D) + j k ′ x zz xy y(where j iterates from 1 to m , and k ′ = D − m +1 . Also note that this definition of u ( j ) is local to Case 2.III.m+1 .)First consider the evaluation when we have u (1) . Suppose that F ( u (1) ) = x . Butthen, in ( E u with preferences xzy would form a coalition ofsize k ′ , and misreport as zxy , inducing profile u (1) and an improved outcome x .This would contradict WSP, so, F ( u (1) ) = z .For a general j , assume that F ( u ( j ) ) = z . By a similar logic to before, wecan show that F ( u ( j +1) ) = z . Suppose that F ( u ( j +1) ) = x . So then in profile u ( j ) , voters with preferences xzy would form a coalition of size k ′ , and misreportas zxy , inducing profile u ( j +1) and an improved outcome x over z . This wouldcontradict WSP, so, F ( u ( j +1) ) = z , for any j .This gives us the following profile and evaluation: u ( m ) = D - m k ′ (1 - D) + m k ′ x zz xy y F ( u ( m ) ) = z But then, note that in profile u , the voters with preferences zxy would forma coalition of size k ′ , and misreport as xzy , inducing profile u ( m ) and improvedoutcome z . Again, the coalition size is less than ε , as in .This would contradict WSP. This in turn contradicts the original hypothesis that F ( u ) = z .This concludes Step 2 . We see that for a general ε , knowing that F ( u ) = y ,then there is nothing that F can evaluate to on u . This means that there is novoting rule F that satisfies the properties P, A, N, and WSP on an expansion ofthe Condorcet domain. Step 3
We first consider the following rich domain: U I = ( x y y zy z x yz x z x ) Consider the voting rule F satisfying P, A, N, and WSP on U I . We want toshow that there is no voting rule F = F C that satisfies these axioms. Fix ε > u I. = a b c 1 - a - b - cx y y zy z x yz x z x(Note that the profiles and weights here have no relation to the weights andprofiles that were named in previous steps). Suppose that the result of Condorcet F C ( u I. ) is x . So, a > . Suppose that: F ( u I. ) = y ( I u I = a 1 - ax yy xz z We want to show that: F ( u I ) = y Assume that a is fixed, and the conditions for the following cases are based on c (and implicitly on b ). Case 3.I.1.1.0.1: ≤ − a − c < ε If F ( u I ) was not y , then those in profile u I with preferences yxz would forma coalition of size 1 − a − c , where a weight of b would misrepresent as yzx , anda weight of 1 − a − b − c would misrepresent as zyx , inducing profile u I. andevaluation ( I y . This would contradict WSP. So,when 1 − a − c < ε , F ( u I ) = y . Case 3.I.1.1.0.2: ε ≤ − a − c < ε Again, by assumption for this step, we are supposing that: F ( u I. ) = y ( I Consider the following profile: u I = a k 1 - a - k - m mx y y zy z x yz x z x(Here, k = b ; m = − a − b − c ).Suppose that F ( u I ) = y .Then, people with preferences yxz in u I would want to induce initial profile u I. . So, a group of size b − k would misrepresent their preferences as yzx , and agroup of size 1 − a − b − c − m would misrepresent their preferences as zyx . Thiswould create profile u I. , and improved outcome y . Note that the coalition size isless than ε : b − k + 1 − a − b − c − m = 1 − a − c < ε So, to satisfy WSP, this tells us that F ( u I ) = y .Consider renaming the weights on u I in the following manner.Where: a ∗ = a ; b ∗ = k ; c ∗ = 1 − a − k − m ; 1 − a ∗ − b ∗ − c ∗ = m.u I = a ∗ b ∗ c ∗ − a ∗ − b ∗ − c ∗ x y y zy z x yz x z xThis is still the same profile but with different labels on the weights. We seethat b ∗ + 1 − a ∗ − b ∗ − c ∗ = k + m < ε . So, knowing the F ( u I ) = y , we see that u I is just an instance of Case 3.I.1.1.0.1 . This tells us that F ( u I ) = y .Now, for the inductive hypothesis, we assume that all cases up to Case 3.I.1.1.0.n hold. In other words, given some profile: u ∗ = a b ∗ c ∗ − a − b ∗ − c ∗ x y y zy z x yz x z xwhere the following conditions are true: F ( u ∗ ) = y ( n − ε ≤ − a − c ∗ < nε ( W F ( u I ) = y . Now, we consider the following case:
Case 3.I.1.1.0.n+1: nε ≤ − a − c < ( n + 1) ε We are still supposing that: F ( u I. ) = y ( I u I ( n ) = a k ′ ′ - m ′ m ′ x y y zy z x yz x z x(Here, k ′ = nbn +1 , m ′ = n (1 − a − b − c ) n +1 ).We consider the result F ( u I ( n ) ). Suppose that F ( u I ( n ) ) = y . But then, voterswith preferences yxz in u I ( n ) would want to induce initial profile u I. . A weight of b − k ′ would misreport as yzx and a weight of 1 − a − b − c − m ′ would misreporttheir preferences as zyx . This would create profile u I. and improved result y ,contradicting WSP. So, this tells us that F ( u I ( n ) ) = y .But then, note that the weights on profile u I ( n ) fulfill the following: F ( u I ( n ) ) = y ( n − ε ≤ k ′ + m ′ = n (1 − a − c ) n + 1 < nε So, this we see that profile u I ( n ) is a case of a profile u ∗ that satisfies a case of , where b ∗ = k ′ , c ∗ = 1 − a − k ′ − m ′ , − a − b ∗ − c ∗ = m ′ . By theinductive hypothesis, this tells us that: F ( u I ) = y This is true in general. This tells us that for any a > , and for any c thatsatisfies 1 − a − c >
0, if the winner of F on u I. is y , then it also follows that: F a 1 - ax yy xz z = y ( I Now, we will show that a voting rule F = F C that satisfies the axioms,where F ( u I ) = y , is impossible.Apply permutation σ ∗ to profile u I , where σ ∗ ( x ) = y, σ ∗ ( y ) = x , and σ ∗ ( z ) = z .This gives us the following profile and evaluation: u I = a 1 - ay xx yz z F ( u I ) = x ( I Case 3.I.1.1.1.1 ≤ | a − | < ε If a − (1 − a ) < ε , then voters in profile u I with preferences yxz would form acoalition of size 2 a −
1, and misreport their preferences as xyz . This would createprofile u I and evaluation ( I y .This would contradict WSP, meaning that on this range of a and b , our originalhypothesis that F = F C cannot be true. So, this tells us that when a ∗ fulfills thefollowing inequality, then F must agree with F C :12 < a ∗ <
12 + ε W F a ∗ − a ∗ x yy xz z = x Case 3.I.1.1.1.2 ε ≤ | a − | < ε (i.e., ε + < a ≤ ε + )Note that a = + ε − δ for some 0 < δ ≤ ε .Again, we have: F ( u I ) = y ( I u I = a - k 1 - a + kx yy xz zWhere: k = a − ε − δ < ε Note that a − k = + ε − δ < + ε Additionally, note that: δ < ε ε − δ > The weight on preferences xyz in profile u I also satisfies the following inequal-ity: a − k = 12 + ε − δ ε ε − δ >
12 + ε < a − k <
12 + ε F = F C from Case 3.I.1.1.1.1 . So, F ( u I ) = x .But then, note that voters in profile u I with preferences xyz would misreportas yxz to induce profile u I and improved result x . This again would contradictWSP, meaning that on this range of ( a, b ), we have that F = F C .A similar logic will hold for all a > . So, in general, we see that our assumptionfor the evaluation F ( u I. ) in fails to hold, and we move on to . Now we suppose that: F ( u I. ) = z ( I u I. = a b c 1 - a - b - cx y y zy z x yz x z xConsider the following profile: u I = a 1 - ax zy yz x Now we want to show that: F ( u I ) = z Assume that a is fixed, and the conditions for the following cases are based on b and c . Case 3.I.1.2.0.1: ≤ b + c < ε We consider the evaluation of F ( u I ). By supposition, we have: F ( u I. ) = z ( I Suppose that F ( u I ) = z . But then, voters in profile u I with preferences zyx would want to induce outcome z and evaluation ( I yzx with weight b , and misreport yxz with weight c , forming profile u I. , and result z . So, this would improve their result, contradicting WSP. For thiscase, we have that F ( u I ) = z . Case 3.I.1.2.0.2: ε ≤ b + c < ε We are still assuming that: F ( u I. ) = z ( I u I = a k m 1 - a - k - mx y y zy z x yz x z xLet k = b , and m = c . Note that the definitions of k and m are local to thiscase.Suppose that F ( u I ) = z . But then, voters in profile ( u I ) with preferences zyx would want to induce evaluation ( I
4) and outcome z . They would deviate, andmisreport yzx with weight b − k , and misreport yxz with weight c − m . Thiswould create profile u I. and evaluation ( I z . So, this wouldimprove their result, contradicting WSP. This tells us that F ( u I ) = z .But then, we consider the following relabeling of the weights in profile u I : a = a ; b ′ = k ; c ′ = m ; 1 − a ′ − b ′ − c ′ = 1 − a − k − m Note that b ′ + c ′ < ε . We have shown that it must be true that F ( u I ) = z . So,profile u I is just a case of , meaning that: F ( u I ) = z For the inductive hypothesis, we assume that all cases up to and including
Case3.I.1.2.0.n hold. So, in other words, given some profile: u ∗ = a b ∗ c ∗ − a − b ∗ − c ∗ x y y zy z x yz x z xwhere: F ( u ∗ ) = z and the following inequality holds: ( n − ε ≤ b ∗ + c ∗ < nε ( W F ( u I ) = z Case 3.I.1.2.0.n+1 nε ≤ b + c < ( n + 1) ε Again we have: F ( u I. ) = z ( I F ( u I ) = z Consider the following profile: u I ( n ) = a k m 1 - a - k - mx y y zy z x yz x z xHere, we let k = nbn +1 , and m = ncn +1 . Also note that this definition of u I ( n ) islocal to this case.Suppose that F ( u I ( n ) ) = z . But then, voters with preferences zyx would wantto induce profile u I. and outcome z .Voters in profile u I ( n ) with preferences zyx would deviate, and misreport yzx with weight b − k , and misreport yxz with weight c − m . This would be a coalitionof size b + cn +1 < ε . So, this would create profile u I. and evaluation ( I z . This would result in an improved outcome for the zyx coalition, andwould contradict WSP. So, F ( u I ( n ) ) = z .Then, note that the weights in profile u I ( n ) fulfill inequality ( W n − ε ≤ n ( b + c ) n + 1 < nε So, profile u I ( n ) reduces to a profile u ∗ that satisfies Case 3.I.1.2.0.n (where b ∗ = k and c ∗ = m ). This implies that F ( u I ) = z . This is true in general. F ( u I ) = z ( I Now, we will show that a voting rule F = F C that satisfies the axioms,where F ( u I ) = z , is impossible.Apply permutation σ ∗∗ to u I , where σ ∗∗ ( x ) = z, σ ∗∗ ( z ) = x, σ ∗∗ ( y ) = y . u I = a 1 - ax zy yz x F ( u I ) = z ( I u I = a 1 - az xy yx z = 1 - a ax zy yz xBy A and N, we have: F ( u I ) = x ( I Case 3.I.1.2.1.1 ≤ | a − | < ε If a − (1 − a ) < ε , then voters in profile u I with preferences xyz would form acoalition of size 2 a −
1, and misreport their preferences as zyx . This would createprofile u I and evaluation ( I x . This would contradictWSP, meaning that Case 3.I.1.2.1.1 fails to hold.
Case 3.I.1.2.1.n+1 nε ≤ | a − | < ( n + 1) ε Again, we have: F ( u I ) = z ( I F ( u I ) = x ( I u I ( j ) = a - jk 1 - a + jkx zy yz xHere, j iterates from 1 to n , and k = a − n +1 . Note that the definition of u I ( j ) islocal to this case.We first consider the evaluation F ( u I (1) ). Suppose that F ( u I (1) ) = z . But then,voters in profile u I with preferences xyz would form a coalition of size k , andmisreport their preferences as zyx . This would induce profile u I (1) and an improvedresult over z , contradicting WSP. So, we have that F ( u I (1) ) = z . Now, assume for a general j that F ( u I ( j ) ) = z . We can show that F ( u I ( j +1) ) = z .Suppose instead that F ( u I ( j +1) ) = z . But then, voters in profile u I ( j ) with preferences xyz would form a coalition of size k , and misreport their preferences as zyx . Thiswould induce profile u I ( j +1) and an improved result over z , contradicting WSP. So,in general, F ( u I ( j +1) ) = z . In particular, this is true for the following profile: u I ( n ) = a - nk 1 - a + nkx zy yz x = 1 - a + k a - kx zy yz xBut then, note that voters in profile u I ( n ) with preferences xyz would form acoalition of size k , and misreport their preferences as zyx . This would induceprofile u I , and improved result x over z , contradicting WSP. This case fails to hold.This concludes and . We next consider . Suppose that the result of Condorcet F C ( u I. ) is y. So, a < , and a + b + c > . Suppose that: F ( u I. ) = x ( I u I = a b + c 1 - a - b - cx y zy x yz z x Suppose that F ( u I ) = y ( I Case 3.I.2.1.1.1 ≤ b < ε But then, note that voters in profile u I. with preferences yzx will misreport as yxz with a coalition weight of b < ε . This would induce profile u I and improvedoutcome y , violating WSP. So, this fails to hold. Case 3.I.2.1.1.n+1 nε ≤ b < ( n + 1) ε We are assuming that: F ( u I ) = y ( I u I ( j ) = a jk b + c - jk 1 - a - b - cx y y zy z x yz x z xHere, j iterates from 1 to n , and k = bn +1 . Also note that this definition of u I ( j ) is local to this case.Suppose that F ( u I (1) ) = y . But then, voters in profile u I (1) with preferences yzx will misreport as yxz with a coalition weight of k < ε . This would induce profile u I and improved outcome y , violating WSP. So, F ( u I (1) ) = y .Now, assume for a general j that F ( u I ( j ) ) = y . We can show that F ( u I ( j +1) ) = y . Suppose instead that F ( u I ( j +1) ) = y . But then, voters in profile u I ( j +1) withpreferences yzx will misreport as yxz with a coalition weight of k < ε . Thiswould induce profile u I ( j ) and improved outcome y , violating WSP. So, F ( u I ( j +1) ) = y . In particular, this is true for: u I ( n ) = a nk b + c - nk 1 - a - b - cx y y zy z x yz x z x = a b - k b + c + k 1 - a - b - cx y y zy z x yz x z xSo, voters in profile u I. with preferences yzx will misreport as yxz with acoalition weight of k < ε . This would induce profile u I ( n ) and improved outcome y ,violating WSP. So, this case fails to hold. Next, suppose that F ( u I ) = z ( I Case 3.I.2.1.2.1 ≤ b < ε But then, voters in profile u I with preferences yxz will misreport as yzx with a weight of b < ε . This will induce profile u I. and evaluation ( I x over z . This would contradict WSP, so this case fails to hold. Case 3.I.2.1.2.n+1 nε ≤ b < ( n + 1) ε By assumption for this step, we have: F ( u I. ) = x ( I F ( u I ) = z ( I Consider the following profile: u I ( j ) = a jk b + c - jk 1 - a - b - cx y y zy z x yz x z xHere, j iterates from 1 to n , and k = bn +1 . Note that the definition of u I ( j ) islocal to this case.Suppose that F ( u I (1) ) = z . But then, note that voters in profile u I with prefer-ences yxz will misreport as yzx with a weight of k < ε . This will induce profile u I (1) and an improved result over z , contradicting WSP. So, F ( u I (1) ) = z .Now, assume for a general j that F ( u I ( j ) ) = z . We can likewise show that F ( u I ( j +1) ) = z . Suppose instead that F ( u I ( j +1) ) = z . But then, voters in profile u I ( j ) with preferences yxz will misreport as yzx with a weight of k < ε . Thiswill induce profile u I ( j +1) and an improved result over z , contradicting WSP. So, ingeneral, F ( u I ( j +1) ) = z .In particular, this is true for the following profile: u I ( n ) = a nk b + c - nk 1 - a - b - cx y y zy z x yz x z x = a b - k c + k 1 - a - b - cx y y zy z x yz x z xBut then, note that voters in profile u I ( n ) with preferences yxz will misreportas yzx with a weight of k < ε . This will induce profile u I. and evaluation ( I x over z , contradicting WSP. So, this case fails to hold. So, instead, it must be the case that F ( u I ) = x ( I u I = a 1 - ax yy xz zNote that z is Pareto dominated here, so cannot be the result of the evaluation F ( u I ). Suppose that F ( u I ) = y ( I Case 3.I.2.1.3.1.1 ≤ − a − b − c < ε In this case, voters in profile u I with preferences zyx will misreport as yxz with a weight of 1 − a − b − c < ε . This would induce profile u I and improved result y over x , contradicting WSP. So, Case 3.I.2.1.3.1.1 fails to hold.
Case 3.I.2.1.3.1.n+1 nε ≤ − a − b − c < ( n + 1) ε Again, for , we have that: F ( u I ) = x ( I F ( u I ) = y ( I u I ( j ) = a b + c +jk 1 - a - b - c - jkx y zy x yz z xHere, j iterates from 1 to n , and k = − a − b − cn +1 . Note that this definition of u I ( j ) is local to this case.We first consider u I (1) . Suppose that F ( u I (1) ) = x . But then, voters in profile u I with preferences zyx will misreport as yxz with a weight of k < ε . This wouldinduce profile u I (1) and an improvement over x , contradicting WSP. So, F ( u I (1) ) = x .Now, assume that for some j that F ( u I ( j ) ) = x . We can show that F ( u I ( j +1) ) = x .Suppose instead that F ( u I ( j +1) ) = x . But then, note that voters in profile u I ( j ) with preferences zyx will misreport as yxz with a weight of k < ε . This wouldinduce profile u I ( j +1) and an improvement over x , contradicting WSP. So, in general, F ( u I ( j +1) ) = x .This is true in particular for the following profile: u I ( n ) = a b + c +nk 1 - a - b - c - nkx y zy x yz z x = a 1 - a - k kx y zy x yz z xBut then, note that voters in profile u I ( n ) with preferences zyx will misreportas yxz with a weight of k < ε . This will induce profile u I , achieving improvedoutcome y over x , contradicting WSP. So, fails to hold. Instead, it must be true that F ( u I ) = x ( I u I = a 1 - ax yy xz zBy assumption for , the Condorcet winner F C ( u I. ) = y , implying that a < , and 1 − a > .Consider the following relabeling of the weights on profile u I , and apply permu-tation σ ∗ , where σ ∗ ( x ) = y , σ ∗ ( y ) = x , and σ ∗ ( z ) = z . u I = A 1 - Ay xx yz z F ( u I ) = x ( I A = 1 − a > ). u I = A 1 - Ax yy xz zBy N and A: F ( u I ) = y ( I . So, this case fails to hold. Suppose that F ( u I. ) = z . But, note the symmetry between x and z ,meaning that the impossibility of also implies the impossibility of . Suppose that the result of Condorcet F C ( u I. ) is z . But, we note that x and z are symmetric in this domain, meaning that this case has been ruled out byconsideration of . So, we move on to the next domain. STEP 3, Domain II (3.II)
Consider the next following rich domain: U II = ( x y yy z xz x z ) Now we assume that F is some voting rule that satisfies P, A, N, and WSP on U II . We want to show that F must equal F C . So, we proceed by contradiction.Fix ε >
0, and fix a .We consider the following generic profile, where F ( u II. ) = F C ( u II. ). u II. = a b 1 - a - bx y yy z xz x zWe see that z is Pareto-dominated by y - so, the result of voting rules F and F C on u I. cannot be z . Suppose that the Condorcet winner F C ( u II. ) = x .This means that a > by definition of the Condorcet method. As we areassuming that F = F C , we have that: F ( u II. ) = y ( I u II = a 1 - ax yy xz z We want to show that for a general b , F ( u II ) = y . Case 3.II.1.0.1 ≤ b < ε Suppose that F ( u II ) = x . But then, voters in profile u II with preferences yxz would misreport as yzx with a coalition size of b < ε . This would induce profile u II. and evaluation ( I y , contradicting WSP.So, for this case, it must be true that F ( u II ) = x . Case 3.II.1.0.2 ε ≤ b < ε By assumption for , we have that: F ( u II. ) = y ( I u II = a b - k 1 - a - b + kx y yy z xz x zHere, k = b .For F to satisfy WSP, then F ( u II ) = y . Suppose instead that F ( u II ) = x . Butthen, voters in profile u II with preferences yxz would form a coalition of size k < ε , and misreport their preferences as yzx . This would induce profile u II. and evaluation ( I y , contradicting WSP. So, we knowthat F ( u II ) = y .Also note that: u II = a b - k 1 - a - b + kx y yy z xz x z = a k 1 - a - kx y yy z xz x zThis will imply that F ( u II ) = y . Suppose instead that F ( u II ) = x . But then,voters in profile u II with preferences yxz would form a coalition of size k < ε , andmisreport their preferences as yzx . This would induce profile u II and improvedresult y , contradicting WSP. So, we have for this case that F ( u II ) = x .We can also see that F ( u II ) = y by applying . Note that k < ε . So,profile u II is just a case of the profiles that we considered in Case 3.II.1.0.1 toshow that F ( u II ) = y . Because we have shown that F ( u II ) = y , then this in turnimplies that F ( u II ) = y .Now, for the inductive hypothesis, we assume that all cases up to Case 3.II.1.0.n hold. In other words, given some profile: u ∗ = a b ∗ − a − b ∗ x y yy z xz x zwhere the following conditions are true: F ( u ∗ ) = y ( n − ε ≤ b ∗ < nε ( W F ( u II ) = y .Now, we consider the following case: Case 3.II.1.0.n+1 nε ≤ b < ( n + 1) ε Again, by assumption for , we have that: F ( u II. ) = y ( I u IIk = a b - k 1 - a - b + kx y yy z xz x z(Here, k = bn +1 . Note that this definition of k is local to this step). Suppose that F ( u IIk ) = x . But then, voters in profile u IIk with preferences yxz would form a coalition of size k < ε , and misreport their preferences as yzx . Thiswould induce profile u II. and evaluation ( I y .This would contradict WSP, meaning that F ( u IIk ) = y .But then, consider the weights in profile u IIk . Note that: b − k = nbn + 1And: ( n − ε ≤ nbn + 1 < nε These weights fulfill inequality ( W F ( u IIk ) = y , by the inductivehypothesis, we can conclude that in this case, F ( u II ) = y , as we wanted. Thus, ingeneral, for F to satisfy WSP, it must be true that: F ( u II ) = y ( I Now, we will show that evaluation ( I
15) results in a violation of WSP.Apply permutation σ ∗ to profile u II , where σ ∗ ( x ) = y, σ ∗ ( y ) = x , and σ ∗ ( z ) = z .This gives us the following profile and evaluation: u II = a 1 - ay xx yz z F ( u II ) = x ( I Case 3.II.1.1.1 ≤ | a − | < ε If a − (1 − a ) < ε , then voters in profile u II with preferences yxz would form acoalition of size 2 a −
1, and misreport their preferences as xyz . This would createprofile u II and evaluation ( I y .This would contradict WSP, meaning that on this range of a and b , our originalhypothesis that F = F C cannot be true. So, this tells us that when a ∗ fulfills theinequality, F agrees with F C :12 < a ∗ <
12 + ε W F a ∗ − a ∗ x yy xz z = x Case 3.II.1.1.2 ε ≤ | a − | < ε (i.e., ε + < a ≤ ε + )Note that a = + ε − δ for some 0 < δ ≤ ε .Again, we have: F ( u II ) = y ( I u II = a - k 1 - a + kx yy xz zWhere: k = a − ε − δ < ε Note that a − k = + ε − δ < + ε Additionally, note that: δ < ε ε − δ > xyz in profile u II : a − k = 12 + ε − δ ε ε − δ >
12 + ε < a − k <
12 + ε W
7) where F = F C from Case 3.II.1.1.1 .So, F ( u II ) = x .But then, note that voters in profile u II with preferences xyz would misreportas yxz with a weight of k to induce profile u II and improved result x . This wouldcontradict WSP, meaning that on this range of ( a, b ), F = F C .In other words, for a profile, where:12 < a ∗ <
12 + ε We can say that: F a ∗ − a ∗ x zy yz x = x A similar logic will apply for all a > . So, we see that for , there is novoting rule F = F C that satisfies the properties that we want. Now suppose that the Condorcet winner F C ( u II. ) = y .This means that a < .We are assuming that F = F C , so, we have that: F ( u II. ) = x ( I u II = a 1 - ax yy xz zWe want to show that F ( u II ) = x . Case 3.II.2.1 ≤ b < ε Assume that the following is true, and proceed by contradiction. F a 1 - ax yy xz z = F ( u II ) = y Then, those with true preference ranking yzx in profile u II. would form a coali-tion of size b < ε and misreport their true preferences as yxz , which would formprofile u II , inducing result y , meaning that this would be a profitable deviation.This contradicts WSP, and tells us for the case where b < ε , then F ( u II ) = x . Case 3.II.2.2 ε ≤ b < ε Consider the profile: u II = a m 1 - a - mx y yy z xz x z(where m = b ).Then, F ( u II ) must equal x , because otherwise, from profile u II. , a coalition ofsize b − m would deviate from yzx and misreport as yxz to form profile u II , andachieve result y . This would contradict WSP (because the coalition size b − m < ε ),so F ( u II ) = x . Also note that m < ε .Then, we consider F ( u II ). If F ( u II ) = y , then all voters in u II with preferences yzx would misreport as yxz to form profile u II . So, this would contradict WSP,meaning that F ( u II ) = x . A similar logic applies for all subsequent b values. So, in general, for any a < it must be true that: F a 1 - ax yy xz z = x If we rename 1 − a and call it A , where A > , then we have: F A 1 - Ay xx yz z = x Then, we can apply the permutation σ ∗ , where σ ∗ ( x ) = y, σ ∗ ( y ) = x, σ ∗ ( z ) = z .By N and A, we have: F A 1 - Ax yy xz z = y This is the same pair of profiles that we had in , so the same resultsfollow, leading to a contradiction that tells us that F = F C . So, this shows thatfor Domain II , there is no voting rule F other than the Condorcet method thatsatisfies the axioms on this domain. STEP 3, Domain III
We will now consider the final domain: U III = ( x x y yy z z xz y x z ) Now we assume that F is some voting rule that satisfies the P, A, N, and WSP,where F = F C .So, F must differ from F C for some profile u III. : u III. = a b c 1 - a - b - cx x y yy z z xz y x zNote that the Condorcet winner on profile u III. can never be z . Also note that x and y are symmetric in the profile. So, we assume without loss of generality thatthe Condorcet winner of u III. is x . This means that: a + b > − a − b Suppose that F ( u III. ) = y ( I u III = a + b c 1 - a - b - cx y yy z xz x z Suppose that F ( u III ) = x Case 3.III.1.1.1 ≤ b < ε In this case, voters in profile u III. with preferences xzy will misreport as xyz with a coalition size of b , inducing profile u III and improved result x . This wouldcontradict WSP, so Case 3.III.1.1.1 fails to hold.
Case 3.III.1.1.n+1 nε ≤ b < ( n + 1) ε Consider the following profile: u III ( j ) = a + b - jk jk c 1 - a - b - cx x y yy z z xz y x zHere, k = bn +1 , and j iterates from 1 to n . Note that the definition of u III ( j ) islocal to this case.We first consider F ( u III (1) ). Suppose that F ( u III (1) ) = x . But then, voters in profile u III (1) with preferences xzy will misreport as xyz with a coalition size of k < ε ,inducing profile u III and improved result x . This would contradict WSP, so weknow that F ( u III (1) ) = x .Now, assume for a general j that F ( u III ( j ) ) = x . We can show that it followsthat F ( u III ( j +1) ) = x . Suppose instead that F ( u III ( j +1) ) = x . But then, voters inprofile u III ( j +1) with preferences xzy will misreport as xyz with a coalition size of k < ε , inducing profile u III ( j ) , and improved outcome x . This would contradict WSP,meaning that F ( u III ( j +1) ) = x . This is true for any general j . In particular, this istrue for: u III ( n ) = a + b - nk nk c 1 - a - b - cx x y yy z z xz y x z = a + k b - k c 1 - a - b - cx x y yy z z xz y x z But now, note that voters in profile u III. with preferences xzy will misreportas xyz with a coalition size of k , inducing profile u III ( n ) and improved outcome x over y . So, fails to hold. Now suppose that F ( u III ) = z Case 3.III.1.2.1
In this case, some voters in profile u III with preferences xyz will misreport as xzy with a coalition size of b , inducing profile u III. and evaluation ( I y over z . This would contradict WSP, so Case 3.III.1.2.1 fails tohold.
Case 3.III.1.2.n+1 nε ≤ b < ( n + 1) ε Consider the following profile: u III ( j ) = a + b - jk jk c 1 - a - b - cx x y yy z z xz y x zHere, k = bn +1 , and j iterates from 1 to n . Note that the definition of u III ( j ) islocal to this case.We first consider F ( u III (1) ). Suppose that F ( u III (1) ) = z . But then, voters in profile u III with preferences xyz will misreport as xzy with a coalition size of k < ε ,inducing profile u III (1) and improved result over z . This would contradict WSP, sowe know that F ( u III (1) ) = z .Now, assume for a general j that F ( u III ( j ) ) = z . We can show that it follows that F ( u III ( j +1) ) = z . Suppose instead that F ( u III ( j +1) ) = z . But then, voters in profile u III ( j ) with preferences xyz will misreport as xzy with a coalition size of k < ε ,inducing profile u III ( j +1) , and improved outcome x . This would contradict WSP,meaning that F ( u III ( j +1) ) = z . This is true for any general j . In particular, this istrue for: u III ( n ) = a + b - nk nk c 1 - a - b - cx x y yy z z xz y x z = a + k b - k c 1 - a - b - cx x y yy z z xz y x z But now, note that voters in profile u III ( n ) with preferences xyz will misreport as xzy with a coalition size of k , inducing profile u III. and evaluation ( I y over z . So, fails to hold. Instead, we have that: F ( u III ) = y Where: u III = a + b c 1 - a - b - cx y yy z xz x z a + b > − a − b But then, consider the following relabeling of the weights on profile u III . u III = a + b c 1 - a - b - cx y yy z xz x z = A B 1 - A - Bx y yy z xz x zWhere: a + b = A > .But then, note that in , we showed that for a generic profile u ∗ , and votingrule F (satisfying P, A, N, and WSP), where: u ∗ = a ∗ b ∗ − a ∗ − b ∗ x y yy z xz x z F ( u ∗ ) = ya ∗ > F a ∗ − a ∗ x yy xz z = y But, as we showed in , this led to a contradiction with WSP.So, is impossible.
Instead we suppose that F ( u III. ) = y ( I u III = a + b c 1 - a - b - cx y yy z xz x z Suppose that F ( u III ) = x Case 3.III.2.1.1
But then, voters in profile u III. with preferences xzy will misreport as xyz with a coalition size of b , inducing profile u III and improved result x . This wouldcontradict WSP, so Case 3.III.2.1.1 fails to hold.
Case 3.III.1.1.n+1 nε ≤ b < ( n + 1) ε Consider the following profile: u III ( j ) = a + b - jk jk c 1 - a - b - cx x y yy z z xz y x zHere, k = bn +1 , and j iterates from 1 to n . Note that the definition of u III ( j ) islocal to this case.We first consider F ( u III (1) ). Suppose that F ( u III (1) ) = x . But then, voters in profile u III (1) with preferences xzy will misreport as xyz with a coalition size of k < ε ,inducing profile u III and improved result x . This would contradict WSP, so weknow that F ( u III (1) ) = x .Now, assume for a general j that F ( u III ( j ) ) = x . We can show that it followsthat F ( u III ( j +1) ) = x . Suppose instead that F ( u III ( j +1) ) = x . But then, voters inprofile u III ( j +1) with preferences xzy will misreport as xyz with a coalition size of k < ε , inducing profile u III ( j ) , and improved outcome x . This would contradict WSP,meaning that F ( u III ( j +1) ) = x . This is true for any general j . In particular, this istrue for: u III ( n ) = a + b - nk nk c 1 - a - b - cx x y yy z z xz y x z = a + k b - k c 1 - a - b - cx x y yy z z xz y x z But now, note that voters in profile u III. with preferences xzy will misreportas xyz with a coalition size of k , inducing profile u III ( n ) and improved outcome x over z . So, fails to hold. Suppose that F ( u III ) = y But, using the same logic as with , this will fail to satisfy WSP.
So, we are left with: F ( u III ) = z Consider the following profile: u III = a + c 1 - a - cy yz xx zWe see that y must be the winner (by property P). Case 3.III.2.3.1 ≤ a + b < ε Here, voters in profile u III with preferences xyz will misreport as yzx with aweight of a , and will misreport as yxz with a weight of b . This will create profile u III , resulting in improved outcome y over z . This would contradict WSP, meaningthat Case 3.III.2.3.1 fails to hold.
Case 3.III.2.3.n+1 nε ≤ a + b < ( n + 1) ε Now, consider the following profile: u III ( j ) = a + b - j(k + m) c + jk 1 - a - b - c + jmx y yy z xz x zHere, k = an +1 and m = bn +1 . Also, j iterates from 1 to n . Note that thisdefinition of u III ( j ) is local to this case.We consider the evaluation F ( u III (1) ). Suppose that F ( u III (1) ) = z . But then, votersin profile u III with preferences xyz will misreport as yzx with a weight of k ,and will misreport as yxz with a weight of m . This would induce profile u III (1) andwould result in an improved outcome over z . This would contradict WSP, meaningthat F ( u III (1) ) = z . Now, assume for some general j that F ( u III ( j ) ) = z . We can show that F ( u III ( j +1) ) = z . Suppose instead that F ( u III ( j +1) ) = z . But then, voters in profile u III ( j ) withpreferences xyz will misreport as yzx with a weight of k , and will misreport as yxz with a weight of m . This would induce profile u III ( j +1) and would result inan improved outcome over z . So, for a general j , we have that F ( u III ( j +1) ) = z . Inparticular, this is true for the following profile: u III ( n ) = a + b - n(k + m) c + nk 1 - a - b - c + nmx y yy z xz x z= (k + m) a + c - k 1 - a - c - mx y yy z xz x zBut then, note that voters in profile u III ( n ) with preferences xyz will misreport as yzx with a weight of k , and will misreport as yxz with a weight of m . This willinduce profile u III , resulting in improved outcome y over z . This would contradictWSP, meaning that Case 3.III.2.3.n+1 fails to hold in general.So, we see that there is no voting rule F other than the Condorcet method thatsatisfies the axioms P, A, N, and WSP on a rich domain. This concludes Step 3 ,and concludes the proof of the theorem. Acknowledgements
First and foremost, I thank my advisor Professor Eric Maskin for suggesting thisincredible thesis topic, and for his guidance, patience, and all of his help through-out this project. I am also grateful for the intellectually fascinating environmentand courses of the Mathematics Department, and in particular, for the support ofProfessors Clifford Taubes and Dusty Grundmeier. Finally, thank you to ProfessorsJerry Green, Christopher Avery, and Scott Kominers for their enlightening coursesand for all of their advice and encouragement of my pursuit of economics. References
Arrow, Kenneth J.
Social Choice and Individual Values.
Economic Theory.
M´emoire del’Academie Royale des Sciences.