Manipulability of Voting Systems

Theoretical background Coalition manipulability Optimal voting systems

Manipulability of voting systems

François Durand, Fabien Mathieu, Ludovic Noirie

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Coalition manipulability Optimal voting systems Introduction

Origin: politics. Applications: any situation of collective choice. Can we trust the electors? If not, is the result stable?

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Coalition manipulability Optimal voting systems Table of contents

1 Theoretical background Preferences Gibbard’s theorem

2 Coalition manipulability Deﬁnition and examples Simulation results

3 Optimal voting systems Position of the problem Reformulation

4 Conclusion

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Table of contents

1 Theoretical background Preferences Gibbard’s theorem

2 Coalition manipulability Deﬁnition and examples Simulation results

3 Optimal voting systems Position of the problem Reformulation

4 Conclusion

m C = {cj }j=1: candidates.

Preferred candidate: c1 ∈ C → very poor description.   c1 Strict preference ordering:  c2  ∈ Sm → better. c3 But does not describe strength of preferences.

Proﬁle 1 Proﬁle 2

The elector attributes a real number uj called utility to each candidate cj .

Proﬁle 1 Proﬁle 2 c1 10 10 c2 5 -5 c3 -10 -10

Von Neumann-Morgenstern theorem (1944) gives suﬃcient and necessary conditions for such a representation to make sense. m When a utility vector in R is without ties, it induces a strict preference ordering in Sm.

n E = {ei }i=1: electors. m n Population: an element of (R ) .

m n Culture: a probability law over (R ) .

Uniform spherical culture: each elector is drawn independently with a uniform distribution over the unit hypersphere.

Each elector almost surely has a strict preference order in Sm. n It induces a uniform distribution over (Sm) .

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems 65 40 35 75 60 25

cj is Condorcet winner iﬀ for any other candidate ck , cj is preferred to ck by more than half the electors. No Condorcet winner in the above example: Condorcet paradox.

Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Condorcet winner

Majority matrix (matrix of duels): Mjk is the number of electors preferring cj to ck .

Electors 40 35 25 Duels c1 c2 c3 c1 c2 c3 c1 Preferences c2 c3 c1 c2 c3 c1 c2 c3

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems 40 75 60 25

cj is Condorcet winner iﬀ for any other candidate ck , cj is preferred to ck by more than half the electors. No Condorcet winner in the above example: Condorcet paradox.

Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Condorcet winner

Majority matrix (matrix of duels): Mjk is the number of electors preferring cj to ck .

Electors 40 35 25 Duels c1 c2 c3 c1 c2 c3 c1 65 Preferences c2 c3 c1 c2 35 c3 c1 c2 c3

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems 75 25

cj is Condorcet winner iﬀ for any other candidate ck , cj is preferred to ck by more than half the electors. No Condorcet winner in the above example: Condorcet paradox.

Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Condorcet winner

Majority matrix (matrix of duels): Mjk is the number of electors preferring cj to ck .

Electors 40 35 25 Duels c1 c2 c3 c1 c2 c3 c1 65 40 Preferences c2 c3 c1 c2 35 c3 c1 c2 c3 60

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems cj is Condorcet winner iﬀ for any other candidate ck , cj is preferred to ck by more than half the electors. No Condorcet winner in the above example: Condorcet paradox.

Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Condorcet winner

Majority matrix (matrix of duels): Mjk is the number of electors preferring cj to ck .

Electors 40 35 25 Duels c1 c2 c3 c1 c2 c3 c1 65 40 Preferences c2 c3 c1 c2 35 75 c3 c1 c2 c3 60 25

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Condorcet winner

Majority matrix (matrix of duels): Mjk is the number of electors preferring cj to ck .

Electors 40 35 25 Duels c1 c2 c3 c1 c2 c3 c1 65 40 Preferences c2 c3 c1 c2 35 75 c3 c1 c2 c3 60 25

cj is Condorcet winner iﬀ for any other candidate ck , cj is preferred to ck by more than half the electors. No Condorcet winner in the above example: Condorcet paradox.

1 Theoretical background Preferences Gibbard’s theorem

2 Coalition manipulability Deﬁnition and examples Simulation results

3 Optimal voting systems Position of the problem Reformulation

4 Conclusion

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Game form

n E = {ei }i=1: electors. m C = {cj }j=1: candidates. For i ∈ {1,..., n}, Si a non-empty set: elector ei ’s strategies. Game form: g : S1 × ... × Sn → C.

In the usual two-round system, a strategy is given by: The name of a candidate; For each pair of candidates, the name of one of them.

Preference ordering: a total preorder over C (allow ties).

A strategy s ∈ Si is dominant for elector ei and preference ordering P iﬀ, whatever the other electors do, strategy s for ei produces an outcome at least as high in P as does any other strategy in Si .

A game form is straightforward iﬀ for every elector ei and preference ordering P, some strategy is dominant for ei and P.

If the game form is not straightforward, it means that even if you have a preference ordering, you may not know what to vote! It may depend on the other electors...

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Preferences Coalition manipulability Gibbard’s theorem Optimal voting systems Dictatorship

An elector ei is a dictator for game form g iﬀ, for every possible outcome c ∈ g(S1 × ... × Sn), elector ei has a strategy that ensures c to win, whatever the other electors do. A game form is dictatorial if there is a dictator for g.

ei being a dictator doesn’t mean that g depends only on its i-th variable. Simply, ei can choose the winner if she wants to.

Every straightforward game form with at least three possible outcomes is dictatorial.

Very general: No symmetry assumption between the electors or candidates. No assumption that the sets of strategies are the same. No assumption about what a “sincere” strategy should be. No assumption that an elector has a preference ordering.

Only strong assumption: the game form is deterministic. But cf. Gibbard 1977.

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Deﬁnition and examples Coalition manipulability Simulation results Optimal voting systems Table of contents

1 Theoretical background Preferences Gibbard’s theorem

2 Coalition manipulability Deﬁnition and examples Simulation results

3 Optimal voting systems Position of the problem Reformulation

4 Conclusion

m For i ∈ {1,..., m}, σi : R → Si : sincerity functions. Voting system: game form + sincerity functions. A voting system is subject to manipulation if the sincere strategy is not always dominant.

In most systems, we have a cultural idea of sincerity, but not always. Example: each elector votes black or white. If the number of black ballots is odd, candidate c1 wins: otherwise, c2 wins... Gibbard’s theorem ensures that, whatever the sincerity functions are, a voting system that is not subject to manipulation and has at least three possible outcomes is dictatorial.

m n Given a voting system, a population in (R ) is coalition-manipulable iﬀ there is a candidate c such that electors preferring c to the sincere winner can cast their ballots so that c wins.

It’s the worst case scenario: Perfect information about other electors, Ability to compute the manipulation, Perfect coordination between members of the coalition, No response from other electors.

Gibbard’s theorem: CM by one elector (individual manipulation)!

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Sincere ballot c1 c2 c3 CM c1 c3 c3

Theoretical background Deﬁnition and examples Coalition manipulability Simulation results Optimal voting systems Example of voting system: Plurality rule

Strategies: C. Sincerity: vote for one’s preferred candidate. Winner: the candidate with most votes.

Electors 40 35 25 c1’s utility 10 -10 -4 c2’s utility 1 10 -10 c3’s utility -10 9 10

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems CM c1 c3 c3

Theoretical background Deﬁnition and examples Coalition manipulability Simulation results Optimal voting systems Example of voting system: Plurality rule

Strategies: C. Sincerity: vote for one’s preferred candidate. Winner: the candidate with most votes.

Electors 40 35 25 c1’s utility 10 -10 -4 c2’s utility 1 10 -10 c3’s utility -10 9 10 Sincere ballot c1 c2 c3

Strategies: C. Sincerity: vote for one’s preferred candidate. Winner: the candidate with most votes.

Electors 40 35 25 c1’s utility 10 -10 -4 c2’s utility 1 10 -10 c3’s utility -10 9 10 Sincere ballot c1 c2 c3 CM c1 c3 c3

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Sincere ballot c3 c1 c2

N.B.: there is no coalition-manipulation (CM) here.

Theoretical background Deﬁnition and examples Coalition manipulability Simulation results Optimal voting systems Example of voting system: Veto rule

Strategies: C. Sincerity: vote for one’s least liked candidate. Winner: the candidate with least votes.

Electors 40 35 25 c1’s utility 10 -10 4 c2’s utility 1 10 -10 c3’s utility -10 9 10

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems N.B.: there is no coalition-manipulation (CM) here.

Theoretical background Deﬁnition and examples Coalition manipulability Simulation results Optimal voting systems Example of voting system: Veto rule

Strategies: C. Sincerity: vote for one’s least liked candidate. Winner: the candidate with least votes.

Electors 40 35 25 c1’s utility 10 -10 4 c2’s utility 1 10 -10 c3’s utility -10 9 10 Sincere ballot c3 c1 c2

Strategies: C. Sincerity: vote for one’s least liked candidate. Winner: the candidate with least votes.

Electors 40 35 25 c1’s utility 10 -10 4 c2’s utility 1 10 -10 c3’s utility -10 9 10 Sincere ballot c3 c1 c2

N.B.: there is no coalition-manipulation (CM) here.

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Electors 40 35 25 Duels c1 c2 c3 Minimum c1 c3 c3 c1 65 40 40 CM c2 c2 c1 c2 35 40 35 c3 c1 c2 c3 60 60 60

Theoretical background Deﬁnition and examples Coalition manipulability Simulation results Optimal voting systems Example of voting system: Maximin

Strategies: Sm (strict orders). Sincerity: preference ordering. Winner: greatest minimum in the majority matrix.

Electors 40 35 25 Duels c1 c2 c3 Minimum c1 c2 c3 c1 65 40 40 Sincere ballot c2 c3 c1 c2 35 75 35 c3 c1 c2 c3 60 25 25

Strategies: Sm (strict orders). Sincerity: preference ordering. Winner: greatest minimum in the majority matrix.

Electors 40 35 25 Duels c1 c2 c3 Minimum c1 c2 c3 c1 65 40 40 Sincere ballot c2 c3 c1 c2 35 75 35 c3 c1 c2 c3 60 25 25

Electors 40 35 25 Duels c1 c2 c3 Minimum c1 c3 c3 c1 65 40 40 CM c2 c2 c1 c2 35 40 35 c3 c1 c2 c3 60 60 60

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Electors 40 35 25 Duels c1 c2 c3 Borda c1 c2 c1 c1 65 65 130 CM c2 c3 c3 c2 35 75 110 c3 c1 c2 c3 35 25 60

Theoretical background Deﬁnition and examples Coalition manipulability Simulation results Optimal voting systems Example of voting system: Borda count

Strategies: Sm (strict orders). Sincerity: preference ordering. Winner: greatest sum in the majority matrix.

Electors 40 35 25 Duels c1 c2 c3 Borda c1 c2 c3 c1 65 40 105 Sincere ballot c2 c3 c1 c2 35 75 110 c3 c1 c2 c3 60 25 85

Strategies: Sm (strict orders). Sincerity: preference ordering. Winner: greatest sum in the majority matrix.

Electors 40 35 25 Duels c1 c2 c3 Borda c1 c2 c3 c1 65 40 105 Sincere ballot c2 c3 c1 c2 35 75 110 c3 c1 c2 c3 60 25 85

Electors 40 35 25 Duels c1 c2 c3 Borda c1 c2 c1 c1 65 65 130 CM c2 c3 c3 c2 35 75 110 c3 c1 c2 c3 35 25 60

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Electors 40 35 25 c1 c3 c3 CM c2 c2 c1 c3 c1 c2 c1 c2 c3 Round 1 c1 c3 c3 c1 c2 c1 Round 2 c1 c3 c3 c1 Winner c3

Theoretical background Deﬁnition and examples Coalition manipulability Simulation results Optimal voting systems Example of voting system: Single transferable vote (STV)

Strategies: Sm (strict orders). Sincerity: preference ordering. m − 1 rounds: at each round, candidate with least top votes is eliminated.

Electors 40 35 25 c1 c2 c3 Sincere ballot c2 c3 c1 c3 c1 c2 Round 1 Round 2 Winner

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Electors 40 35 25 c1 c3 c3 CM c2 c2 c1 c3 c1 c2 Round 1 c1 c3 c3 c1 c2 c1 Round 2 c1 c3 c3 c1 Winner c3

Theoretical background Deﬁnition and examples Coalition manipulability Simulation results Optimal voting systems Example of voting system: Single transferable vote (STV)

Strategies: Sm (strict orders). Sincerity: preference ordering. m − 1 rounds: at each round, candidate with least top votes is eliminated.

Electors 40 35 25 c1 c2 c3 Sincere ballot c2 c3 c1 c3 c1 c2 Round 1 c1 c2 c3 Round 2 Winner

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Electors 40 35 25 c1 c3 c3 CM c2 c2 c1 c3 c1 c2 Round 1 c1 c3 c3 Round 2 c1 c3 c3 Winner c3

Theoretical background Deﬁnition and examples Coalition manipulability Simulation results Optimal voting systems Example of voting system: Single transferable vote (STV)

Strategies: Sm (strict orders). Sincerity: preference ordering. m − 1 rounds: at each round, candidate with least top votes is eliminated.

Electors 40 35 25 c1 c2 c3 Sincere ballot c2 c3 c1 c3 c1 c2 Round 1 c1 c2 c3 Round 2 c1 c2 c1 Winner c1

Strategies: Sm (strict orders). Sincerity: preference ordering. m − 1 rounds: at each round, candidate with least top votes is eliminated.

Electors 40 35 25 Electors 40 35 25 c1 c2 c3 c1 c3 c3 Sincere ballot c2 c3 c1 CM c2 c2 c1 c3 c1 c2 c3 c1 c2 Round 1 c1 c2 c3 Round 1 c1 c3 c3 Round 2 c1 c2 c1 Round 2 c1 c3 c3 Winner c1 Winner c3

Strategies: {0, 1}m. Sincerity: vote 1 for candidates with positive utility (there are other reasonable models). Winner: the candidate with most votes 1.

Electors 40 35 25 c1’s utility 10 -10 -4 c2’s utility 1 10 -10 c3’s utility -10 9 10

Electors 40 35 25 Total Electors 40 35 25 Total 1 0 0 40 1 0 1 65 Sincere 1 1 0 75 CM 0 1 0 35 0 1 1 60 0 1 1 60

Strategies: [−10, 10]m. Sincerity: utility vector stretched into this interval (there are other reasonable models). Winner: greatest average.

Electors 40 35 25 Average 10 -10 -4 -0.5 Sincere ballot 1 10 -10 1.4 -10 9 10 1.65

Electors 40 35 25 Average 10 -10 -4 -0.5 CM 1 10 -10 1.4 -10 -10 10 -5

Strategies: [−10, 10]m. Sincerity: utility vector stretched into this interval (there are other reasonable models). Winner: greatest median.

Electors 40 35 25 Median 10 -10 -4 -4 Sincere ballot 1 10 -10 1 -10 9 10 9

Electors 40 35 25 Median 10 -10 -4 -4 CM 10 10 -10 10 -10 -10 10 -10

1 Theoretical background Preferences Gibbard’s theorem

2 Coalition manipulability Deﬁnition and examples Simulation results

3 Optimal voting systems Position of the problem Reformulation

4 Conclusion

m n µ a culture: a probability law over (R ) . Given a voting system, we call its CM rate for µ the probability that the population drawn by µ is coalition-manipulable.

Computational approach: simulations.

1 Draw the preferences of the population according to µ. 2 For each voting system studied, evaluate the sincere winner and determine if a coalition-manipulation (CM) is possible. Then loop and do statistics.

0.9 Approval Borda 0.8 Range average Range median 0.7 Coombs Plurality 0.6 Schulze Nanson 0.5 Maximin CM rate Bucklin 0.4 Ranked pairs 0.3 Veto STV-loft 0.2 Baldwin Two-round 0.1 STV

0 1 2 3 4 10 10 10 10 Number of electors n

0.9 Approval 0.8 Borda Range average 0.7 Range median Coombs 0.6 Plurality Schulze 0.5 Nanson CM rate Maximin 0.4 Bucklin 0.3 Ranked pairs Veto 0.2 STV-loft Baldwin 0.1 Two-round STV 0 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Number of candidates m

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Position of the problem Coalition manipulability Reformulation Optimal voting systems Table of contents

1 Theoretical background Preferences Gibbard’s theorem

2 Coalition manipulability Deﬁnition and examples Simulation results

3 Optimal voting systems Position of the problem Reformulation

4 Conclusion

Conjecture µ: a culture with “nice” properties. n n ((Si )i=1, (σi )i=1, g): a voting system. Then there exists g 0 such that the voting system deﬁned by 0 Si = Sm, 0 m σi : R → Sm the canonical surjection (with a tie-break rule), 0 n g :(Sm) → C n n has a CM rate for µ at most as high as ((Si )i=1, (σi )i=1, g). The uniform spherical culture meets these “nice” properties. In what follows, voting systems depend only on strict preference orderings.

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Duels c1 c2 c3 c1 65 60 c2 35 60 c3 40 40

→ c3 wins under Plurality rule... but c1 is the Condorcet winner. → Plurality rule does not meet the Condorcet criterion.

Example: Plurality rule

Electors 40 35 25 c3 c2 c1 Preferences c1 c1 c2 c2 c3 c3

Theoretical background Position of the problem Coalition manipulability Reformulation Optimal voting systems Condorcet criterion

A voting system meets Condorcet criterion iﬀ whenever there exists a Condorcet winner, she is elected (with sincere ballots).

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Duels c1 c2 c3 c1 65 60 c2 35 60 c3 40 40

but c1 is the Condorcet winner. → Plurality rule does not meet the Condorcet criterion.

Theoretical background Position of the problem Coalition manipulability Reformulation Optimal voting systems Condorcet criterion

A voting system meets Condorcet criterion iﬀ whenever there exists a Condorcet winner, she is elected (with sincere ballots).

Example: Plurality rule

Electors 40 35 25 c3 c2 c1 Preferences c1 c1 c2 c2 c3 c3

→ c3 wins under Plurality rule...

A voting system meets Condorcet criterion iﬀ whenever there exists a Condorcet winner, she is elected (with sincere ballots).

Example: Plurality rule

Electors 40 35 25 Duels c1 c2 c3 c3 c2 c1 c1 65 60 Preferences c1 c1 c2 c2 35 60 c2 c3 c3 c3 40 40

→ c3 wins under Plurality rule... but c1 is the Condorcet winner. → Plurality rule does not meet the Condorcet criterion.

A voting system meets majority coalition criterion iﬀ, given the strategies of less than half the electors, the other electors can always choose the winner by using appropriate strategies.

Almost all usual voting systems meet this criterion. Exception: Veto rule. In a voting system meeting this criterion, if the winner is not a Condorcet winner (whether there exists one or not), then the situation is manipulable.

Given a voting system, we can “condorcify” it: whenever there is a Condorcet winner, elect her; otherwise, use the original system.

Theorem If a voting system meets majority coalition criterion, its condorciﬁed system has a CM rate at most as high.

Goal: a voting system meeting majority coalition criterion, whose CM rate for µ is minimal.

Thanks to the theorem, is it suﬃcient to solve the following problem.

Goal: a voting system meeting Condorcet criterion, whose CM rate for µ is minimal.

In what follows, we always consider that the voting system meets Condorcet criterion.

1 Theoretical background Preferences Gibbard’s theorem

2 Coalition manipulability Deﬁnition and examples Simulation results

3 Optimal voting systems Position of the problem Reformulation

4 Conclusion

n X , Y ∈ (Sm) . v, c ∈ {1,..., m}. We say that X is (v, c)-connected to Y iﬀ: v is Condorcet winner in X . X and Y diﬀer only by some electors preferring c to v in X .

e1 e2 e3 e1 e2 e3 1 2 4 (1, 4)-connection 1 2 3 X = 4 1 1 Y = 4 1 2 3 4 3 3 4 1 2 3 2 2 3 4

We wouldn’t like c to be declared winner in Y : in that case, X would be manipulable!

1 Any non-Condorcet situation Y is manipulable.

2 If X1 is (v, c)-connected to X2 and if X2 has a Condorcet winner too, then this winner cannot be c. Thus, this connection doesn’t lead to a manipulation opportunity for c. 3 If X is a Condorcet situation, it is manipulable if and only there exists a non-Condorcet situation Y whose winner is c and such that X is (v, c)-connected to Y .

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Y1.3

X4 Y2.2 X5

Y3.1

Theoretical background Position of the problem Coalition manipulability Reformulation Optimal voting systems An open problem...

X1 Y1.1 For each Yj , choose one Y1.2 Y1 X2 winner Yj .c. Y1.3 If Yj .c is chosen and if Xi is X3 connected to Yj .c, then Xi is Y2.1 manipulable. X4 Y2.2 Y2 Goal: minimize the X5 µ-measure of Xi that are Y2.3 manipulable. (In uniform X6 Y3.1 culture, minimize the X number of X that are 7 Y .2 Y i 3 3 manipulable.) X8 Y3.3

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Coalition manipulability Optimal voting systems Table of contents

1 Theoretical background Preferences Gibbard’s theorem

2 Coalition manipulability Deﬁnition and examples Simulation results

3 Optimal voting systems Position of the problem Reformulation

4 Conclusion

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Coalition manipulability Optimal voting systems Conclusion

Voting systems can be used in any situation where multiple agents need to make a decision together. All voting systems are susceptible to manipulation, but some of them are manipulable more often than the other ones.

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Coalition manipulability Optimal voting systems Frame count

This is the 41-th frame.

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Coalition manipulability Optimal voting systems A story of paradoxes and impossibility theorems

1785: Condorcet paradox. More than half the electors can prefer candidate a to b, b to c, but c to a. 1951: Arrow impossibility theorem. A social choice function cannot meet a set of reasonable criteria (unrestricted domain, non-dictatorship, Pareto eﬃciency, independence of irrelevant alternatives). 1973–1975: Gibbard-Satterthwaite impossibility theorem. Any deterministic non-dictatorial voting system is subject to manipulation. 1977: Gibbard impossibility theorem. Generalization to non-deterministic voting systems. 1991: Bartholdi and Orlin. Deciding manipulation in Single transferable vote is NP-hard.

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Coalition manipulability Optimal voting systems Von Neumann-Morgenstern theorem (1)

m Pm L = {(p1,..., pm) ∈ R+, j=1 pj = 1}: lotteries over C. ≺ a binary relation on L: agent’s preferences over the lotteries. Completeness: ∀(L, M) ∈ L2, L ≺ M or M ≺ L. Transitivity: ∀(L, M, N) ∈ L3, L ≺ M and M ≺ N ⇒ L ≺ N. Archimedeanness: ∀(L, M, N) ∈ L3, L ≺ M and M ≺ N ⇒ ∃ε ∈]0, 1[ such that (1 − ε)L + εN ≺ M ≺ εL + (1 − ε)N. Independance of irrelevant alternatives: ∀(L, M, N) ∈ L3, ∀r ∈ [0, 1], L ≺ M ⇒ rL + (1 − r)N ≺ rM + (1 − r)N.

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Coalition manipulability Optimal voting systems Von Neumann-Morgenstern theorem (2)

The following conditions are equivalent. 1 ≺ is complete, transitive, archimedean and independent of irrelevant alternatives. m 2 There exists U = (u1,..., um) ∈ R such that for any two lotteries L = (p1,..., pm) and M = (q1,..., qm):

m m X X L ≺ M ⇔ pj uj ≤ qj uj j=1 j=1

When they are met, U is deﬁned up to two constants: V = (v1,..., vm) is another utility vector representing ≺ if and only if ∃a ∈ R+\{0}, b ∈ R such that ∀j ∈ {1,..., m}, vj = auj + b.

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems c2

→ c2 wins.

Actually, whatever the tie-breaking rule chosen, Plurality rule is susceptible to individual manipulation (since it is not dictatorial).

Electors e1 e2 e3 c1 c2 c3 Preferences c2 c1 c2 c3 c3 c1 Ballot c1 c2

Theoretical background Coalition manipulability Optimal voting systems Individual manipulability with Plurality rule

Example: tie-breaking rule “lowest index candidate”.

Electors e1 e2 e3 c1 c2 c3 Preferences c2 c1 c2 c3 c3 c1 Sincere ballot c1 c2 c3

→ c1 wins.

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems c2

→ c2 wins.

Actually, whatever the tie-breaking rule chosen, Plurality rule is susceptible to individual manipulation (since it is not dictatorial).

Theoretical background Coalition manipulability Optimal voting systems Individual manipulability with Plurality rule

Example: tie-breaking rule “lowest index candidate”.

Electors e1 e2 e3 Electors e1 e2 e3 c1 c2 c3 c1 c2 c3 Preferences c2 c1 c2 Preferences c2 c1 c2 c3 c3 c1 c3 c3 c1 Sincere ballot c1 c2 c3 Ballot c1 c2

→ c1 wins.

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Actually, whatever the tie-breaking rule chosen, Plurality rule is susceptible to individual manipulation (since it is not dictatorial).

Theoretical background Coalition manipulability Optimal voting systems Individual manipulability with Plurality rule

Example: tie-breaking rule “lowest index candidate”.

Electors e1 e2 e3 Electors e1 e2 e3 c1 c2 c3 c1 c2 c3 Preferences c2 c1 c2 Preferences c2 c1 c2 c3 c3 c1 c3 c3 c1 Sincere ballot c1 c2 c3 Ballot c1 c2 c2

→ c1 wins. → c2 wins.

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Coalition manipulability Optimal voting systems Individual manipulability with Plurality rule

Example: tie-breaking rule “lowest index candidate”.

Electors e1 e2 e3 Electors e1 e2 e3 c1 c2 c3 c1 c2 c3 Preferences c2 c1 c2 Preferences c2 c1 c2 c3 c3 c1 c3 c3 c1 Sincere ballot c1 c2 c3 Ballot c1 c2 c2

→ c1 wins. → c2 wins.

Actually, whatever the tie-breaking rule chosen, Plurality rule is susceptible to individual manipulation (since it is not dictatorial).

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Example: If ballots are orders: vote Candidate c a b v (c, a, b, v). Sincere utility 1 2 -2 -1 Approval: vote only for c. Pretended utility +∞ -1 -5 -6 Etc.

Theoretical background Coalition manipulability Optimal voting systems Trivial strategy

Deﬁnition: trivial strategy of e for c against v Intuitively, the elector e acts as if: c has inﬁnite utility, v has a very low utility, other candidates have negative utility but keep their relative positions (sincere for e).

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems If ballots are orders: vote (c, a, b, v). Approval: vote only for c. Etc.

Theoretical background Coalition manipulability Optimal voting systems Trivial strategy

Example:

Candidate c a b v Sincere utility 1 2 -2 -1 Pretended utility +∞ -1 -5 -6

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Coalition manipulability Optimal voting systems Trivial strategy

Example: If ballots are orders: vote Candidate c a b v (c, a, b, v). Sincere utility 1 2 -2 -1 Approval: vote only for c. Pretended utility +∞ -1 -5 -6 Etc.

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems TM attempt c v v → a wins → TM failure CM attempt c a v → c wins → CM success

Examples: all the previous ones. Negative example: Veto rule.

Electors 25 35 40 v c c Preferences a v v c a a Sincere ballot c a a → v wins

Theoretical background Coalition manipulability Optimal voting systems Trivial manipulability (TM)

Deﬁnition There exists a non-winning candidate c so that if all electors preferring c to v use their trivial strategy for c against v, then c is elected.

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems TM attempt c v v → a wins → TM failure CM attempt c a v → c wins → CM success

Theoretical background Coalition manipulability Optimal voting systems Trivial manipulability (TM)

Deﬁnition There exists a non-winning candidate c so that if all electors preferring c to v use their trivial strategy for c against v, then c is elected. Examples: all the previous ones. Negative example: Veto rule.

Electors 25 35 40 v c c Preferences a v v c a a Sincere ballot c a a → v wins

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems CM attempt c a v → c wins → CM success

Theoretical background Coalition manipulability Optimal voting systems Trivial manipulability (TM)

Electors 25 35 40 v c c Preferences a v v c a a Sincere ballot c a a → v wins TM attempt c v v → a wins → TM failure

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Coalition manipulability Optimal voting systems Trivial manipulability (TM)

Electors 25 35 40 v c c Preferences a v v c a a Sincere ballot c a a → v wins TM attempt c v v → a wins → TM failure CM attempt c a v → c wins → CM success

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Examples: most of the previous ones. Negative example: previous one with Veto rule. Diﬀerences with TM: Generally, TM is not unison (keep sincere preference order about candidates apart from c and v); Generally, UM does not use the trivial strategy.

Theoretical background Coalition manipulability Optimal voting systems Unison manipulability (UM)

Deﬁnition There exists a non-winning candidate c so that all electors preferring c to v can cast a same ballot so that c is elected.

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Diﬀerences with TM: Generally, TM is not unison (keep sincere preference order about candidates apart from c and v); Generally, UM does not use the trivial strategy.

Theoretical background Coalition manipulability Optimal voting systems Unison manipulability (UM)

Deﬁnition There exists a non-winning candidate c so that all electors preferring c to v can cast a same ballot so that c is elected.

Examples: most of the previous ones. Negative example: previous one with Veto rule.

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Coalition manipulability Optimal voting systems Unison manipulability (UM)

Deﬁnition There exists a non-winning candidate c so that all electors preferring c to v can cast a same ballot so that c is elected.

Examples: most of the previous ones. Negative example: previous one with Veto rule. Diﬀerences with TM: Generally, TM is not unison (keep sincere preference order about candidates apart from c and v); Generally, UM does not use the trivial strategy.

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Deﬁnition: Coalition manipulability resistant to simple counterthreat (CMSCT) There exists a candidate c such that the electors preferring c to v can cast their ballots so that: If other electors vote sincerely, c wins; Whatever other electors vote, v cannot win.

Theoretical background Coalition manipulability Optimal voting systems Coalition manipulability, simple counterthreat (CMSCT)

In CM, we suppose that electors preferring c to v lie, but the other ones still vote sincerely. Counter-threats, counter-counter-threats... very complicated! → Need for more tractable notions.

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Coalition manipulability Optimal voting systems Coalition manipulability, simple counterthreat (CMSCT)

In CM, we suppose that electors preferring c to v lie, but the other ones still vote sincerely. Counter-threats, counter-counter-threats... very complicated! → Need for more tractable notions.

Deﬁnition: Coalition manipulability resistant to simple counterthreat (CMSCT) There exists a candidate c such that the electors preferring c to v can cast their ballots so that: If other electors vote sincerely, c wins; Whatever other electors vote, v cannot win.

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems CM attempt v a c c → c wins → CM success Counterthreat v v c c → v wins → CMSCT failure

Theoretical background Coalition manipulability Optimal voting systems Coalition manipulability, simple counterthreat (CMSCT)

Negative example: Plurality rule.

Electors 40 15 15 30 v a a c Preferences c v c v a c v a Sincere ballot v a a c → v wins

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Counterthreat v v c c → v wins → CMSCT failure

Theoretical background Coalition manipulability Optimal voting systems Coalition manipulability, simple counterthreat (CMSCT)

Negative example: Plurality rule.

Electors 40 15 15 30 v a a c Preferences c v c v a c v a Sincere ballot v a a c → v wins CM attempt v a c c → c wins → CM success

Negative example: Plurality rule.

Electors 40 15 15 30 v a a c Preferences c v c v a c v a Sincere ballot v a a c → v wins CM attempt v a c c → c wins → CM success Counterthreat v v c c → v wins → CMSCT failure

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems CM attempt v c c → c wins → CM success Counterthreat ? c c → c wins → CMSCT success

Theoretical background Coalition manipulability Optimal voting systems Coalition manipulability, simple counterthreat (CMSCT)

Example: Plurality rule.

Electors 40 35 25 v c a Preferences c v c a a v Sincere ballot v c a → v wins

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Counterthreat ? c c → c wins → CMSCT success

Theoretical background Coalition manipulability Optimal voting systems Coalition manipulability, simple counterthreat (CMSCT)

Example: Plurality rule.

Electors 40 35 25 v c a Preferences c v c a a v Sincere ballot v c a → v wins CM attempt v c c → c wins → CM success

Example: Plurality rule.

Electors 40 35 25 v c a Preferences c v c a a v Sincere ballot v c a → v wins CM attempt v c c → c wins → CM success Counterthreat ? c c → c wins → CMSCT success

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems Theoretical background Coalition manipulability Optimal voting systems Overview of criteria

1 CM 0.9 TM 0.8 UM CMSCT 0.7 IM 0.6

0.5

0.4

0.3

0.2

0.1

Proportion of experiments that are manipulable 0

STV Approval Maximin Veto rule Range votingBorda countPlurality rule Exhaustive ballot Figure: Uniform culture, n = 50 electors and m = 4 candidates

François Durand, Fabien Mathieu, Ludovic Noirie Manipulability of voting systems