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Social Choice and Welfare

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29-03-2018 | Original Paper

A concept of sincerity for combinatorial voting

Authors: Francesco De Sinopoli, Claudia Meroni

Published in: Social Choice and Welfare | Issue 3/2018

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Abstract

A basic problem in voting theory is that all the strategy profiles in which nobody is pivotal are Nash equilibria. We study elections where voters decide simultaneously on several binary issues. We extend the concept of conditional sincerity introduced by Alesina and Rosenthal (Econometrica 64(6):1311–1341, 1996) and propose an intuitive and simple criterion to refine equilibria in which players are not pivotal. This is shown to have a foundation in a refinement of perfection that takes into account the material voting procedure. We prove that in large elections the proposed solution is characterized through a weaker definition of Condorcet winner and always survives sophisticated voting.

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An example is given by the November 2016 California ballot, in which citizens have been asked to vote on seventeen issues, including marijuana legalization, gun control, drug prices, and condoms in porn.

In the example above, for instance, a voter may want marijuana legalization and gun control to pass together, but prefer neither issue to be implemented rather than each issue passing alone for fear of armed stoned people.

Such a concept has been employed also in Ingberman and Rosenthal (1997).

Oddness of the number of players is assumed just to guarantee that a pure strategy profile induces a pure outcome. In this case preference orders over outcomes are sufficient to study the main concepts that we introduce. We could replace this assumption with any deterministic tie-breaking rule.

A player’s best response depends only on the aggregate behavior of the opponents, i.e., on the sum of the positive votes that each issue gets from the others. Henceforth, we will often use a vector to summarize this information, and we will refer to it as “pivotal event” if the player is decisive for some issue.

Lacy and Niou (2000) and Ahn and Oliveros (2012) observe that if preferences are separable in every issue then voting accordingly to the most preferred outcome is a dominant strategy for a player. This is an immediate corollary of the characterization in Proposition 1.

Separable preferences do not imply the existence of a Condorcet winner. Take for example three players with the following separable preferences:

$$\begin{aligned} A&\succ _1\varnothing \succ _1 AB \succ _1 B,\\ B&\succ _2\varnothing \succ _2 AB \succ _2 A,\\ AB&\succ _3 A\succ _3 B\succ _3\varnothing . \end{aligned}$$

The dominant strategy profile $s=((1,0),(0,1),(1,1))$ induces the local Condorcet winner AB, which however is not a Condorcet winner as the majority of the players prefer outcome $\varnothing $ to it.

The issue sincere strategy profile with respect to outcome B is $s=((0,0),(0,0),(1,1))$, so B is not an issue sincere outcome.

If every player plays according to $s'$, strategies (0, 0) and (0, 1) are equivalent for the first two players and they are strictly better than strategies (1, 0) and (1, 1), which would induce outcome AB. Hence, close-by $s'$, the first two strategies are the only best replies for them. Clearly, these strategies induce a different outcome only when the first two players are pivotal for issue B, that is, B takes one vote from the opponents. In this case, if issue A takes two votes then they strictly prefer strategy (0, 1), otherwise they strictly prefer strategy (0, 0). For $s'$ to be perfect, then, the probability of the pivotal event (2, 1) has to be sufficiently greater than the probability of the events (0, 1) and (1, 1).

We use the term b-strategy as a reminiscence of behavioral strategy. In fact, b-strategies would precisely be behavioral strategies if we described a combinatorial voting game as an extensive form one.

In particular, to the b-strategy (x, y) with $x,y\in (0,1)$ corresponds the completely-mixed strategy $xy(1,1)+x(1-y)(1,0)+y(1-x)(0,1)+(1-x)(1-y)(0,0)$.

This result is in general not new, as Wichardt (2008) shows that an extensive-form game without perfect recall may not have any Nash equilibrium in behavioral strategies. However, his example cannot be framed in the context of combinatorial voting.

The only strategies that survive iterated dominance are strategy (1, 1) for players 1 and 2 and strategies (1, 1) and (1, 0) for player 3.

In particular, outcome $\varnothing $ is a local Condorcet winner of the game, while the outcome that survives the process, AB, is the Condorcet winner. See the example in Table 5 in Lacy and Niou (2000) for a game in which iterated dominance eliminates the Condorcet winner and selects a local Condorcet winner.

The same relation would hold if uncertainty were introduced in the vote counting. As a matter of fact, under our assumption of strict preferences an equivalent definition of b-perfection could be stated based on a vanishing probability of misrecording votes, in a way similar to Laslier and Van der Straeten (2016).

Note that to the b-strategy (x, x) corresponds the mixed strategy $x^2(1,1)+x(1-x)(1,0)+x(1-x)(0,1)+(1-x)^2(0,0)$, which gives positive weight also to the pure strategies that are not best replies.

If player 3 plays (0, 0), player 2 weakly prefers strategy (0, 0) to both strategies (1, 0) and (0, 1). The condition on player 1’s strategy assures that player 2 prefers strategy (0, 0) also to strategy (1, 1) (i.e., $9\ge 3x+9z+10(1-x-y-z)$).

Among the equivalent definitions of perfect equilibrium proposed in the literature, we will use the extension of Definition 6 to mixed strategies (obtained just substituting in that definition s with $\sigma $).

In particular, the utility of the first three strategies is $\frac{112\varepsilon -992\varepsilon ^2+2352\varepsilon ^3-1664\varepsilon ^4}{3(1 - 8 \varepsilon + 11\varepsilon ^2)}$, while the utility of strategy (1, 1) is $\frac{22\varepsilon -92\varepsilon ^2-78\varepsilon ^3+316\varepsilon ^4}{3(1 - 8 \varepsilon + 11\varepsilon ^2)}$.

Ahn DS, Oliveros S (2012) Combinatorial voting. Econometrica 80(1):89–141CrossRef

Alesina A, Rosenthal H (1996) A theory of divided government. Econometrica 64(6):1311–1341CrossRef

Brams SJ, Kilgour DM, Zwicker WS (1997) Voting on referenda: the separability problem and possible solutions. Electoral Stud 16(3):359–377CrossRef

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Title: A concept of sincerity for combinatorial voting
Authors: Francesco De Sinopoli
Claudia Meroni
Publication date: 29-03-2018
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 3/2018
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-018-1125-5

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