metadata: authors: 0: author: Pongou,Roland
1: author: Sidie,Ghislain Junior
count: World
docna: 0: docna: Voting when Rankings Matter : Truthful
            Equilibria, Efficiency, and Abstention
docty: Policy Research Working Paper
abstracts: cdata!: Ranked voting is an election format in
            which each voter ranks candidates on a ballot, and
            individual rankings are aggregated using a general rule to
            produce a social ranking. This paper proposes a
            non-cooperative model of this electoral system. The setting
            allows for unequal voting rights, abstention, and social
            incomparability of candidates, and each voter's utility
            is measured by how close his or her true preferences are to
            the social ranking. The analysis uncovers three main
            findings. First, it proves the existence of a pure strategy
            Nash equilibrium. Second, it shows that truthtelling is
            always a Nash equilibrium regardless of the voting rule and
            the structure of individual preferences. Third, under mild
            conditions, truthtelling is Pareto-efficient when voters
            have strict preferences.  Extending the analysis to
            majoritarian elections with costly voluntary participation
            shows that truthtelling is an equilibrium if and only if the
            costs of participation are not too high and the election is
            tight. The findings have implications for the design of
            ranked voting systems that are compatible with truthtelling
            and efficiency while allowing unrestricted freedom in the
            choice of the voting rule. A reinterpretation of the model
            in the context of intrapersonal bargaining, where the
            decision-maker has multiple rational selves, has
            implications for the occurrence of cyclic individual choices
            that reflect stable and efficient behavioral patterns.
display_title: Voting when Rankings Matter :
            Truthful Equilibria, Efficiency, and Abstention
listing_relative_url: /research/2024/07/34359934/voting-rankings-matter-truthful-equilibria-efficiency-abstention
topicv3: Social Analysis m1326036 744,Political Economy of Reform m1326263 836,Social Cohesion m1327189 738,Political Participation m1327389 926,Political Systems and Government(PSG)
            m1327721 839
publishtoextweb_dt: 2024-07-02T00:00:00Z
dois: 10.1596/1813-9450-10837
origu: Off of Sr VP Dev Econ/Chief Econ (DECVP)
no_of_pages: 43
author_keys: 0: author_key: 1065540
1: author_key: 2381484
model_extractions: Policy Research Working Paper 10837 Voting when Rankings Matter Truthful Equilibria, Efficiency, and Abstention _Roland Pongou Ghislain Junior Sidie_ Africa Region Office of the Chief Economist July 2024
Policy Research Working Paper 10837

# **Abstract**

Ranked voting is an election format in which each voter ranks candidates on a ballot, and individual rankings are aggregated using a general rule to produce a social ranking. This paper proposes a non-cooperative model of this electoral system. The setting allows for unequal voting rights, abstention, and social incomparability of candidates, and each voter’s utility is measured by how close his or her true preferences are to the social ranking. The analysis uncovers three main findings. First, it proves the existence of a pure strategy Nash equilibrium. Second, it shows that truthtelling is always a Nash equilibrium regardless of the voting rule and the structure of individual preferences. Third, under mild conditions, truthtelling is Pareto-efficient when voters have strict preferences. Extending the analysis to majoritarian elections with costly voluntary participation shows that truthtelling is an equilibrium if and only if the costs of participation are not too high and the election is tight. The findings have implications for the design of ranked voting systems that are compatible with truthtelling and efficiency while allowing unrestricted freedom in the choice of the voting rule. A reinterpretation of the model in the context of _intrapersonal bargaining_ , where the decision-maker has multiple rational selves, has implications for the occurrence of cyclic individual choices that reflect stable and efficient behavioral patterns.

This paper is a product of the Office of the Chief Economist, Africa Region. It is part of a larger effort by the World Bank to provide open access to its research and make a contribution to development policy discussions around the world. Policy Research Working Papers are also posted on the Web at http://www.worldbank.org/prwp. The authors may be contacted at gsidie@worldbank.org and rpongou@uottawa.ca.

_The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent._ Produced by the Research Support Team
# **Voting When Rankings Matter: Truthful Equilibria,**

# **Efficiency, and Abstention**

Roland Pongou<sup>∗</sup> Ghislain Junior Sidie<sup>∗</sup>

**Keywords:** Ranked voting; Truthtelling; Abstention; Social incomparability; Tie-breaking rules; Intrapersonal bargaining; Cyclic individual choice.

**JEL Classification:** D72, C72, D61

> ∗Department of Economics, University of Ottawa, Canada. Roland Pongou : rpongou@uottawa.ca; Ghislain Junior Sidie (corresponding author): gsidi040@uottawa.ca.
# **1 Introduction**

Ranked voting is an election format in which voters rank candidates on a ballot. In a series of relatively recent popular contributions (Maskin, 2022, 2018b, 2018a), Eric Maskin argues that this voting system is better than other popular systems such as first-past-thepost, where voters only vote for one of the competing candidates. Maskin and Sen (2017) also argue that it leads to outcomes that better represent society’s preferences. Other acknowledged advantages of this voting system include the fact that it reduces the vote splitting problem<sup>1</sup> and leads to less polarized electoral outcomes. Despite being used or advocated in several societies, we still know little about how ranked voting affects voters’ incentives and strategic behavior in contexts where each individual votes independently.

In this paper, our goal is threefold. First, we propose a non-cooperative model of ranked voting and analyze the existence of an equilibrium. Second, we analyze voters’ strategic behavior, with a special focus on truthtelling. Third, we analyze the efficiency of truthtelling as an equilibrium behavior. In addition, the model has several applications. In particular, we show an application to the 2016 Republican presidential primaries in the United States (Dasgupta & Maskin, 2020) and highlight how the equilibrium outcome differs under ranked voting and first-past-the-post (or plurality). We also develop an application to majoritarian elections with voluntary participation (Börgers, 2004) and discover additional results. To our knowledge, our analysis and results are new, and they have clear policy implications for the design of voting systems that are compatible with truthtelling and outcome efficiency. Moreover, we reinterpret our model in the context of _intrapersonal bargaining_ and shed light on the pertinence of certain individual choices that violate rationality.

Our working model is more general than the traditional model of ranked voting, which basically consists of a finite set of candidates, a finite set of voters, with each voter holding a strict preference relation over candidates, and majority rule (Maskin, 2022). Our framework generalizes several aspects of this classical model as follows. First, whereas the classical model of ranked voting only considers _majority rule_ as already mentioned, our findings hold under a much larger class of voting rules. We consider the whole class 1Vote splitting occurs when the presence of one or more alternatives decreases the likelihood of another alternative in competition to be elected.

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of voting rules with abstention (May (1952), Rubinstein (1980), Tchantcho, Diffo Lambo, Pongou, and Engoulou (2008), Freixas and Zwicker (2009)), where voters may have unequal voting rights and are allowed to abstain.<sup>2</sup> Second, voters do not have to rank candidates strictly in our setting, as they may choose to abstain between certain candidates; in other words, a voter’s ranking is a weak ordering. Third, following Arrow (1951), in our setting, votes are aggregated using a general rule to produce a social ranking.<sup>3</sup> Another distinguishing feature of our model is that each voter derives utility from how close his or her true preferences are to the social ranking. Therefore, a voter’s utility function represents a hyper-preference relation or a “meta-ranking" (Sen (1974)).<sup>4</sup> To move from our basic model to a strategic form game, we assume that voters form the set of players, each player’s strategy set is the set of all weak orderings of the candidates, and each player’s utility function is given as mentioned above. Given a profile of players’ votes, the social ranking is obtained by proceeding to a pairwise comparison of candidates under the prevailing voting rule. If the social ranking fails to rank two candidates, a tie-breaking rule is used to rank them; this tie-breaking rule assigns a probability to the event that one candidate is ranked above the other.<sup>5</sup> Each voter has an expected utility

> 2Majority rule does not count abstention and gives an equal voting right to all voters. For this reason, it is not used in several real-life organizations and legislative bodies whose rules are designed to give greater voting power to more important members. Such organizations include the World Bank, the International Monetary Fund, the United Nations Security Council, the European Union, the United States Electoral College, and most economic corporations, to name a few. The class of voting rules with abstention addresses these limitations of majority rule.

> 3Arrow (1951) requires the social ranking to be a strict ordering of candidates. While our findings hold under this restriction, they also hold when it is removed. It is this latter feature of our analysis that makes it possible to reinterpret our model in the context of _intrapersonal bargaining_ and shed light on certain individual behaviors.

> 4A hyper-preference relation is a ranking of the rankings over a set of alternatives. Hyper-preferences are generally used when it is important to assess an individual’s utility from a social state or consensus by evaluating how close their preferences are to this state (Sen, 1974).

> 5Our class of tie-breaking rules embeds well-known rules, including the alphabetical tie-breaking rule, the status quo rule, the chairperson’s vote tie-breaking rule, and the coin toss tie-breaking rule. Importantly, in our model, the tie-breaking rule is _state-dependent_ , meaning that a tie on a pair of candidates { _a,b_ } and a tie on another pair of candidates { _c,d_ } may be broken using very different rules; for example, the tie on { _a,b_ } may be broken using the alphabetical tie-breaking rule whereas the tie on { _c,d_ } may be broken using the coin toss tie-breaking rule. Our class, therefore, generalizes the class of uniform rules.

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associated with the social ranking of each pair of candidates. This expected utility is generalized at the social ranking level by taking the weighted average of the expected utility over the set of all pairs of candidates, with the weight distribution being informative of the extent to which the voter values each pair of candidates.

We uncover three main findings. First, we show that a Nash equilibrium in pure strategies always exists (Theorem 1). Second, we show that truthtelling is always a Nash equilibrium, which means that no voter has an incentive to manipulate the outcome of a ranked-choice election by submitting a ballot that is different from his or her true preferences (Theorem 1). Third, we find that truthtelling is Pareto-efficient when there are exactly two candidates and voters’ true preferences do not include ties between the candidates (Theorem 3). When there are more than two candidates, we show that truthtelling is Pareto-efficient if voters have strict and neutral preferences, ties are broken using a fair coin toss, and the social ranking evaluated at the profile of true preferences is complete (Theorem 4).<sup>6</sup> All these results hold under a general class of voting rules— of which majority rule is a particular case. In addition, we provide an extension of our model to voluntary majority voting rules with costly participation. This particular application is reminiscent of the model analyzed in Börgers (2004), though we address a different question. We find that truthtelling is a Nash equilibrium if and only if participation costs are not too high and the election is tight enough (Theorem 5). This result highlights the need to lower the cost of participating in popular elections in polarized societies.

As noted earlier, an appeal of our study is that the findings hold regardless of the nature of the social ranking. In particular, they apply both when the social ranking is acyclic (or linear (Arrow (1951)) and when it is cyclic. It is well known that whether the social ranking is acyclic or cyclic depends both on the nature of the voting rule and the extent of heterogeneity in individual preferences.<sup>7</sup> Cyclic social rankings have been traditionally viewed as undesirable (Arrow, 1951), although it is now argued that it may provide insights into why political parties tend to rotate in power in polarized societies, even under the assumption that the distribution of voters’ preferences is stable over time (Regenwetter & Grofman, 1998). The classical approach to addressing cyclic social

> 6The assumption regarding the completeness of the social ranking is similar to the assumption imposed by Arrow (1951), but ours is a bit weaker.

> 7For instance, majority rule may produce a Condorcet cycle, but not for all profiles of individual preferences.

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rankings has been to either restrict the set of individual preferences or restrict the set of voting rules (see, e.g., Black (1948); Dasgupta and Maskin (2008, 2020)). We have good news: truthful voting remains a Nash equilibrium for the classes of preferences and voting rules that produce acyclic (or even transitive) social outcomes (Theorem 2, Corollary 1).

We reinterpret our basic model in the context of _intrapersonal bargaining_ to shed light on the occurrence of cyclic individual choice, which constitutes a violation of rationality. Following the psychological literature of “choice under conflicts” or “reason-based choice” (Simonson, 1989; Shafir, Simonson, & Tversky, 1993), economists studying intrapersonal bargaining problems generally assume that an individual facing a choice problem has multiple rational inner selves holding preferences that may be conflicting (Bernheim and Rangel (2004); Benhabib and Bisin (2005); Eliaz and Spiegler (2006); Fudenberg and Levine (2006); De Clippel and Eliaz (2012); Ambrus and Rozen (2015)). In this literature, several reasons are advanced to explain systematic violations of rationality.<sup>8</sup> Our analysis complements this line of research by offering a non-cooperative solution to an intrapersonal bargaining problem where the decision-maker’s rational selves can be viewed as different criteria for ranking the alternatives in the choice set. We then show that cyclic individual choice can emerge as the _equilibrium_ outcome of interactions between these selves, especially if they hold conflicting preferences or priorities over the choice menu. Our analysis explains why such irrational choices can exist in reality (see, e.g., Section 6), especially when the number of selves is small.<sup>9</sup> Our findings imply that ranked voting addresses several shortcomings of traditional voting systems such as plurality and range voting<sup>10</sup> , used in most presidential elections, parliamentary institutions, and international organizations. Vote-splitting and manipulability are the most well-known limitations of these traditional systems. For example, evidence suggests that vote-splitting has tainted the outcome of many elections, including the U.S. presidential elections in 1992 and 2002, the 2016 primary elections of the

> 8For a comprehensive survey on bounded rationality in choice theory, interested readers can refer to de Clippel and Rozen (2023).

> 9The occurrence of a cyclic social ranking is rare when the number of voters (or selves) is large (Balasko & Crès, 1997).

> 10These voting rules consist of selecting the alternative, if any, which receives the most support from the electoral body. The support can be in terms of the number of “yes” votes (plurality) or a score factor (range voting). A tie-breaking rule is generally applied in case there are many candidates who receive the same support.

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Republican Party in the U.S., and the French presidential election of 2017 (Maskin & Sen, 2017). A voting rule is vulnerable to manipulation when voters have an incentive to cast a strategic vote, that is, a vote that is different from their true preferences. As a result, the outcome of such elections will be distorted. This is a major concern that goes against the very purpose of voting, which is to select the candidate who best reflects voters’ true preferences.<sup>11</sup> In light of these limitations, majority rule, introduced by de Caritat and De Condorcet (1785), has proven itself as the best replacement for these voting rules (Dasgupta & Maskin, 2020), particularly with regard to the vote-splitting issue. The robustness of majority rule to vote-splitting is essentially attributable to its format (Dasgupta & Maskin, 2008, 2020). However, whether this format is not enough to fix the problem of its vulnerability to strategic manipulation is an open question. Our paper builds on these studies, but we significantly depart from them by proposing a political competition model that is compatible with truthtelling regardless of whether we restrict preferences (Black, 1948; Dasgupta & Maskin, 2008, 2020) or not. We therefore address the question on the vulnerability of majority-like rules to manipulation using a _non-cooperative_ approach. We also show that truthtelling is Pareto-efficient under some mild conditions.

This paper relates to the literature that seeks to understand voter behavior. While ranked voting has been studied, we believe that our paper is the first that develops a non-cooperative model of this electoral system under a very large class of voting and aggregation rules. Some contributions to this line of research include Cain (1978), AustenSmith (1987), Austen-Smith and Banks (1996), Dellis (2009), Freixas and Parker (2015), Acharya and Meirowitz (2017), and Pongou and Tchantcho (2021). In particular, the latter study provides a necessary and sufficient condition under which truthful voting is a Nash equilibrium in a round-robin political tournament. We diverge from early studies in our scope, analysis, and results. Our analysis rationalizes truthful voting behavior in ranked voting without imposing any condition on preferences. In addition, we investigate the efficiency of truthtelling in our context. Our study also complements Börgers (2004) by identifying a necessary and sufficient condition under which truthtelling is a Nash equilibrium in majoritarian elections with costly voluntary participation.

> 11Theoretical evidence of the vulnerability of plurality and range voting rules to strategic manipulation can be found in Gibbard (1973), Satterthwaite (1975), and Sanver (2009).

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By showing that truthtelling is an equilibrium outcome of ranked voting irrespective of preference structure, we view our analysis as establishing a _possibility_ result. In this sense, we contribute to a recent literature that has discovered other possibility results (see, e.g., Maskin (2020), Dasgupta and Maskin (2008, 2020), Bahel and Sprumont (2020)). This literature tends to contrast with classical impossibility results such as those obtained by Arrow (1951), Gibbard (1973), and Satterthwaite (1975). Some recent studies argue that some of the assumptions in these pioneering works are too strong and so can be relaxed. Relaxing them has indeed led to possibility results (Maskin, 2020). While we view our study as contributing to this latter trend, our setting, analysis, and results are completely different. Our focus is on ranked voting and its equilibrium and efficiency properties in a non-cooperative context.

The rest of this paper is organized as follows. Section 2 defines the model of rankedchoice election. Section 3 formalizes a ranked-choice election as a strategic form game. In Section 4, we develop our main finding. An extension of our model to voluntary majority rule with costly voting is provided in Section 5. We reinterpret our model in the context of intrapersonal bargaining in Section 6. Section 7 concludes.

# **2 A model of ranked-choice election**

A ranked-choice election is a system of political competition between a finite set of candidates under a given voting rule, where each voter submits a ballot consisting of their ranking of the candidates. It is described as a list ( _N,X,_ ( _R_<sup>_i_</sup> ) _i_ ∈ _N ,W,F_ ), where _N_ = {1 _,_ 2 _,...,n_ } is a finite set of voters, _X_ = { _x_ 1 _,x_ 2 _,...,xm_ } is a finite set of candidates or alternatives, ( _R_<sup>_i_</sup> ) _i_ ∈ _N_ is a profile of voters’ preferences over candidates, _W_ is a voting rule, and _F_ is an aggregation rule that combines individual votes to produce the final ranking of the candidates.

The following subsections aim to clarify the key elements defining the model of ranked-choice election.

## **2.1 Individual preferences**

In a ranked-choice election, each voter _i_ ∈ _N_ has preferences _R_<sup>_i_</sup> over candidates. _R_<sup>_i_</sup> is assumed to be a binary relation over _X_ that is reflexive, transitive, and complete.

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Given any two candidates _xp_ and _xq_ , _R_ {<sup>_i_</sup> _xp,xq_ }<sup>denotestherestrictionof</sup><sup>_Ri_tothepair</sup> of candidates { _xp,xq_ }. A voter _i_ may strictly prefer _xp_ to _xq_ , in which case we have _R_<sup>_i_ormaybeindifferentbetweenthetwocandidates,inwhichcasewehave</sup> { _xp,xq_ }<sup>=</sup><sup>_xpxq_,</sup> _R_ {<sup>_i_</sup> _xp,xq_ }<sup>=(</sup><sup>_xpxq_).Itfollowsthat</sup><sup>_R_</sup> {<sup>_i_</sup> _xp,xq_ }<sup>∈{</sup><sup>_xpxq,xqxp,_(</sup><sup>_xpxq_)}forany</sup><sup>_xp,xq_∈</sup><sup>_X_.A</sup> collection of individual preferences ( _R_<sup>_i_</sup> ) _i_ ∈ _N_ is called a preference profile and denoted by _R_ = ( _R_<sup>_i_</sup> ) _i_ ∈ _N_ .

# **2.2 Voting rule**

The classical model of ranked-choice election is generally studied under majority rule (Dasgupta and Maskin (2008), Dasgupta and Maskin (2020)). We use a class of rules that is much larger; it is the class of voting rules with abstention, where voters may have unequal voting rights and may abstain.

Each voter casts a ballot that consists of a ranking of the candidates. Let _x_ and _y_ be two candidates. A voter may rank _x_ before _y_ , rank _y_ before _x_ , or rank them equally, which means abstaining. Let _T_ 1 be the set of voters who rank _x_ before _y_ , _T_ 2 the set of voters who abstain, and _T_ 3 the set of voters who rank _y_ before _x_ . The restriction of the voting profile to _x_ and _y_ can be summarized by _T_ = ( _T_ 1 _,T_ 2 _,T_ 3), where the sets _T_ 1, _T_ 2, and _T_ 3 are mutually disjoint and cover _N_ . _T_ is called a tripartition of _N_ and can be viewed as a function T from _N_ to {−1 _,_ 0 _,_ 1} such that _T_ 1 = _T_<sup>−1</sup> {1}, _T_ 2 = _T_<sup>−1</sup> {0} and _T_ 3 = _T_<sup>−1</sup> {−1}. We denote by T ( _N_ ) the set of all tripartitions of _N_ . For _T,T_<sup>′</sup> ∈T ( _N_ ), we write _T_ ⊆ _T_<sup>′</sup> if _T_ can be transformed into _T_<sup>′</sup> by shifting one or more voters to higher levels of approval, and _T_ ⊂ _T_<sup>′</sup> if _T_ ⊆ _T_<sup>′</sup> and _T_ ≠ _T_<sup>′</sup> .

**Definition 1 :** _A voting rule for N is described by a subset W of_ T ( _N_ ) _such that the following conditions hold._

_1._ ( _N,_ ∅ _,_ ∅) ∈ _W ._ _2. For all tripartitions T,T_<sup>′</sup> ∈T ( _N_ ) _such that T_ ⊆ _T_<sup>′</sup> _, if T_ ∈ _W , then T_<sup>′</sup> ∈ _W ._ _3._ ( _T_ 3 _,T_ 2 _,T_ 1) ∉ _W whenever_ ( _T_ 1 _,T_ 2 _,T_ 3) ∈ _W ._ Elements of _W_ are called winning tripartitions, and elements not in _W_ are called losing tripartitions. The first condition is a natural condition that requires a candidate to be socially ranked above a candidate if he or she receives the support of all the voters.

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The second condition is a monotonicity condition that requires any tripartition containing a winning tripartition to be winning. In other words, if a tripartition is winning, then any tripartition that results from any voter increasing his or her level of approval is also winning. The last condition is a stability condition that rules out the possibility of two winning disjoint tripartitions supporting different candidates and creating unnecessary social instability.

In the literature, the voting rule _W_ is referred to as a voting rule with abstention. This class of rules was introduced by Rubinstein (1980), and it generalizes several classes of voting rules, including majority rule (Dasgupta & Maskin, 2008) and the class of simple games where voters can only vote yes and no (Shapley, 1962).<sup>12</sup>

# **2.3 Aggregation rule and social ranking**

An aggregation rule is defined as a function from the set of individual preferences to the set of probability distributions over the set of complete and antisymmetric binary relations over the set of candidates. Below, we clarify this definition.

In a ranked-choice election, voters’ reported preferences are aggregated to produce a social ranking of candidates, which is a binary relation.<sup>13</sup> We will need the following notation: denote by P( _X_ ) = {{ _x,y_ } ∶ _x,y_ ∈ _X_ } = {{ _xp,xq_ } ∶ 1 ≤ _p_ < _q_ ≤ _m_ } the set of pairs of candidates.

Let ( _N,X,_ ( _R_<sup>_i_</sup> ) _i_ ∈ _N ,W,F_ ) be a ranked-choice election. A reported vote or a voting _m_ <u>(</u> _m_ −1) strategy for an individual _i_ ∈ _N_ is given by a sequence _si_ = ( _s_<sup>_pq_</sup> _i_<sup>)</sup> 1≤ _p_ < _q_ ≤ _m_<sup>∈{−1</sup><sup>_,_0</sup><sup>_,_1}</sup> 2 , where for all 1 ≤ _p_ < _q_ ≤ _m_ , _s_<sup>_pq_</sup> _i_ equal 1, -1, or 0 if individual _i_ has respectively reported preferring _xp_ over _xq_ , _xq_ over _xp_ , or is indifferent between the two candidates. Following the literature, we interpret indifference as abstention (Freixas & Zwicker, 2009). A voting

> 12Some contributions to the literature on voting rules with abstention include, among others, Tchantcho et al. (2008), and Freixas and Zwicker (2009).

> 13As noted earlier, Arrow (1951) requires that the social ranking be transitive. Other studies have required acyclicity instead. Our main findings hold regardless of whether or not we restrict the preference domain and voting rules to those that produce transitive or acyclic social rankings. As argued later, cyclic social rankings are not uninteresting, even in a political setting, as they may partially explain the cycle of policies and governments (even assuming that the distribution of preferences does not change over time). We also use cyclic social rankings to explain some behavioral patterns that systematically violate rationality. For these reasons, we impose no strong restrictions on the domain of preferences and the set of voting rules.

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profile is a sequence _s_ = ( _si_ ) _i_ ∈ _N_ of vote strategies for all individuals. The truthful vote for an individual _i_ ∈ _N_ , denoted by _s_<sup>∗</sup> _i_<sup>(</sup><sup>_R_) = (</sup><sup>_s_∗</sup> _i_<sup>_pq_</sup> ( _R_ ))1≤ _p_ < _q_ ≤ _m_ ), is defined for all 1 ≤ _p_ < _q_ ≤ _m_ by: Let _s_ = ( _si_ ) _i_ ∈ _N_ be a voting strategy. For all _p,q_ with _p_ < _q_ , we denote by _N_<sup>1</sup> ( _s_<sup>_pq_</sup> ) = { _i_ ∈ _N_ ∶ _s_<sup>_pq_</sup> _i_<sup>= 1}thesetofindividualswhohaveranked</sup><sup>_xp_above</sup><sup>_xq_inthevotingstrategy</sup> _s_ , by _N_<sup>−1</sup> ( _s_<sup>_pq_</sup> ) = { _i_ ∈ _N_ ∶ _s_<sup>_pq_</sup> _i_<sup>= −1}thesetofindividualswhohaveranked</sup><sup>_xq_above</sup><sup>_xp_,</sup> and by _N_<sup>0</sup> ( _s_<sup>_pq_</sup> ) = { _i_ ∈ _N_ ∶ _s_<sup>_pq_</sup> _i_<sup>= 0}thesetofvoterswhoareindifferentbetween</sup><sup>_xp_and</sup><sup>_xq_.</sup> The social ranking under the voting rule _W_ induced by a voting strategy _s_ , which we denote by _F_ ( _s_ ), is given by: where _xpxq_ ∈ _F_ ( _s_ ) means that _xp_ is socially ranked above _xq_ , so he or she is ranked above _xq_ in the social ranking _F_ ( _s_ ); and _xp_ ∗ _xq_ ∈ _F_ ( _s_ ) means that _xp_ and _xq_ are socially incomparable, that is, neither of the two candidates is socially preferable to the other. Note that the third condition in our definition of _W_ rules out the possibility of social indifference between two candidates _xp_ and _xq_ . In fact, we cannot simultaneously have ( _N_<sup>1</sup> ( _s_<sup>_pq_</sup> ) _,N_<sup>0</sup> ( _s_<sup>_pq_</sup> ) _,N_<sup>−1</sup> ( _s_<sup>_pq_</sup> )) ∈ _W_ and ( _N_<sup>1</sup> ( _s_<sup>_qp_</sup> ) _,N_<sup>0</sup> ( _s_<sup>_qp_</sup> ) _,N_<sup>−1</sup> ( _s_<sup>_qp_</sup> )) = ( _N_<sup>−1</sup> ( _s_<sup>_pq_</sup> ) _,N_<sup>0</sup> ( _s_<sup>_pq_</sup> ) _,N_<sup>1</sup> ( _s_<sup>_pq_</sup> )) ∈ _W_ .

We assume a probabilistic tie-breaking rule in case of social incomparability between two candidates. More precisely, if _xp_ and _xq_ are socially incomparable (i.e., _xp_ ∗ _xq_ ∈ _F_ ( _s_ )), then _xp_ is socially ranked above _xq_ with some probability Λ _pq_ and _xq_ is socially ranked above _xp_ with probability Λ _qp_ = 1 − Λ _pq_ . This tie-breaking rule generalizes the lexicographical tie-breaking rule where the probability Λ _pq_ of _xp_ is always equal to 1 whenever _p_ < _q_ .

**Example 1 :** _Consider a ranked-choice election where N_ = {1 _,_ 2 _,_ 3} _, X_ = { _x_ 1 _,x_ 2 _,x_ 3} _, and the voting rule is simple majority rule. Under this rule, a tripartition_ ( _T_ 1 _,T_ 2 _,T_ 3) _is_

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_winning if and only if_ ∣ _T_ 1 ∣>∣ _T_ 3 ∣ _. Voters’ preferences are defined as follows:_ _It follows that the social ranking induced by the truthful vote s_<sup>∗</sup> ( _R_ ) _over is given by: F_ ( _s_<sup>∗</sup> ( _R_ )) = { _x_ 1 _x_ 2 _,x_ 1 ∗ _x_ 3 _,x_ 2 _x_ 3} _Therefore, the outcomes of the social ranking are the following: x_ 1 _is ranked above x_ 2 _, x_ 2 _is ranked above x_ 3 _, and x_ 1 _and x_ 3 _are socially incomparable. A probabilistic tie-breaking rule can consist of a fair coin toss, which gives an equal probability to either x_ 1 _or x_ 3 _being socially ranked above the other. It may also consist of choosing the candidate that is given priority following a certain criterion (determined, for example, by the alphabetical order of names, the preference of the election chairperson,_<sup>14</sup> _candidate seniority, or a random or non-random order chosen before the vote takes place). Note that because the tie-breaking rule is state-dependent in our model, a different rule may be used for each pair of candidates._

# **3 A ranked-choice election as a strategic form game**

In this section, we model a ranked-choice election as a strategic form game. Before diving into it, we first need to recall some preliminary definitions.

A strategic form game is a triplet ( _N,_ ( _Si_ ) _i_ ∈ _N ,_ ( _ui_ ) _i_ ∈ _N_ ), where _N_ is a finite set of players, _Si_ is a non-empty set of strategies of player _i_ ∈ _N_ , and _ui_ ∶∏<sup>→Risthe</sup> _j_ ∈ _N_<sup>_Sj_</sup> payoff function of player _i_ ∈ _N_ . We denote by _S_ = ∏<sup>thesetofstrategyprofilesand</sup> _j_ ∈ _N_<sup>_Sj_</sup> _s_ = ( _si,s_ − _i_ ) ∈ _S_ the strategy profile where player _i_ chooses the strategy _si_ and the other players choose the strategies _s_ − _i_ .

A strategy profile _s_<sup>∗</sup> ∈ _S_ is a pure strategy Nash equilibrium of a game ( _N,_ ( _Si_ ) _i_ ∈ _N ,_ ( _ui_ ) _i_ ∈ _N_ ) if, for all _i_ ∈ _N_ , _ui_ ( _s_<sup>∗</sup> _i_<sup>_,s_</sup> −<sup>∗</sup> _i_<sup>) ≥</sup><sup>_ui_(</sup><sup>_si,s_</sup> −<sup>∗</sup> _i_<sup>),forall</sup><sup>_si_∈</sup><sup>_Si_.Thatis,forall</sup><sup>_i_∈</sup><sup>_N_,wehave</sup> A strategy profile _s_<sup>∗</sup> ∈ _S_ is Pareto-efficient if there is no other strategy profile _s_ ∈ _S_ such that _ui_ ( _s_ ) ≥ _ui_ ( _s_<sup>∗</sup> ) for all _i_ ∈ _N_ with _ui_ ( _s_ ) > _ui_ ( _s_<sup>∗</sup> ) for some _i_ .

> 14In the U.S. Senate, the tie-breaker is the Vice-President.

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# **3.1 Utility function**

Let ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ) be a ranked-choice election. For a given voting strategy profile _s_ , each individual _i_ gains utility from the social ranking _F_ ( _s_ ), measured by how close the latter is to his or her preference _R_<sup>_i_</sup> . Such a utility is equivalent to any decreasing function of the distance between _R_<sup>_i_</sup> and _F_ ( _s_ ), capturing voter _i_ ’s dissatisfaction with the social ranking _F_ ( _s_ ).

Inspired by the intuition underlying the well-known Kemeny distance Kemeny (1959) defined on the set of asymmetric weak orders,<sup>15</sup> and more in line with the locally generated distance introduced in Diffo Lambo, Tchantcho, and Moulen (2012) and defined on the set of all binary relations, the level of dissatisfaction of voters with respect to the social ranking is defined over the set P( _X_ ). More precisely, given a pair { _x,y_ } between candidates _x_ and _y_ , the level of dissatisfaction of a voter _i_ with respect to the social ranking _F_ ( _s_ ) is specified across three possible scenarios. First, the individual is fully unsatisfied, _R_<sup>_i_and</sup><sup>_F_(</sup><sup>_s_){</sup><sup>_x,y_} =</sup><sup>_yx_;inthiscasethelevelofdissatisfactionisnormalizedto1.</sup> { _x,y_ }<sup>=</sup><sup>_xy_</sup> Second, the individual is fully satisfied, _R_ {<sup>_i_</sup> _x,y_ }<sup>=</sup><sup>_xy_=</sup><sup>_F_(</sup><sup>_s_){</sup><sup>_x,y_};inthiscasethelevelof</sup> dissatisfaction is equal to 0. Lastly, we have an intermediate case between the two previous scenarios, which corresponds to the situation where the voter _i_ is indifferent between the two candidates ( _R_ {<sup>_i_</sup> _x,y_ }<sup>= (</sup><sup>_xy_))andthesocialrankingranksthetwocandidatesinany</sup> way ( _F_ ( _s_ ){ _x,y_ } ∈{ _xy,yx_ }). In this case, the level of dissatisfaction is equal to any fixed real number _αi_ ∈]0 _,_ 1]. Setting _α_ = ( _αi_ ) _i_ ∈ _N_ ∈]0 _,_ 1]<sup>_n_</sup> as a profile of individuals’ parameters, we denote by _d_<sup>{</sup> _α_<sup>_x,y_}</sup> ( _R_<sup>_i_</sup> _,F_ ( _s_ )) the dissatisfaction level of voter _i_ over the pair { _x,y_ } with the social ranking _F_ ( _s_ ) induced by the voting strategy profile _s_ . Table 1 summarizes our previous discussion.

|_R_<sup>_i_</sup>|_xy_|(_xy_)|_xy_|
|---|---|---|---|
|_F_(_s_)|_xy_|_xy_ or _yx_|_yx_|
|_d_<sup>{</sup><sup>_x,y_}</sup><br>_α_<br>(_R_<sup>_i_</sup>_,F_(_s_))|0|_αi_|1|

**Table 1:** Individual i’s dissatisfaction level with the social ranking F(s) induced by the voting strategy profile _s_ relative to a pair of candidates _x_ and _y_ . _α_ = ( _αi_ ) _i_ ∈ _N_ ∈]0 _,_ 1]<sup>_n_</sup> is a profile of individuals’ parameters.

> 15A weak order is a complete, reflexive, and transitive binary relation.

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Before defining the utility that each voter derives from a voting strategy profile, we need to introduce some notations. For any strategy _s_ , we denote by _F_ ( _s_ ) and _F_ <u>(</u> _s_ <u>)</u> the complete relation defined for all pair of candidates { _xp,xq_ } by: Both _F_ ( _s_ ) and _F_ <u>(</u> _s_ <u>)</u> agree with the social ranking _F_ ( _s_ ) over any pair of candidates whenever the society is able to rank them. In the case of social incomparability between a pair of candidates, the former (the latter) ranks the candidate with the lower (higher) subscript first (second).

Also, let Λ _pq_ ( _F_ ( _s_ )) denote the function defined by: Given the measure of dissatisfaction _dα_ , we measure the utility that a voter _i_ derives from a voting strategy profile _s_ = ( _s_ 1 _,s_ 2 _,...sn_ ) by the function: _Ui_<sup>_pq_(</sup><sup>_s_) represents the expected utility that voter</sup><sup>_i_derives from the social ranking of the</sup> candidates _xp_ and _xq_ . It incorporates both situations where the society is able and unable to rank the two candidates. For instance, if for a given _s_ , the society ranks _xp_ above _xq_ , i.e., _xpxq_ ∈ _F_ ( _s_ ), then _Ui_<sup>_pq_(</sup><sup>_s_) = 1−</sup><sup>_d_</sup> _α_<sup>{</sup><sup>_xp,xq_}</sup> ( _R_<sup>_i_</sup> _,F_ ( _s_ )). In this case, _dα_<sup>{</sup><sup>_xp,xq_}</sup> ( _R_<sup>_i_</sup> _,F_ ( _s_ )) measures how far individual _i_ ’s preference over the two candidates is from the social ranking. It is interpreted as the level of dissatisfaction (or disutility) that individual _i_ derives from the social ranking of the candidates _xp_ and _xq_ . Since the maximum possible value for 13
this measure of dissatisfaction is normalized to 1, it is natural to measure the utility the individual derives from the given social ranking by _Ui_<sup>_pq_(</sup><sup>_s_) = 1 −</sup><sup>_d_</sup> _α_<sup>{</sup><sup>_xp,xq_}</sup> ( _R_<sup>_i_</sup> _,F_ ( _s_ )).<sup>16</sup> Our study does not include negative utilities since each individual has the choice to participate or not in the ranked-choice election. We assume that an individual is not penalized if he or she chooses to not participate in a vote.

On the other hand, if for a given _s_ , the society fails to rank the candidates _xp_ and _xq_ , then the probabilistic tie-breaking rule applies, and the utility is defined by Equation (2). In this case, Λ _pq_ represents the probability that _xp_ is socially ranked above _xq_ when social incomparability between the two candidates occurs. Λ _pq_ can be determined by nature, candidates past experience, or by lobby groups. It is common knowledge.

We assume that voters may value different pairs of candidates differently. This means that how _xp_ and _xq_ are socially compared might be more or less meaningful to an individual than how _xs_ and _xt_ are socially compared, with { _xp,xq_ } and { _xs,xt_ } representing different pairs of candidates. _θi_<sup>_pq_</sup> is a non-negative real number representing the weight voter _i_ puts on the social ranking of candidates _xp_ and _xq_ . Therefore, it is logical to define the utility _Ui_<sup>_d_(</sup><sup>_s_)thatavoter</sup><sup>_i_derivesfromthesocialranking</sup><sup>_F_(</sup><sup>_s_)astheweightedsumofutilities</sup> _Ui_<sup>_pq_(</sup><sup>_s_)overthesetofcandidatepairsP(</sup><sup>_X_).Finally,wesaidthatvotershaveneutral</sup> preferences if they value each pair of candidates similarly (although they might have very different preferences), that is _θi_<sup>_pq_=</sup><sup>_θ_</sup> _j_<sup>_pq_</sup> for all _i,j_ ∈ _N_ and _xp,xq_ ∈ _A_ .

# **3.2 Strategic form game**

We now model a ranked-choice election as a strategic form game. Assume that in a ranked-choice election ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ), each voter _i_ ’s utility is measured by a utility function _Ui_<sup>_dα_</sup> as defined in Equation (2). We associate to this model the strategic form game Γ _R_ = ( _N,_ ( _Si_ ) _,_ ( _Ui_<sup>_dα_)) = (</sup><sup>_N,S,U_)where:</sup>

- for all _i_ ∈ _N_ , _Si_ = { _si_ = ( _s_<sup>_pq_</sup> _i_<sup>)1≤</sup><sup>_p_<</sup><sup>_q_≤</sup><sup>_m_:</sup><sup>_spq_</sup> _i_<sup>∈{−1</sup><sup>_,_0</sup><sup>_,_1}},isthestrategysetofindividual</sup> _i_ ; and

> 16Our main result will continue to hold if the utility is any decreasing function of the dissatisfaction measure _dα_ .

14
is the payoff function of individual _i_ .

# **4 Main findings: Equilibrium existence, truthtelling, and efficiency**

In this section, we state our main findings. First, we show that a ranked-choice election always admits a pure-strategy Nash equilibrium. Second, we show that truthtelling is always a Nash equilibrium, meaning that no voter has an incentive to cast a ballot that misrepresents their true preferences. Third, we show that truthtelling is always Pareto-efficient when there are exactly two candidates and voters’ preferences do not have ties.

## **4.1 Equilibrium existence and truthtelling**

In this section, we study equilibrium existence in a ranked-choice election and truthtelling behavior. Our main result below shows that truthful voting is always a Nash equilibrium of the game Γ _R_ .

**Theorem 1 :** _Let_ ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ) _be a ranked-choice election and α_ ∈]0 _,_ 1]<sup>_n_</sup> _a profile of individual parameters. A Nash equilibrium in pure strategies exists. In addition, the truthful voting profile s_<sup>∗</sup> ( _R_ ) _is a Nash equilibrium for the associated strategic form game_ Γ _R_ = ( _N,X,U_<sup>_dα_</sup> ) _._

For expositional reasons, the proofs of Theorem 1 and all subsequent results are presented in the appendix. Remark that Theorem 1 holds under general conditions. We do not need to impose any condition on the parameters _αi_ , _i_ ∈ _N_ , or the probabilities Λ _pq_ , 1 ≤ _p_ < _q_ ≤ _n_ , for a candidate _xp_ to be socially ranked above another candidate _xq_ when both are socially incomparable.

The example below illustrates Theorem 1.

**Example 2 :** _Consider a ranked-choice election_ ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ) _where N_ = {1 _,_ 2 _,_ 3} _, A_ = { _x_ 1 _,x_ 2} _, and the voting rule is simple majority rule, where a tripartition_ ( _T_ 1 _,T_ 2 _,T_ 3) _is winning if and only if_ ∣ _T_ 1 ∣>∣ _T_ 3 ∣ _. Let R be a preference profile defined as follows: R_<sup>1</sup> = ( _x_ 1 _x_ 2) _, R_<sup>2</sup> = _x_ 1 _x_ 2 _and R_<sup>3</sup> = _x_ 2 _x_ 1 _._

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_The truthful voting profile is s_<sup>∗</sup> ( _R_ ) = (0 _,_ 1 _,_ −1) _and the corresponding social ranking is F_ ( _s_<sup>∗</sup> ( _R_ )) = { _x_ 1 ∗ _x_ 2} _. The utilities of voters from this social ranking at the preference profile R are :_ _- If voter 1 deviates from_ 0 _to_ 1 _, then F_ ((1 _,_ 1 _,_ −1)) = { _x_ 1 _x_ 2} _and his or her utility is u_ 1((1 _,_ 1 _,_ −1)) = 1− _α_ 1 _. If he or she deviates from_ 0 _to_ −1 _, then F_ ((−1 _,_ 1 _,_ −1)) = { _x_ 2 _x_ 1} _and his utility is u_ 1((−1 _,_ 1 _,_ −1)) = 1 − _α_ 1 _. Hence, voter 1 has no incentive to deviate from his or her true preferences._ _- If voter 2 deviates from_ 1 _to either_ 0 _or_ −1 _, then F_ ((0 _,_ 0 _,_ −1)) = _F_ ((0 _,_ −1 _,_ −1)) = { _x_ 2 _x_ 1} _and his or her utility is u_ 2((0 _,_ 0 _,_ −1)) = _u_ 2((0 _,_ −1 _,_ −1)) = 0 _. Hence, voter 2 has no incentive to deviate from his or her true preferences._ _- If voter 3 deviates from_ −1 _to either_ 1 _or_ 0 _, then F_ ((0 _,_ 1 _,_ 1)) = _F_ ((0 _,_ 1 _,_ 0)) = { _x_ 1 _x_ 2} _and his or her utility is u_ 3((0 _,_ 1 _,_ 1)) = _u_ 2((0 _,_ 1 _,_ 0)) = 0 _. Hence, voter 3 has no incentive to deviate from his or her true preferences._ _It follows that no voter has an incentive to deviate from their truth preferences. Therefore, truthtelling is a pure-strategy Nash equilibrium for the associated strategic form game_ Γ _R_ = ( _N,Z,U_<sup>_dα_</sup> ) _._ It is worth mentioning that the strategic form game associated to a ranked-choice election may admit other Nash equilibria different from the truthful vote. For instance, the voting profile (1 _,_ 1 _,_ −1) is another Nash equilibrium of the game defined in the previous example.

# **4.2 Acyclic and cyclic social rankings**

Our analysis has an implication for truthtelling behavior when the social ranking is acyclic or cyclic. Whether the social ranking is acyclic or cyclic depends on both the nature of the voting rule and the extent of heterogeneity in voters’ preferences. Consequently, scholars have naturally resorted to preference domain restriction (see, e.g., Black (1948); Dasgupta and Maskin (2008, 2020) or have instead restricted the set of voting rules (Arrow, 1951) to address the issue of cyclic social outcomes. One important question 16
is whether our finding that truthtelling is a Nash equilibrium continues to hold when preferences and voting rules are restricted such that the social ranking is acyclic or even transitive. Fortunately, the answer is affirmative. In fact, the proof of Theorem 1 does not rely on any assumption on the internal structure of individual preferences or the nature of the voting rule. Our finding would be significantly weak if it depended on whether cyclic social rankings are allowed in our setting. But this is not the case. This is reassuring, even if in reality, the occurrence of a cyclic social ranking is a rare event, especially in large electorates. Empirical studies indeed find that the probability of a Condorcet paradox is negligible (see, e.g., Tideman and Plassmann (2011)).

More formally, for any ranked-choice election, ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ), let R _W_ be the domain of individual preferences such that the social ranking is acyclic, given the voting rule _W_ .<sup>17</sup> Similarly, let W _R_ be the collection of voting rules such that the social ranking is acyclic given the preference profile _R_ = ( _Ri_ ) _i_ ∈ _N_ . The following result holds:

**Theorem 2 :** _Let_ ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ) _be a ranked-choice election such that R_ = ( _Ri_ ) _i_ ∈ _N_ ⊆ R _W or W_ ∈W _R, and α_ ∈]0 _,_ 1]<sup>_n_</sup> _a profile of individual parameters. A Nash equilibrium in pure strategies exists. In addition, the truthful voting profile s_<sup>∗</sup> ( _R_ ) _is a Nash equilibrium for the associated strategic form game_ Γ _R_ = ( _N,X,U_<sup>_dα_</sup> ) _._

In particular, if the voting rule is the majority rule, then following Dasgupta and Maskin (2008, 2020), let R be the set of linear orderings over _X_ that satisfies the _noCondorcet-cycle property_ , that is for each triple { _x_ 1 _,x_ 2 _,x_ 3} ⊆ _X_ , there exist no orderings, _R,R_<sup>′</sup> _,R_<sup>′′</sup> in R such that _R_ { _x_ 1 _,x_ 2 _,x_ 3} = _x_ 1 _x_ 2 _x_ 3 _,R_ {<sup>′</sup> _x_ 1 _,x_ 2 _,x_ 3}<sup>=</sup><sup>_x_2</sup><sup>_x_3</sup><sup>_x_1</sup><sup>_,R_</sup> {<sup>′′</sup> _x_ 1 _,x_ 2 _,x_ 3}<sup>=</sup><sup>_x_3</sup><sup>_x_1</sup><sup>_x_2</sup><sup>_._It</sup> follows that the social ranking of the ranked-choice election ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ), where _W_ is the majority rule and _R_ = ( _Ri_ ) _i_ ∈ _N_ ⊆R is acyclic or produce no Condorcet cycle. The following result is then a direct consequence of the previous one.

**Corollary 1 :** _Let_ ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ) _be a ranked-choice election where W is majority rule, R_ = ( _Ri_ ) _i_ ∈ _N_ ⊆R _, and α_ ∈]0 _,_ 1]<sup>_n_</sup> _is a profile of individual parameters. A Nash equilibrium in pure strategies exists. In addition, the truthful voting profile s_<sup>∗</sup> ( _R_ ) _is a Nash equilibrium for the associated strategic form game_ Γ _R_ = ( _N,X,U_<sup>_dα_</sup> ) _._

> 17A binary relation _R_ on _X_ is said to be acyclic if, for any _x_ 1 _,x_ 2 _,x_ 3 ∈ _X_ , we cannot have _x_ 1 _Rx_ 2 _,x_ 2 _Rx_ 3, and _x_ 3 _Rx_ 1.

17
We note that while cyclic social rankings have traditionally been viewed as undesirable, others have argued that their occurrence provides partial insights into why policies and governments tend to rotate (Regenwetter & Grofman, 1998). For preference domains and voting rules that can produce a cyclic social ranking, our main finding continues to hold. To illustrate, consider a ranked-choice election, ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ), where _N_ = {1 _,_ 2 _,_ 3}, _X_ = { _x_ 1 _,x_ 2 _,x_ 3}, and the voting rule is simple majority rule. For simplicity, assume every pairwise comparison of candidates has equal worth to each individual and that social incomparability between any two candidates is broken by a fair coin toss. Let _R_ be a preference profile defined as follows: _R_<sup>1</sup> = _x_ 1 _x_ 2 _x_ 3, _R_<sup>2</sup> = _x_ 2 _x_ 3 _x_ 1 and _R_<sup>3</sup> = _x_ 3 _x_ 1 _x_ 2. The truthful voting profile is _s_<sup>∗</sup> ( _R_ ) = ( _s_<sup>∗</sup> 1<sup>(</sup><sup>_R_)</sup><sup>_,s_∗</sup> 2<sup>(</sup><sup>_R_)</sup><sup>_,s_∗</sup> 3<sup>(</sup><sup>_R_)),where</sup><sup>_s_∗</sup> 1<sup>(</sup><sup>_R_) = (</sup><sup>_s_∗</sup> 1<sup>12</sup><sup>_,s_∗</sup> 1<sup>13</sup><sup>_,s_∗</sup> 1<sup>23) =</sup> (1 _,_ 1 _,_ 1), _s_<sup>∗</sup> 2<sup>(</sup><sup>_R_) = (−1</sup><sup>_,_−1</sup><sup>_,_1),and</sup><sup>_s_∗</sup> 3<sup>(</sup><sup>_R_) = (1</sup><sup>_,_−1</sup><sup>_,_−1).Thecorrespondingsocialrankingis</sup> _F_ ( _s_<sup>∗</sup> ( _R_ )) = { _x_ 1 _x_ 2 _,x_ 2 _x_ 3 _,x_ 3 _x_ 1}. Here, we see that truthful voting produces a Condorcet cycle. Remark that it is not profitable for any voter to deviate from his or her true preference. For instance, if voter 1 deviates from his or her true preference and casts a ballot such as (1 _,_ 1 _,_ 0) while the other two voters are truthful, then his or her utility will be<sup><u>1</u>whichissmallerthanhisutilityof</sup><sup><u>2</u>heorshevotestruthfully.</sup> 2<sup>,</sup> 3<sup>when</sup> In the example above, truthtelling is a Nash equilibrium that leads to a cyclic social ranking. This shows that a cyclic social choice can emerge as a _stable_ pattern in a political economy in which political actors and voters are rational. Assuming the distribution of preferences is stable over time, this example provides some insight into why political parties tend to rotate in power. However, there exist other Nash equilibria that do not lead to a cyclic social outcome. For example, the voting profile that consists of _s_ = ( _s_ 1 _,s_ 2 _,s_ 3), with _s_ 1 = _s_ 2 = _s_ 3 = (1 _,_ 1 _,_ 1) is a Nash equilibrium that leads to the social ranking _F_ ( _s_ ) = _x_ 1 _x_ 2 _x_ 3. So it can be argued that when a social ranking that is cyclic is not desirable, society may still coordinate on an equilibrium that is not cyclic, although this will not reflect truthful preferences. Although this result can be easily generalized to a ranked-choice election with a finite number of voters in which the voting rule is the simple majority rule, it is not clear whether it holds for any voting rule.

# **4.3 Truthtelling and efficiency**

In this section, we address the question of when truthtelling is Pareto-efficient. In the example above, we note that the truthful equilibrium _s_<sup>∗</sup> ( _R_ ) = (0 _,_ 1 _,_ −1) is Pareto optimal.

18
However, this need not be the case in general as it is shown in the following examples.

**Example 3 :** _Consider a ranked-choice election_ ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ) _where N_ = {1 _,_ 2 _,_ 3} _, A_ = { _x_ 1 _,x_ 2} _, and the voting rule is unanimity rule, where a tripartition_ ( _T_ 1 _,T_ 2 _,T_ 3) _is winning if and only if T_ 1 = _N . Let R be a preference profile defined as follows: R_<sup>1</sup> = _x_ 1 _x_ 2 _, R_<sup>2</sup> = _x_ 1 _x_ 2 _and R_<sup>3</sup> = ( _x_ 1 _x_ 2) _. The truthful voting profile is s_<sup>∗</sup> ( _R_ ) = (1 _,_ 1 _,_ 0) _and the corresponding social ranking is F_ ( _s_<sup>∗</sup> ( _R_ )) = { _x_ 1 ∗ _x_ 2} _. The utilities of voters from this social ranking at the preference profile R are :_

_The truthful voting profile is not Pareto-efficient since it is Pareto-dominated by the strategy profile s_ = (1 _,_ 1 _,_ 1) _. In fact, we have that F_ ((1 _,_ 1 _,_ 1)) = { _x_ 1 _x_ 2} _. It follows that u_ 1((1 _,_ 1 _,_ 1)) = 1 ≤ _u_ 1( _s_<sup>∗</sup> ( _R_ )) _, u_ 2((1 _,_ 1 _,_ 1)) = 1 ≤ _u_ 2( _s_<sup>∗</sup> ( _R_ )) _, and u_ 3((1 _,_ 1 _,_ 1)) = 1 − _α_ 3 = _u_ 1( _s_<sup>∗</sup> ( _R_ )) _. Remark that at least one of the first two inequalities has to be strict. Therefore, the truthful vote s_<sup>∗</sup> ( _R_ ) _is not Pareto-efficient._ Consider the following example as well.

**Example 4 :** _Consider a ranked-choice election_ ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ) _where N_ = {1 _,_ 2} _, A_ = { _x_ 1 _,x_ 2 _,x_ 3} _, and the voting rule is simple majority rule. For simplicity, assume the social ranking of each pair of candidates has equal worth to each individual and that social incomparability between any two candidates is broken by a fair coin toss. Let R be a preference profile defined as follows: R_<sup>1</sup> = _x_ 1 _x_ 2 _x_ 3 _and R_<sup>2</sup> = _x_ 1( _x_ 2 _x_ 3) _. The truthful voting profile is s_<sup>∗</sup> ( _R_ ) = ( _s_<sup>∗</sup> 1<sup>(</sup><sup>_R_)</sup><sup>_,s_∗</sup> 2<sup>(</sup><sup>_R_))</sup><sup>_,wheres_∗</sup> 1<sup>(</sup><sup>_R_)=(</sup><sup>_s_∗</sup> 1<sup>12</sup><sup>_,s_∗</sup> 1<sup>13</sup><sup>_,s_∗</sup> 1<sup>23)=(1</sup><sup>_,_1</sup><sup>_,_1)</sup><sup>_and_</sup> _s_<sup>∗=(</sup><sup>_s_∗12=(1</sup><sup>_,_1</sup><sup>_,_0)</sup><sup>_.ThecorrespondingsocialrankingisF_(</sup><sup>_s_∗(</sup><sup>_R_))=</sup> 2<sup>(</sup><sup>_R_)</sup> 2<sup>_,s_∗</sup> 2<sup>13</sup><sup>_,s_∗</sup> 2<sup>23)</sup> { _x_ 1 _x_ 2 _,x_ 1 _x_ 3 _,x_ 2 ∗ _x_ 3} _. The utilities of voters from this social ranking at the preference profile R are :_

_The truthful voting profile is not Pareto-efficient since it is Pareto-dominated by the strategy profile s_ = ( _s_ 1 _,s_ 2) _, where s_ 1 = _s_<sup>∗</sup> 1<sup>(</sup><sup>_R_)</sup><sup>_and s_2 = (1</sup><sup>_,_1</sup><sup>_,_1)</sup><sup>_.Infact,wehavethat F_(</sup><sup>_s_) =</sup><sup>_x_1</sup><sup>_x_2</sup><sup>_x_3</sup><sup>_._</sup> _It follows that u_ 1( _s_ ) = 1 > _u_ 1( _s_<sup>∗</sup> ( _R_ )) _and u_ 2((1 _,_ 1 _,_ 1)) =<sup><u>1</u></sup> 3<sup>(3 −</sup><sup>_α_2) =</sup><sup>_u_2(</sup><sup>_s_∗(</sup><sup>_R_))</sup><sup>_.Therefore,_</sup> _the truthful vote s_<sup>∗</sup> ( _R_ ) _is not Pareto-efficient._ A common feature in the elections studied in the last two examples is that voter preferences include ties. The following result shows that truthtelling is Pareto-efficient 19
if there are exactly two candidates and individual preferences are strict orderings.<sup>18</sup> Interestingly, strict ordering is a fairly weak condition and is often imposed by design in societies that use ranked voting to select policies or leaders (see Dasgupta and Maskin (2020), Maskin (2022, 2018a, 2018b)).

**Theorem 3 :** _Let_ ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ) _be a ranked-choice election such that_ ∣ _X_ ∣= 2 _and voters’ preferences are linear orderings over the set of candidates. Then, the truthful voting profile s_<sup>∗</sup> ( _R_ ) _is a Pareto-efficient Nash equilibrium for the associated strategic form game_ Γ _R_ = ( _N,X,U_<sup>_dα_</sup> ) _._

It is worth noting that the fact that a voter has a linear ordering over the set of candidates does not preclude casting a strategic vote involving indifference between candidates. This is consistent with the behavior of a voter during a presidential election who may prefer a candidate from the opposition over the incumbent but decides to stay home and not vote. In practice, choosing not to vote in an election is generally counted as abstention, which, in most societies, is also equivalent to casting a blank ballot.

Also, Theorem 3 has an implication for citizens’ decision to participate in elections in polarized societies where it is very unlikely that a voter will be indifferent between the two opposing groups. It essentially provides strong support toward participation in elections, which is beneficial to both the individual and society as a whole. This is because truthtelling in such societies is not only an equilibrium, but it is also Pareto-efficient.

We illustrate Theorem 3 with the following example.

**Example 5 :** _Consider a ranked-choice election_ ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ) _where N_ = {1 _,_ 2 _,_ 3} _, A_ = { _x_ 1 _,x_ 2} _, and the voting rule W is defined as follows: a tripartition_ ( _T_ 1 _,T_ 2 _,T_ 3) ∈ _W if and only if_ 1 ∉ _T_ 3 _and_ ∣ _T_ 1 ∣≥∣ 2 ∣ _. This means that an alternative x is socially preferred to another alternative y if x is not the worst preferred candidate of voter 1 and at least two voters prefer x to y. So, voter_ 1 _has veto power and so wields more power than each of the two other voters. In the case of social incomparability, we assume that the tie is broken by a fair coin toss. Let R be a preference profile defined as follows: R_<sup>1</sup> = _x_ 1 _x_ 2 _, R_<sup>2</sup> = _x_ 2 _x_ 1 _and R_<sup>3</sup> = _x_ 2 _x_ 1 _._

> 18The assumption that there are only two candidates has been followed in several studies of majority-like rules (see, e.g, May (1952) and Börgers (2004)). While a two-candidate election format accommodates a wide variety of electoral settings, several interesting election settings involve more than two candidates. So, we offer a more general result for any number of candidates later.

20
_The truthful voting profile is s_<sup>∗</sup> ( _R_ ) = (1 _,_ −1 _,_ −1) _and the corresponding social ranking is F_ ( _s_<sup>∗</sup> ( _R_ )) = { _x_ 1 ∗ _x_ 2} _. The utilities of voters from this social ranking at the preference profile R are :_ _- If voter 1 deviates from_ 1 _to t_ 1 ∈{−1 _,_ 0} _, then F_ (( _t,_ −1 _,_ −1)) = { _x_ 2 _x_ 1} _and his or her utility is u_ 1(( _t,_ 1 _,_ −1)) = 0 _. Hence, voter 1 has no incentive to deviate from his or her true preferences._ _- If voter 2 deviates from_ −1 _to_ 0 _, then F_ ((1 _,_ 0 _,_ −1)) = { _x_ 1 ∗ _x_ 2} _and his or her utility is u_ 2((1 _,_ 0 _,_ −1)) =<sup><u>1</u></sup> 2<sup>_.Ifheorshedeviatesto_1</sup><sup>_,thenF_((1</sup><sup>_,_1</sup><sup>_,_−1)) = {</sup><sup>_x_1</sup><sup>_x_2}</sup><sup>_,andhis_</sup> _or her utility is u_ 2((1 _,_ 1 _,_ −1)) = 0 _. Hence, voter 2 has no incentive to deviate from his or her true preferences._ _- Similarly, voter 3 has no incentive to deviate from his or her true preferences._ _Because no voter has an incentive to deviate from his or her true preferences, truthtelling is a pure-strategy Nash equilibrium for the associated strategic form game_ Γ _R_ = ( _N,Z,U_<sup>_dα_</sup> ) _. In addition, no voting strategy profile can guarantee each voter a utility equal to at least_ <u>1</u><sup>_withonevoterbeingstrictlybetteroff.Hence,truthtellingisaPareto-efficientNash_</sup> 2<sup>_,_</sup> _equilibrium._ Theorem 3 guarantees the existence of a Pareto-efficient truthful voting equilibrium under the assumption that voters’ preferences are linear orderings and the number of candidates is two. The former assumption is innocuous and widely adopted both in theory and practice. In contrast, the two-candidate assumption is less trivial and can be seen as limiting in some election systems that involve more than two candidates. The next example shows that Theorem 3 fails to hold if the latter condition is not satisfied.<sup>19</sup>

**Example 6 :** _Consider a ranked-choice election_ ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ) _where N_ = {1 _,_ 2 _,_ 3} _, A_ = { _x_ 1 _,x_ 2 _,x_ 3} _, and the voting rule is simple majority rule. Assume social incomparability between any two candidates is broken by a fair coin toss. Let R be a preference profile defined as follows: R_<sup>1</sup> = _x_ 1 _x_ 2 _x_ 3 _, R_<sup>2</sup> = _x_ 2 _x_ 3 _x_ 1 _, and R_<sup>3</sup> = _x_ 3 _x_ 1 _x_ 2 _. Now suppose the_

> 19We thank Gabriel Carroll for suggesting this example to us.

21
_voters’ weights θ_ 1 _, θ_ 2 _, and θ_ 3 _are such that voter 1 cares mostly about the comparison between x_ 1 _and x_ 3 _(θ_ 1<sup>13= 1</sup><sup>_);voter2caresmostlyaboutthecomparisonbetweenx_1</sup><sup>_andx_2</sup> _(θ_ 2<sup>12= 1</sup><sup>_);andvoter3caresmostlyaboutthecomparisonbetweenx_2</sup><sup>_andx_3</sup><sup>_(θ_</sup> 3<sup>23= 1</sup><sup>_).The_</sup> _truthful voting profile is s_<sup>∗</sup> ( _R_ ) = ( _s_<sup>∗</sup> 1<sup>(</sup><sup>_R_)</sup><sup>_,s_</sup> 2<sup>∗(</sup><sup>_R_)</sup><sup>_,s_∗</sup> 3<sup>(</sup><sup>_R_))</sup><sup>_,wheres_∗</sup> 1<sup>(</sup><sup>_R_) = (</sup><sup>_s_</sup> 1<sup>∗12</sup><sup>_,s_</sup> 1<sup>∗13</sup><sup>_,s_</sup> 1<sup>∗23) =</sup> (1 _,_ 1 _,_ 1) _, s_<sup>∗</sup> 2<sup>(</sup><sup>_R_) = (−1</sup><sup>_,_−1</sup><sup>_,_1)</sup><sup>_,ands_∗</sup> 3<sup>(</sup><sup>_R_) = (1</sup><sup>_,_−1</sup><sup>_,_−1)</sup><sup>_.Thecorrespondingsocialranking_</sup> _is F_ ( _s_<sup>∗</sup> ( _R_ )) = { _x_ 1 _x_ 2 _,x_ 3 _x_ 1 _,x_ 2 _x_ 3} _. The utilities of voters from this social ranking at the preference profile R are :_ _The truthful voting profile is not Pareto-efficient since it is Pareto-dominated by the strategy profile s_ = ( _s_ 1 _,s_ 2 _,s_ 3) _, where s_ 1 = (−1 _,_ −1 _,_ −1) _, s_ 2 = (1 _,_ 1 _,_ −1) _, and s_ 3 = (−1 _,_ 1 _,_ 1) _. In fact, we have that F_ ( _s_ ) = { _x_ 2 _x_ 1 _,x_ 1 _x_ 3 _,x_ 3 _x_ 2} _. It follows that: u_ 1( _s_<sup>∗</sup> ( _R_ )) = _u_ 2( _s_<sup>∗</sup> ( _R_ )) = _u_ 3( _s_<sup>∗</sup> ( _R_ )) = 1 _. Therefore, the truthful vote s_<sup>∗</sup> ( _R_ ) _is Pareto-dominated by the strategy profile s._ For such elections involving at least three candidates, it would be interesting to investigate conditions under which truthful voting is Pareto-efficient. This is the focus of the last part of this section.

We first observe that having a ranked-choice election with only two candidates, say _x_ 1 and _x_ 2, is equivalent to having a ranked-choice election with _m_ candidates, _x_ 1 _,x_ 2 _,....,xm_ , ( _m_ > 2), wherein each voter _i_ ∈ _N_ care mostly about the social comparison between candidates _x_ 1 and _x_ 2. That is, the voters’ weights _θi_ , _i_ ∈ _N_ , are such that _θi_<sup>12</sup> = 1 for all _i_ ∈ _N_ . Therefore, a more general reformulation of Theorem 3 is as follows. Given a ranked-choice election, if voters have strict preferences and care mostly about the social comparison between two candidates, then truthtelling is a Pareto-efficient equilibrium. However, there are instances of voting systems where voters may be concerned about the social comparison of more than one pair of candidates. In these cases, truthtelling may not be Pareto-efficient, as illustrated in Example 6. Our next result builds upon the limitation of Theorem 3 by weakening some assumptions and providing additional conditions that guarantee the Pareto-efficiency of truthtelling in ranked-choice elections where voters are concerned about the social comparisons of more than one pair of candidates.

**Theorem 4 :** _Let_ ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ) _be a ranked-choice election such that voters’ preferences are linear and neutral, social incomparability is broken by a fair coin toss, and_

22
_the social ranking evaluated at the preference profile_ ( _Ri_ ) _i_ ∈ _N is complete. Then, the truthful voting profile s_<sup>∗</sup> ( _R_ ) _is a Pareto-efficient Nash equilibrium for the associated strategic form game_ Γ _R_ = ( _N,X,U_<sup>_dα_</sup> ) _._ The conditions of Theorem 4 are quite innocuous. In particular requiring the social ranking evaluated at the truthful voting profile to be complete means that the social ranking always produces a ranking between any pair of candidates under truthful voting, although social incomparabilities between candidates may occur if some voters engage in strategic voting. This condition is a bit weaker than the one imposed by Arrow (1951), who requires that the social ranking be strict and transitive. If Arrow’s condition is satisfied, then ours also will be satisfied. Theorem 4 shows that truthful voting is a Pareto-efficient Nash equilibrium when these four conditions are satisfied.

Our next result is a direct consequence of Theorem 4, and illustrates a ranked-choice election in which the completeness condition of the social ranking evaluated at the truthful voting profile is satisfied.

**Corollary 2 :** _Let_ ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ) _be a ranked-choice election such that voters’ preferences are linear and neutral, W is simple majority rule, and the number of voters is odd. Assume social incomparability is broken by a fair coin toss, then the truthful voting profile s_<sup>∗</sup> ( _R_ ) _is a Pareto-efficient Nash equilibrium for the associated strategic form game_ Γ _R_ = ( _N,X,U_<sup>_dα_</sup> ) _._

In fact, it is well known that majority rule always produces a strict and transitive ordering of the candidates when the number of voters is odd. It therefore follows from Theorem 4 that truthful voting is a Pareto-efficient Nash equilibrium under this rule if the number of voters is odd.

We illustrate our findings with an example taken from Dasgupta and Maskin (2020), which analyzes the 2016 primary elections in the Republican Party in the United States.

**Example 7 :** _Consider the ranked-choice election_ ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ) _reflecting a pattern in the early 2016 Republican presidential primaries (Dasgupta & Maskin, 2020). Initially, there were about 16 candidates in the race. However, only a handful of candidates, including Donald Trump, Marco Rubio, and John Kasich, survived beyond a couple of weeks. Let us imagine that voters’ preferences broke down into three groups, as shown in the following table:_

23
|_40%_|_35%_|_25%_|
|---|---|---|
|_Trump_|_Rubio_|_Kasich_|
|_Kasich_|_Kasich_|_Rubio_|
|_Rubio_|_Trump_|_Trump_|

_So, we have three candidates: Trump, Rubio, and Kasich, denoted by x_ 1 _, x_ 2 _, and x_ 3 _, respectively. Voters are members of the Republican Party, of which 40% have preferences R_<sup>1</sup> = _x_ 1 _x_ 3 _x_ 2 _, 35% have preferences R_<sup>2</sup> = _x_ 2 _x_ 3 _x_ 1 _, and 25% have preferences R_<sup>3</sup> = _x_ 3 _x_ 2 _x_ 1 _. So, we have 3 types of voters, namely type 1 voter, type 2 voter, and type 3 voter. The voting rule is simple majority rule. The truthful voting profile is s_<sup>∗</sup> ( _R_ ) = ( _s_<sup>∗</sup> 1<sup>(</sup><sup>_R_)</sup><sup>_,s_∗</sup> 2<sup>(</sup><sup>_R_)</sup><sup>_,s_∗</sup> 2<sup>(</sup><sup>_R_))</sup><sup>_,_</sup> _where s_<sup>∗=(</sup><sup>_s_∗12=(1</sup><sup>_,_1</sup><sup>_,_−1)</sup><sup>_,s_∗=(</sup><sup>_s_∗12=(−1</sup><sup>_,_−1</sup><sup>_,_1)</sup><sup>_,and_</sup> 1<sup>(</sup><sup>_R_)</sup> 1<sup>_,s_</sup> 1<sup>∗13</sup><sup>_,s_</sup> 1<sup>∗23)</sup> 2<sup>(</sup><sup>_R_)</sup> 2<sup>_,s_</sup> 2<sup>∗13</sup><sup>_,s_</sup> 2<sup>∗23)</sup> _s_<sup>∗</sup> 3<sup>(</sup><sup>_R_) = (</sup><sup>_s_∗</sup> 3<sup>12</sup><sup>_,s_∗</sup> 3<sup>13</sup><sup>_,s_∗</sup> 3<sup>23) = (−1</sup><sup>_,_−1</sup><sup>_,_−1)</sup><sup>_.ThecorrespondingsocialrankingisF_(</sup><sup>_s_∗(</sup><sup>_R_)) =</sup> _x_ 3 _x_ 2 _x_ 1 _, where the top-ranked candidate is Kasich, followed by Rubio, and then by Trump. Assume that the social ranking of each pair of candidates is worth equally to voters. Voters’ utilities from the social ranking at the preference profile R are:_ _Theorem 3 implies that truthful voting profile s_<sup>∗</sup> ( _R_ ) _is a Pareto-efficient Nash equilibrium._ _Now assume for the sake of illustration that the candidates’ supporters cast a strategic vote each, corresponding to what they would do under_ **_plurality rule_** _. In this case, Trump’s supporters (type 1 voters) would rank Trump first and be indifferent between Kasich and Rubio, so that the strategic vote is s_ 1 = (1 _,_ 1 _,_ 0) _. Rubio’s supporters (type 2 voters) would rank Rubio first and be indifferent between Kasich and Trump, corresponding to s_ 2 = (−1 _,_ 0 _,_ 1) _. Kasich’s supporters (type 3 voters) would rank Kasich first and be indifferent between Rubio and Trump, corresponding to s_ 3 = (0 _,_ −1 _,_ −1) _._<sup>20</sup> _The social ranking associated to this strategic vote is F_ ( _s_ ) = _x_ 1 _x_ 2 _x_ 3 _, where the top-ranked candidate is Trump, followed by Rubio, and then Kasich. Voters’ utilities from this latter social ranking at the strategic voting profile s are:_

> 20Note that this is indeed a Nash equilibrium under plurality rule, although it is not a strong Nash equilibrium as all of Kasich’s supporters can deviate to Rubio and induce the election of the latter; but such a deviation did not happen in reality.

24
_It follows that Trump’s supporters are better off but at the expense of the supporters of Rubio and Kasich. As a result, we have that the truthful vote s_<sup>∗</sup> ( _R_ ) _is not Paretodominated by the strategic vote s. However, we note that more people are better off under the truthful equilibrium in the ranked-choice election than the equilibrium under plurality rule; this corroborates the view that ranked voting produces outcomes that better represent the view of citizens than traditional voting systems where voters only vote for one candidate._

# **5 An extension to voluntary majority rule with costly voting**

The implicit assumption under which the findings in the previous section hold is that voters’ participation in the election is compulsory and that individual costs are minimal. In this section, we extend the analysis by relaxing these assumptions. As in Börgers (2004), we assume there are only two candidates and that voters have a choice between participating and not participating in an election. If a voter chooses to participate, he or she incurs some cost, while no cost is incurred in case of non-participation. In addition, we assume that voters’ participation costs are common knowledge among voters. This enables us to rule out the possibility of ties between the two candidates in voters’ preferences since voters who are indifferent will never vote if voting is costly. We focus on the majority voting rule and assume that social incomparability between the two candidates is broken by a fair coin toss, which gives an equal probability of winning to each candidate. We derive necessary and sufficient conditions under which truthtelling is a Nash equilibrium. Our result complements those in Börgers (2004), in which the author compares, from an efficiency point of view, the outcomes of majority voting when participation is voluntary and compulsory under the assumption that participation costs are private information.

More precisely, let ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ) be a ranked-choice election such that _X_ = { _x_ 1 _,x_ 2}, voter preferences are linear orderings over _X_ , and _W_ is majority rule. The strategy set of each voter _i_ is _Si_ = {1 _,_ −1 _,_ 0}. In the case of social incomparability, we assume that each candidate is elected with probability<sup><u>1</u></sup> 2<sup>.Avoter</sup><sup>_i_canvotefor</sup><sup>_x_1or</sup> _x_ 2, in which case he or she incurs a cost _ci_ ∈(0 _,_ 1]. He or she can also abstain. Under majority rule, abstention is equivalent to non-participation and therefore entails no cost.

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Given a voting strategy profile _s_ ∈ _S_ = {1 _,_ 0 _,_ −1}<sup>_n_</sup> , the utility of voter _i_ associated to _s_ is given by where _Ui_<sup>12(</sup><sup>_s_)isgivenbyEquation(2).</sup> The next result states that truthtelling is a Nash equilibrium if and only if voters’ participation costs are relatively low and the election is tight, meaning that _x_ 1 and _x_ 2 have the same number of supporters.

**Theorem 5 :** _Let_ ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ) _be a ranked-choice election as described above, and c_ ∈(0 _,_ 1]<sup>_n_</sup> _a profile of voters’ participation costs. The truthful voting profile s_<sup>∗</sup> ( _R_ ) _is a Nash equilibrium if and only if_ ∣ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ ))∣= ∣ _N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ ))∣ _and ci_ ≤<sup><u>1</u></sup> 2<sup>_foralli_∈</sup><sup>_N._</sup> _However, s_<sup>∗</sup> ( _R_ ) _is not Pareto-efficient._

We illustrate this result with two examples. In the first example, all the conditions of Theorem 5 are satisfied, while in the second example, one of the conditions fails to hold.

**Example 8 :** _Let_ ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ) _be a ranked-choice election as described in Theorem 5, and where N_ = {1 _,_ 2} _, R_<sup>1</sup> = _x_ 1 _x_ 2 _and R_<sup>2</sup> = _x_ 2 _x_ 1 _._<sup>21</sup> _Therefore, the truthful vote is s_<sup>∗</sup> ( _R_ ) = ( _s_<sup>∗</sup> 1<sup>(</sup><sup>_R_)</sup><sup>_,s_∗</sup> 2<sup>(</sup><sup>_R_)) = (1</sup><sup>_,_−1)</sup><sup>_.Table2displaysthepossiblestrategyvotingprofile_</sup> _along with the corresponding payoffs._

voter 2

**Table 2:** The voters’ payoffs ( _u_ 1( _s_ ) _,u_ 2( _s_ )) for all _s_ ∈ _S_ of Example 8.

_It is clear that the truthful vote (1, -1) is the unique Nash equilibrium when c_ 1 ≤<sup><u>1</u></sup> 2<sup>_and_</sup> _c_ 2 ≤<sup><u>1</u></sup> 2<sup>_.However,itisParetodominatedbythestrategyprofiles_= (0</sup><sup>_,_0)</sup><sup>_._</sup>

> 21It is important to note that an example of two voters should not be viewed as limiting as these two individuals could be representatives of two significant groups.

26
It is important to note that the uniqueness of the truthful vote is not guaranteed in general under the conditions of Theorem 5. For instance, if we have 4 voters with preferences _R_<sup>1</sup> = _R_<sup>2</sup> = _x_ 1 _x_ 2 and _R_<sup>3</sup> = _R_<sup>4</sup> = _x_ 2 _x_ 1. Then, the strategy profile _s_ = ( _s_ 1 _,s_ 2 _,s_ 3 _,s_ 4) = (1 _,_ 1 _,_ 0 _,_ 0) is another Nash equilibrium different from the truthful vote _s_<sup>∗</sup> ( _R_ ) = (1 _,_ 1 _,_ −1 _,_ −1).

**Example 9 :** _Let_ ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ) _be a ranked-choice election as described in Theorem 5, and where N_ = {1 _,_ 2 _,_ 3} _, R_<sup>1</sup> = _R_<sup>2</sup> = _x_ 1 _x_ 2 _, R_<sup>3</sup> = _x_ 2 _x_ 1 _, and c_ 1 = _c_ 2 = _c_ 3 =<sup><u>1</u></sup> 3<sup>_.Therefore,_</sup> _the truthful vote is s_<sup>∗</sup> ( _R_ ) = ( _s_<sup>∗</sup> 1<sup>(</sup><sup>_R_)</sup><sup>_,s_∗</sup> 2<sup>(</sup><sup>_R_))=(1</sup><sup>_,_1</sup><sup>_,_−1)</sup><sup>_.Table3displaysthepossible_</sup> _strategy voting profiles along with the corresponding payoffs._

voter 2

||1|0|−1|
|---|---|---|---|
|1|( <sup>2</sup><br>3<sup>_,_ 2</sup><br>3<sup>_,_−1</sup><br>3<sup>)</sup>|( <sup>1</sup><br>6<sup>_,_ 1</sup><br>2<sup>_,_ 1</sup><br>6<sup>)</sup>|(−<sup>1</sup><br>3<sup>_,_−1</sup><br>3<sup>_,_ 2</sup><br>3<sup>)</sup>|
|voter 1<br>0|( <sup>1</sup><br>2<sup>_,_ 1</sup><br>6<sup>_,_ 1</sup><br>6<sup>)</sup>|(0_,_0_,_ <sup>2</sup><br>3<sup>)</sup>|(0_,_−<sup>1</sup><br>3<sup>_,_ 2</sup><br>3<sup>)</sup>|
|−1|(−<sup>1</sup><br>3<sup>_,_−1</sup><br>3<sup>_,_ 2</sup><br>3<sup>)</sup>|(−<sup>1</sup><br>3<sup>_,_0</sup><sup>_,_ 2</sup><br>3<sup>)</sup>|(−<sup>1</sup><br>3<sup>_,_−1</sup><br>3<sup>_,_ 2</sup><br>3<sup>)</sup>|

voter 3: _s_ 3 = −1

**Table 3:** The voters’ payoffs ( _u_ 1( _s_ ) _,u_ 2( _s_ )) for all _s_ ∈ _S_ of Example 9.

_As expected, the truthful vote is not a Nash equilibrium since voter 3 can profitably deviate to s_ 3 = 0 _. In addition, we remark that this game does not admit any Nash equilibrium._ This example suggests that a pure strategy Nash equilibrium may not exist if the election is not tight. However, there are instances where a Nash equilibrium may exist 27
even if the election is not tight. For example, if there are 2 voters with preferences _R_<sup>1</sup> = _R_<sup>2</sup> = _x_ 1 _x_ 2 and cost functions _c_ 1 = _c_ 2 =<sup><u>1</u></sup> 3<sup>,thenthestrategyprofiles(1</sup><sup>_,_0)and(0</sup><sup>_,_1)</sup> are the only pure strategy Nash equilibria of the associated strategic form game.

# **6 A reinterpretation: Behavioral anomaly and “stable” patterns of irrationality**

In this section, we reinterpret our basic model to provide a new explanation for certain types of individual choices that can be considered irrational in the sense of violating acyclicity (or transitivity) and, hence, the generalized axiom of revealed preference (GARP). There is a large literature that shows that choices made by individuals can be intransitive or even cyclic (May, 1954; Tversky, 1969; MacCrimmon & Larsson, 1979; Loomes, Starmer, & Sugden, 1989; Bell, 1982). Several explanations have been provided, including the existence of multi-attribute alternatives where differential attribute weighting, withinfactor discrimination, and inter-dependencies among attributes can lead to preference cycles (MacCrimmon & Larsson, 1979; Fishburn, 1991), regret aversion (Bell, 1982; Loomes et al., 1989; Loomes, Starmer, & Sugden, 1991), and event-splitting or framing effect (Starmer & Sugden, 1998; Humphrey, 2001; Harless, 1992). Our study provides a new explanation in a context where the decision-maker is assumed to have multiple “inner selves” (De Clippel & Eliaz, 2012), with each self representing a different criterion for ranking the options in the choice set. In the absence of a single criterion for evaluating the different alternatives available to an individual, Shafir et al. (1993) argue that choices can be explained “in terms of the balance of reasons for and against the various alternatives” (p. 12). This heuristic, which is also known as _reason-based choice_ (Simonson, 1989; Tversky & Shafir, 1992; Shafir et al., 1993), generally leads to choices that violate rationality. This is because the different “reasons” or criteria used to evaluate options generally produce conflicting rankings, making the final choice complicated. In the economics literature, choice under conflicts is generally modeled by assuming that the decision-maker has multiple selves (Bernheim and Rangel (2004); Benhabib and Bisin (2005); Eliaz and Spiegler (2006); Fudenberg and Levine (2006); De Clippel and Eliaz (2012); Ambrus and Rozen (2015)) In this context, our analysis can be viewed as offering a non-cooperative solution to 28
an _intrapersonal bargaining_ problem, where the decision-maker aggregates the rankings of his or her different inner selves using a voting or aggregation rule. Indeed, our basic model, ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ), can be reinterpreted as follows: _N_ is a finite set of selves of the decision-maker; _X_ is a finite choice set; _Ri_ is self _i_ ’s ranking of the alternatives in the choice set; _W_ is the voting rule; and _F_ the corresponding aggregation rule.

To illustrate how a cyclic choice can emerge as a “stable” behavioral pattern, suppose an individual has received job offers from three companies: Company A, Company B, and Company C, located in three different cities. Assume that there are three characteristics that are most important to this individual in assessing the desirability of each offer: pay (the higher the pay, the better), the city’s cost of living(the lower the cost of living, the better), and proximity to a beach (the closer to a beach, the better). Let ‘H’ stand for ‘high’, ‘M’ for ‘medium’, and ‘L’ for ‘low’. These three criteria can be seen as the individual’s different selves. She evaluates the three offers as follows:

||Income|Living cost|Beach proximity|
|---|---|---|---|
|A|H|H|M|
|B|M|L|H|
|C|L|M|L|

Each offer is better than the other two in one dimension but worse than one of the other two in another dimension. If she is unable or unwilling to consider one characteristic (self) as the most important one, she might decide to rank any two offers as follows: _x_ is better than _y_ if _x_ dominates _y_ in at least two characteristics. According to this criterion, Company A is better than Company B (because A ranks higher in terms of pay and proximity to a beach); Company B is better than Company C (because B ranks higher in terms of pay and cost of living), and Company C is better than Company A (because C ranks higher in terms of the living cost and the proximity to a beach). This decision-making process can be translated into an interpersonal bargaining model ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ), where _N_ = {1 _,_ 2 _,_ 3} is the set of the individual’s different selves; _X_ = { _A,B,C_ }; _R_<sup>1</sup> = _ABC_ , _R_<sup>2</sup> = _BCA_ , _R_<sup>3</sup> = _CAB_ ; _W_ is simple majority rule; and _F_ the corresponding aggregation rule. We assume that incomparability between any two job offers is broken by a fair coin toss. Here, we can see that the different selves have conflicting preferences. The truthful profile is a Nash equilibrium, the corresponding social ranking being _F_ ( _s_<sup>∗</sup> ( _R_ )) = { _AB,BC,CA_ }. This means that if the individual has to choose 29
between Company A and Company B, she will choose Company A; if she has to choose between Company B and Company C, she will choose Company B; and if the choice is between Company C and Company A, she will choose Company C. This choice is clearly cyclic. Even more interesting is the fact that it is an outcome of a Nash equilibrium, which implies that it depicts a “stable” pattern of irrationality in decision-making.

# **7 Conclusion**

This study proposes a non-cooperative model of ranked voting and analyzes voting behavior and outcomes. In our model, each voter casts a ballot that consists of his or her ranking of the candidates. These individual votes are then aggregated to produce a social ranking, and each voter derives utility based on how close his or her true preferences are to the social ranking. Our setting is more general than the classical model of ranked voting (Maskin, 2022), as it allows for unequal voting rights, abstention, and social incomparability of candidates. We show that truthtelling is always a pure strategy Nash equilibrium. This result implies that ranked voting, unlike several other electoral systems, provides incentives for voters to reveal their true preferences. In addition, truthtelling is Pareto-efficient if there are exactly two candidates and voters’ true preferences do not include ties. When there are more than two candidates, we show that truthtelling is Pareto-efficient if voters’ preferences do not include ties and are neutral, social incomparability is broken by a fair coin toss, and the social ranking evaluated at the truthful voting profile is complete, a condition a bit weaker than the one imposed by Arrow (1951). Moreover, we extend the analysis to majoritarian elections where participation is voluntary and costly (Börgers, 2004), and derive necessary and sufficient conditions for truthtelling to be an equilibrium. Our findings have an implication for the design of voting systems compatible with truthful voting behaviors and outcome efficiency. Because our model allows unrestricted freedom in the choice of the voting rule, it can find practical applications in a wide range of real-life institutions and organizations.

We reinterpret our analysis to offer a non-cooperative solution to an intrapersonal bargaining problem. In so doing, we show that our findings have implications for the pertinence of cyclic individual choices that describe a _stable_ and efficient behavior when a person has multiple rational selves.

30
# **Proofs**

**Proof (Theorem 1) :** _Let_ ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ) _be a ranked-choice election and α_ ∈ ]0 _,_ 1]<sup>_n_</sup> _a profile of individuals parameter. To show that there exists a Nash equilibrium in pure strategies, it suffices to show that truthtelling is a pure-strategy Nash equilibrium. We will do this, therefore proving the two assertions in the theorem. To show that s_<sup>∗</sup> ( _R_ ) _is a Nash equilibrium of the associated game_ Γ _R_ = ( _N,Z,U_<sup>_dα_</sup> ) _, we need to prove that m_ <u>(</u> _m_ −1) _Ui_<sup>_dα_(</sup><sup>_s_∗(</sup><sup>_R_)) ≥</sup><sup>_U_</sup> _i_<sup>_dα_(</sup><sup>_ti_(</sup><sup>_R_)</sup><sup>_,s_</sup> −<sup>∗</sup> _i_<sup>(</sup><sup>_R_))</sup><sup>_foralli_∈</sup><sup>_Nandti_(</sup><sup>_R_) ∈{−1</sup><sup>_,_0</sup><sup>_,_1}</sup> 2 _. Let i_ ∈ _N and m_ <u>(</u> _m_ −1) _ti_ ( _R_ ) ∈{−1 _,_ 0 _,_ 1} 2 _, since for all profile Q,_

_it suffices to show that Ui_<sup>_pq_(</sup><sup>_s_∗(</sup><sup>_R_)) ≥</sup><sup>_U_</sup> _i_<sup>_pq_(</sup><sup>_ti_(</sup><sup>_R_)</sup><sup>_,s_</sup> −<sup>∗</sup> _i_<sup>(</sup><sup>_R_))</sup><sup>_forall_{</sup><sup>_xp,xq_} ∈P(</sup><sup>_X_)</sup><sup>_._</sup> _Let_ { _xp,xq_ } ∈P( _X_ ) _. Without loss of generality, assume p_ < _q. Three cases are possible for F_ ( _s_<sup>∗</sup> ( _R_ )) _._

**First case:** _xpxq_ ∈ _F_ ( _s_<sup>∗</sup> ( _R_ )) _._

_If R_ {<sup>_i_</sup> _xp,xq_ }<sup>=</sup><sup>_xpxq, then U_</sup> _i_<sup>_pq_(</sup><sup>_s_∗(</sup><sup>_R_)) = 1 ≥</sup><sup>_U_</sup> _i_<sup>_pq_(</sup><sup>_ti_(</sup><sup>_R_)</sup><sup>_,s_</sup> −<sup>∗</sup> _i_<sup>(</sup><sup>_R_))</sup><sup>_, since the maximum utility_</sup> _that individual i can derive from the social ranking of a pair of candidates is equal to 1._ _since_ ( _N_<sup>1</sup> ( _s_<sup>∗</sup><sup>_pq_</sup> ( _R_ )) _,N_<sup>0</sup> ( _s_<sup>∗</sup><sup>_pq_</sup> ( _R_ ) _,N_<sup>−1</sup> ( _s_<sup>∗</sup><sup>_pq_</sup> ( _R_ ))) ∈ _W . Thus,_ 31
_For t_<sup>_pq_</sup> _i_<sup>(</sup><sup>_R_) = −1</sup><sup>_,bymonotonicity,wehave:_</sup> _And if F_ { _xp,xq_ }( _ti_ ( _R_ ) _,s_ −<sup>∗</sup> _i_<sup>(</sup><sup>_R_)) =</sup><sup>_xp_∗</sup><sup>_xq,then_</sup> _Therefore, Ui_<sup>_pq_(</sup><sup>_s_∗(</sup><sup>_R_)) ≥</sup><sup>_U pq_</sup> _i_<sup>(</sup><sup>_ti_(</sup><sup>_R_)</sup><sup>_,s_</sup> −<sup>∗</sup> _i_<sup>(</sup><sup>_R_))</sup><sup>_wheneverxpxq_∈</sup><sup>_F_(</sup><sup>_s_∗(</sup><sup>_R_))</sup><sup>_._</sup>

**Second case:** _xqxp_ ∈ _F_ ( _s_<sup>∗</sup> ( _R_ )) _._

_The proof is similar to that of the first case._

**Third case:** _xp_ ∗ _xq_ ∈ _F_ ( _s_<sup>∗</sup> ( _R_ )) _._

_If R_ {<sup>_i_</sup> _xp,xq_ }<sup>=</sup><sup>_xpxq,thens_∗</sup> _i_<sup>_pq_</sup> ( _R_ ) = 1 _and Ui_<sup>_pq_(</sup><sup>_s_∗(</sup><sup>_R_)) = Λ</sup><sup>_pq._</sup> _If ti_ ( _R_ ) = 0 _, then_ ( _N_<sup>1</sup> ( _ti_ ( _R_ ) _,s_ −<sup>∗</sup> _i_<sup>(</sup><sup>_R_))</sup><sup>_,N_0(</sup><sup>_ti_(</sup><sup>_R_)</sup><sup>_,s_</sup> −<sup>∗</sup> _i_<sup>(</sup><sup>_R_)</sup><sup>_,N_−1(</sup><sup>_ti_(</sup><sup>_R_)</sup><sup>_,s_</sup> −<sup>∗</sup> _i_<sup>(</sup><sup>_R_)))</sup><sup>_isequal_</sup> _to_ ( _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )) ∖{ _i_ } _,N_<sup>0</sup> ( _s_<sup>∗</sup> ( _R_ ) ∪{ _i_ } _,N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ ))) _which doesn’t belong to W ._ _We know that xp_ ∗ _xq_ ∈ _F_ ( _s_<sup>∗</sup> ( _R_ )) _implies that_ ( _N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ )) _,N_<sup>0</sup> ( _s_<sup>∗</sup> ( _R_ ) _,N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ ))) _doesn’t belong to W , but_ ( _N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ )) _,N_<sup>0</sup> ( _s_<sup>∗</sup> ( _R_ ) ∪{ _i_ } _,N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )) ∖{ _i_ }) _might belong to W , which means that it is possible that xqxp_ ∈ _F_ ( _ti_ ( _R_ ) _,s_ −<sup>∗</sup> _i_<sup>(</sup><sup>_R_))</sup><sup>_.Therefore,fortpq_</sup> _i_<sup>(</sup><sup>_R_) = 0</sup><sup>_,_</sup> _F_ { _xp,xq_ }( _ti_ ( _R_ ) _,s_ −<sup>∗</sup> _i_<sup>(</sup><sup>_R_)) ∈{</sup><sup>_xqxp,xp_∗</sup><sup>_xq_}</sup><sup>_,whichleadstoU_</sup> _i_<sup>_pq_(</sup><sup>_ti_(</sup><sup>_R_)</sup><sup>_,s_</sup> −<sup>∗</sup> _i_<sup>(</sup><sup>_R_)) ∈{0</sup><sup>_,_Λ</sup><sup>_pq_}</sup><sup>_._</sup> _Finally, we obtain Ui_<sup>_pq_(</sup><sup>_ti_(</sup><sup>_R_)</sup><sup>_,s_</sup> −<sup>∗</sup> _i_<sup>(</sup><sup>_R_)) ≤</sup><sup>_U pq_</sup> _i_<sup>(</sup><sup>_s_∗(</sup><sup>_R_)) = Λ</sup><sup>_pq._</sup> _When R_<sup>_ithereasoningissimilartothecasewhereRi_</sup> { _xp,xq_ }<sup>=</sup><sup>_xqxp,_</sup> { _xp,xq_ }<sup>=</sup><sup>_xpxq._</sup> _Now assume that R_ {<sup>_i_</sup> _xp,xq_ }<sup>= (</sup><sup>_xpxq_)</sup><sup>_.Inthiscase,s_∗</sup> _i_<sup>(</sup><sup>_R_) = 0</sup><sup>_andU_</sup> _i_<sup>_pq_(</sup><sup>_s_∗(</sup><sup>_R_)) = 1 −</sup><sup>_αi._</sup> 32
_If t_<sup>_pq_</sup> _i_<sup>(</sup><sup>_R_) = 1</sup><sup>_,theneitherF_{</sup><sup>_x_</sup> _p_<sup>_,x_</sup> _q_<sup>}(</sup><sup>_ti_(</sup><sup>_R_)</sup><sup>_,s_</sup> −<sup>∗</sup> _i_<sup>(</sup><sup>_R_)) =</sup><sup>_xp_∗</sup><sup>_xqorF_{</sup><sup>_x_</sup> _p_<sup>_,x_</sup> _q_<sup>}(</sup><sup>_ti_(</sup><sup>_R_)</sup><sup>_,s_</sup> −<sup>∗</sup> _i_<sup>(</sup><sup>_R_)) =</sup> _xpxq, which implies that Ui_<sup>_pq_(</sup><sup>_ti_(</sup><sup>_R_)</sup><sup>_,s_</sup> −<sup>∗</sup> _i_<sup>(</sup><sup>_R_)) = 1 −</sup><sup>_αiorU_</sup> _i_<sup>_pq_(</sup><sup>_ti_(</sup><sup>_R_)</sup><sup>_,s_</sup> −<sup>∗</sup> _i_<sup>(</sup><sup>_R_)) = 1 −</sup><sup>_αi._</sup> _That is, Ui_<sup>_pq_(</sup><sup>_ti_(</sup><sup>_R_)</sup><sup>_,s_</sup> −<sup>∗</sup> _i_<sup>(</sup><sup>_R_)) =</sup><sup>_U_</sup> _i_<sup>_pq_(</sup><sup>_s_∗(</sup><sup>_R_))</sup><sup>_._</sup> _If t_<sup>_pq_</sup> _i_<sup>(</sup><sup>_R_) = −1</sup><sup>_,theneitherF_{</sup><sup>_x_</sup> _p_<sup>_,x_</sup> _q_<sup>}(</sup><sup>_ti_(</sup><sup>_R_)</sup><sup>_,s_</sup> −<sup>∗</sup> _i_<sup>(</sup><sup>_R_)) =</sup><sup>_xp_∗</sup><sup>_xqorF_{</sup><sup>_x_</sup> _p_<sup>_,x_</sup> _q_<sup>}(</sup><sup>_ti_(</sup><sup>_R_)</sup><sup>_,s_</sup> −<sup>∗</sup> _i_<sup>(</sup><sup>_R_)) =</sup> _xqxp, which again implies that_ _Ui_<sup>_pq_(</sup><sup>_ti_(</sup><sup>_R_)</sup><sup>_,s_</sup> −<sup>∗</sup> _i_<sup>(</sup><sup>_R_)) = 1 −</sup><sup>_αiorU_</sup> _i_<sup>_pq_(</sup><sup>_ti_(</sup><sup>_R_)</sup><sup>_,s_</sup> −<sup>∗</sup> _i_<sup>(</sup><sup>_R_)) = 1 −</sup><sup>_αi._</sup> _That is, Ui_<sup>_pq_(</sup><sup>_ti_(</sup><sup>_R_)</sup><sup>_,s_</sup> −<sup>∗</sup> _i_<sup>(</sup><sup>_R_)) =</sup><sup>_U pq_</sup> _i_<sup>(</sup><sup>_s_∗(</sup><sup>_R_))</sup><sup>_._</sup> _It follows that, Ui_<sup>_pq_(</sup><sup>_s_∗(</sup><sup>_R_)) ≥</sup><sup>_U pq_</sup> _i_<sup>(</sup><sup>_ti_(</sup><sup>_R_)</sup><sup>_,s_</sup> −<sup>∗</sup> _i_<sup>(</sup><sup>_R_))</sup><sup>_wheneverxp_∗</sup><sup>_xq_∈</sup><sup>_F_(</sup><sup>_s_∗(</sup><sup>_R_))</sup><sup>_._</sup>

**Proof (Theorem 3) :** _Let_ ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ) _be a ranked-choice election such that X_ = { _x_ 1 _,x_ 2} _and voter preferences are linear orderings over X. Let α_ ∈]0 _,_ 1]<sup>_n_</sup> _be a profile of individual parameters._

_Given that we only have two candidates, x_ 1 _and x_ 2 _, in what follows we drop the superscript_ 12 _in all notations where it is required. Since individual preferences R_ = ( _Ri_ ) _i_ ∈ _N over the two candidates are linear orderings, then each voter belongs either to the set of individuals who truthfully vote for x_ 1 _, N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )) _, or the set of individuals who truthfully vote for x_ 2 _, N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ )) _. That is, N_ = _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )) ∪ _N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ )) _and N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )) ∩ _N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ )) = ∅ _._ _By contradiction, let us assume that the truthful vote is not a Pareto-efficient equilibrium. Then, we can find a strategy voting profile s_ ∈ _S such that:_ _This implies that F_ ( _s_<sup>∗</sup> ( _R_ )) ≠ _F_ ( _s_ ) ∈{ _x_ 1 _x_ 2 _,x_ 2 _x_ 1 _,x_ 1 ∗ _x_ 2} _. We have three distinct scenarios._

**_First case:_** _F_ ( _s_<sup>∗</sup> ( _R_ )) = _x_ 1 _x_ 2 _and F_ ( _s_ ) = _x_ 2 _x_ 1 _._

_F_ ( _s_<sup>∗</sup> ( _R_ )) = _x_ 1 _x_ 2 _implies that_ ( _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )) _,_ ∅ _,N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ ))) ∈ _W . It follows that N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )) ≠∅ _. Let i_ 0 ∈ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )) _. Then we have that_ 1 = _Ui_ 0( _s_<sup>∗</sup> ( _R_ )) > _Ui_ 0( _s_ ) = 0 _, which contradicts equation 3._ 33
**_Second case:_** _F_ ( _s_<sup>∗</sup> ( _R_ )) = _x_ 1 ∗ _x_ 2 _and F_ ( _s_ ) = _x_ 1 _x_ 2 _._

_F_ ( _s_<sup>∗</sup> ( _R_ )) = _x_ 1 ∗ _x_ 2 _implies that_ ( _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )) _,_ ∅ _,N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ ))) ∉ _W . It follows that N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ )) ≠∅ _. Let i_ 0 ∈ _N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ )) _. Then, we have that_ 1 − Λ12 = _Ui_ 0( _s_<sup>∗</sup> ( _R_ )) > _Ui_ 0( _s_ ) = 0 _, unless_ Λ12 = 1 _. If_ Λ12 = 1 _, then for all i_ ∈ _N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ )) _, Ui_ ( _s_<sup>∗</sup> ( _R_ )) = _Ui_ ( _s_ ) = 0 _; and for all i_ ∈ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )) _, Ui_ ( _s_<sup>∗</sup> ( _R_ )) = _Ui_ ( _s_ ) = 1 _. This implies that Ui_ ( _s_<sup>∗</sup> ( _R_ )) = _Ui_ ( _s_ ) _for all i_ ∈ _N . Therefore, we have a contradiction of equation 3 in any case._

**_Third case:_** _F_ ( _s_<sup>∗</sup> ( _R_ )) = _x_ 1 _x_ 2 _and F_ ( _s_ ) = _x_ 1 ∗ _x_ 2 _._

_F_ ( _s_<sup>∗</sup> ( _R_ )) = _x_ 1 _x_ 2 _implies that_ ( _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )) _,_ ∅ _,N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ ))) ∈ _W . It follows that N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )) ≠∅ _. Let i_ 0 ∈ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )) _. Then, we have that_ 1 = _Ui_ 0( _s_<sup>∗</sup> ( _R_ )) > _Ui_ 0( _s_ ) = Λ12 _, unless_ Λ12 = 1 _. If_ Λ12 = 1 _, then for all i_ ∈ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )) _, Ui_ ( _s_<sup>∗</sup> ( _R_ )) = _Ui_ ( _s_ ) = 1 _; and for all i_ ∈ _N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ )) _, Ui_ ( _s_<sup>∗</sup> ( _R_ )) = _Ui_ ( _s_ ) = 0 _. This implies that Ui_ ( _s_<sup>∗</sup> ( _R_ )) = _Ui_ ( _s_ ) _for all i_ ∈ _N . Therefore, we have a contradiction of equation 3 in any case._ _Hence, the Truthful vote is a Pareto-efficient Nash equilibrium of the associated game_ Γ _R_ = ( _N,X,U_<sup>_dα_</sup> ) _._

**Proof (Theorem 4) :** _Let_ ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ) _be a ranked-choice election such that voters’ preferences are linear and neutral, social incomparability is broken by a fair coin toss, and the social ranking evaluated at the truthful voting profile is complete._

_We prove the theorem for three candidates (m=3). The proof of the general case follows easily using the same arguments. So, assume A_ = { _x_ 1 _,x_ 2 _,x_ 3} _and let show that the truthful voting s_<sup>∗</sup> ( _R_ ) _is Pareto-efficient._ _First, we claim that: for all xp,xq,xp_ ′ _,xq_ ′ ∈ _A, if xpxq_ ∈ _F_ ( _s_<sup>∗</sup> ( _R_ )) _and xp_ ′ _xq_ ′ ∈ _F_ ( _s_<sup>∗</sup> ( _R_ )) _, then N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>_pq_</sup> ) ∩ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>_p_′</sup><sup>_q_′</sup> ) ≠∅ _._ _In fact, suppose by contradiction that N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>_pq_</sup> ) ∩ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>_p_′</sup><sup>_q_′</sup> ) = ∅ _. Given that voters’ preferences are strict, then xpxq_ ∈ _F_ ( _s_<sup>∗</sup> ( _R_ )) _and xp_ ′ _xq_ ′ ∈ _F_ ( _s_<sup>∗</sup> ( _R_ )) _implies that_ ( _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>_pq_</sup> ) _,_ ∅ _,N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>_pq_</sup> )) ∈ _W and_ ( _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>_p_′</sup><sup>_q_′</sup> ) _,_ ∅ _,N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>_p_′</sup><sup>_q_′</sup> )) ∈ _W . But N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>_pq_</sup> ) ∩ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>_p_′</sup><sup>_q_′</sup> ) = ∅ _implies that N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>_p_′</sup><sup>_q_′</sup> ) ⊆ _N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>_pq_</sup> ) _. It follows that:_ _This implies that_ ( _N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>_pq_</sup> ) _,_ ∅ _,N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>_pq_</sup> )) ∈ _W , which is a contradiction as_ ( _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>_pq_</sup> ) _,_ ∅ _,N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>_pq_</sup> )) ∈ _W . Given that voters’ preferences are neutral, it follows that θi_<sup>_pq_=</sup><sup>_θpqforalli_∈</sup><sup>_Nandxp,xq_∈</sup><sup>_A.Toshowthatthetruthfulvotings_∗(</sup><sup>_R_)</sup><sup>_isPareto-_</sup> _efficient, assume by contradiction that s_<sup>∗</sup> ( _R_ ) _is not Pareto-efficient. Then, there exists a_ 34
_strategy voting profile s_ ∈ _S such that:_ _The social ranking F_ ( _s_<sup>∗</sup> ( _R_ )) _and F_ ( _s_ ) _must be different. The scenarios where they differ on only one pair of candidates are akin to the two-candidate case demonstrated in Theorem 3. Therefore, our focus is on cases where they differ on two or three pairs of candidates. Given that F_ ( _s_<sup>∗</sup> ( _R_ )) _is complete, then focusing on only non-symmetrical cases leads to the following possible scenarios:_

**_Case 1:_** _F_ ( _s_<sup>∗</sup> ( _R_ )) = { _x_ 1 _x_ 2 _,x_ 2 _x_ 3 _,x_ 1 _x_ 3} _and F_ ( _s_ ) = { _x_ 2 _x_ 1 _,x_ 3 _x_ 2 _,x_ 1 _x_ 3}

_Let i_ ∈ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>12</sup> ) ∩ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>23</sup> ) _, then we have that Ui_ ( _s_<sup>∗</sup> ( _R_ )) = 1 _and Ui_ ( _s_ ) = _θ_ 13 _. As a result, Ui_ ( _s_<sup>∗</sup> ( _R_ )) > _Ui_ ( _s_ ) _unless θ_ 13 = 1 _. But, θ_ 13 = 1 _is equivalent to the two-candidate case, covered in Theorem 3, where all voters mostly care about the social comparison between x_ 1 _and x_ 3 _. Therefore, we have a contradiction of Equation (4) in any case._

**_Case 2:_** _F_ ( _s_<sup>∗</sup> ( _R_ )) = { _x_ 1 _x_ 2 _,x_ 2 _x_ 3 _,x_ 1 _x_ 3} _and F_ ( _s_ ) = { _x_ 2 _x_ 1 _,x_ 3 _x_ 2 _,x_ 3 _x_ 1}

_Let i_ ∈ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>12</sup> ) ∩ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>23</sup> ) _, then we have that Ui_ ( _s_<sup>∗</sup> ( _R_ )) = 1 _and Ui_ ( _s_ ) = 0 _. Therefore, Ui_ ( _s_<sup>∗</sup> ( _R_ )) > _Ui_ ( _s_ ) _, which contradicts Equation (4)._

**_Case 3:_** _F_ ( _s_<sup>∗</sup> ( _R_ )) = { _x_ 1 _x_ 2 _,x_ 2 _x_ 3 _,x_ 3 _x_ 1} _and F_ ( _s_ ) = { _x_ 2 _x_ 1 _,x_ 3 _x_ 2 _,x_ 1 _x_ 3}

_Let i_ ∈ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>12</sup> ) ∩ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>23</sup> ) _, then we have that Ui_ ( _s_<sup>∗</sup> ( _R_ )) = _θ_ 12 + _θ_ 23 _and Ui_ ( _s_ ) = _θ_ 13 _. Let j_ ∈ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>12</sup> ) ∩ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>31</sup> ) _, then we have that Uj_ ( _s_<sup>∗</sup> ( _R_ )) = _θ_ 12 + _θ_ 13 _and Uj_ ( _s_ ) = _θ_ 23 _. Let k_ ∈ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>23</sup> ) ∩ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>31</sup> ) _, then we have that Uk_ ( _s_<sup>∗</sup> ( _R_ )) = _θ_ 23 + _θ_ 13 _and Uk_ ( _s_ ) = _θ_ 12 _. This implies that Ui_ ( _s_<sup>∗</sup> ( _R_ )) + _Uj_ ( _s_<sup>∗</sup> ( _R_ )) + _Uk_ ( _s_<sup>∗</sup> ( _R_ )) = 2 > 1 = _Ui_ ( _s_ ) + _Uj_ ( _s_ ) + _Uk_ ( _s_ ) _, which contradicts Equation (4)._

**_Case 4:_** _F_ ( _s_<sup>∗</sup> ( _R_ )) = { _x_ 1 _x_ 2 _,x_ 2 _x_ 3 _,x_ 1 _x_ 3} _and F_ ( _s_ ) = { _x_ 2 _x_ 1 _,x_ 2 ∗ _x_ 3 _,x_ 1 _x_ 3}

_Let i_ ∈ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>12</sup> )∩ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>23</sup> ) _, then we have that Ui_ ( _s_<sup>∗</sup> ( _R_ )) = _θ_ 12 + _θ_ 23 + _θ_ 13 = 1 _and Ui_ ( _s_ ) =<sup><u>1</u></sup> 2<sup>_θ_23 +</sup><sup>_θ_13</sup><sup>_.Asaresult,Ui_(</sup><sup>_s_∗(</sup><sup>_R_)) ></sup><sup>_Ui_(</sup><sup>_s_)</sup><sup>_unlessθ_13 = 1</sup><sup>_.But,θ_13 = 1</sup><sup>_isequivalent_</sup> _to the two-candidate case, covered in Theorem 3. Therefore, we have a contradiction of Equation (4) in any case._

**_Case 5:_** _F_ ( _s_<sup>∗</sup> ( _R_ )) = { _x_ 1 _x_ 2 _,x_ 2 _x_ 3 _,x_ 3 _x_ 1} _and F_ ( _s_ ) = { _x_ 2 _x_ 1 _,x_ 2 ∗ _x_ 3 _,x_ 1 _x_ 3}

_Let i_ ∈ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>12</sup> ) ∩ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>31</sup> ) _, then we have that Ui_ ( _s_<sup>∗</sup> ( _R_ )) = _θ_ 12 + _θ_ 13 _and Ui_ ( _s_ ) =<sup><u>1</u></sup> 2<sup>_θ_23</sup><sup>_.Letj_∈</sup><sup>_N_1(</sup><sup>_s_∗(</sup><sup>_R_)12) ∩</sup><sup>_N_1(</sup><sup>_s_∗(</sup><sup>_R_)23)</sup><sup>_,thenwehavethatUj_(</sup><sup>_s_∗(</sup><sup>_R_)) =</sup><sup>_θ_12 +</sup><sup>_θ_23</sup> _and Uj_ ( _s_ ) = 2<sup><u>1</u></sup><sup>_θ_23+</sup><sup>_θ_13</sup><sup>_.Letk_∈</sup><sup>_N_1(</sup><sup>_s_∗(</sup><sup>_R_)23)∩</sup><sup>_N_1(</sup><sup>_s_∗(</sup><sup>_R_)31)</sup><sup>_,thenwehavethatUk_(</sup><sup>_s_∗(</sup><sup>_R_)) =</sup> _θ_ 23 + _θ_ 13 _and Uk_ ( _s_ ) = _θ_ 12 +<sup><u>1</u></sup> 2<sup>_θ_23</sup><sup>_.ThisimpliesthatUi_(</sup><sup>_s_∗(</sup><sup>_R_)) +</sup><sup>_Uj_(</sup><sup>_s_∗(</sup><sup>_R_)) +</sup><sup>_Uk_(</sup><sup>_s_∗(</sup><sup>_R_)) =</sup> 2 > 1 +<sup><u>1</u></sup><sup>_whichcontradictsEquation(4)._</sup> 2<sup>_θ_23 =</sup><sup>_Ui_(</sup><sup>_s_) +</sup><sup>_Uj_(</sup><sup>_s_) +</sup><sup>_Uk_(</sup><sup>_s_)</sup><sup>_,_</sup> 35
**_Case 6:_** _F_ ( _s_<sup>∗</sup> ( _R_ )) = { _x_ 1 _x_ 2 _,x_ 2 _x_ 3 _,x_ 1 _x_ 3} _and F_ ( _s_ ) = { _x_ 1 ∗ _x_ 2 _,x_ 2 ∗ _x_ 3 _,x_ 1 _x_ 3} _Let i_ ∈ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>12</sup> ) ∩ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>23</sup> ) _, then we have that Ui_ ( _s_<sup>∗</sup> ( _R_ )) = _θ_ 12 + _θ_ 23 + _θ_ 13 = 1 _and Ui_ ( _s_ ) =<sup><u>1</u></sup> 2<sup>_θ_12 +</sup><sup><u>1</u></sup> 2<sup>_θ_23 +</sup><sup>_θ_13</sup><sup>_.Asaresult,Ui_(</sup><sup>_s_∗(</sup><sup>_R_)) ></sup><sup>_Ui_(</sup><sup>_s_)</sup><sup>_unlessθ_13= 1</sup><sup>_.Butθ_13= 1</sup> _is equivalent to the two-candidate case, covered in Theorem 3. Therefore, we have a contradiction of Equation (4) in any case._

**_Case 7:_** _F_ ( _s_<sup>∗</sup> ( _R_ )) = { _x_ 1 _x_ 2 _,x_ 2 _x_ 3 _,x_ 3 _x_ 1} _and F_ ( _s_ ) = { _x_ 1 ∗ _x_ 2 _,x_ 2 ∗ _x_ 3 _,x_ 1 _x_ 3}

_Let i_ ∈ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>12</sup> )∩ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>31</sup> ) _, then we have that Ui_ ( _s_<sup>∗</sup> ( _R_ )) = _θ_ 12 + _θ_ 13 _and Ui_ ( _s_ ) = <u>12</u><sup>_θ_12 +</sup><sup><u>1</u></sup> 2<sup>_θ_23</sup><sup>_.Letj_∈</sup><sup>_N_1(</sup><sup>_s_∗(</sup><sup>_R_)12) ∩</sup><sup>_N_1(</sup><sup>_s_∗(</sup><sup>_R_)23)</sup><sup>_,thenwehavethatUj_(</sup><sup>_s_∗(</sup><sup>_R_)) =</sup><sup>_θ_12 +</sup><sup>_θ_23</sup> _and Uj_ ( _s_ ) = 2<sup><u>1</u></sup><sup>_θ_12 +</sup><sup><u>1</u></sup> 2<sup>_θ_23 +</sup><sup>_θ_13</sup><sup>_.Letk_∈</sup><sup>_N_1(</sup><sup>_s_∗(</sup><sup>_R_)23) ∩</sup><sup>_N_1(</sup><sup>_s_∗(</sup><sup>_R_)31)</sup><sup>_,thenwehavethat_</sup> _Uk_ ( _s_<sup>∗</sup> ( _R_ )) = _θ_ 23 + _θ_ 13 _and Uk_ ( _s_ ) =<sup><u>1</u></sup> 2<sup>_θ_12 +</sup><sup><u>1</u></sup> 2<sup>_θ_23</sup><sup>_.ThisimpliesthatUi_(</sup><sup>_s_∗(</sup><sup>_R_)) +</sup><sup>_Uj_(</sup><sup>_s_∗(</sup><sup>_R_)) +</sup> _Uk_ ( _s_<sup>∗</sup> ( _R_ )) = 2 > 1 +<sup><u>1</u></sup> 2<sup>(</sup><sup>_θ_12 +</sup><sup>_θ_23) =</sup><sup>_Ui_(</sup><sup>_s_) +</sup><sup>_Uj_(</sup><sup>_s_) +</sup><sup>_Uk_(</sup><sup>_s_)</sup><sup>_,whichcontradictsEquation(4)._</sup>

**_Case 8:_** _F_ ( _s_<sup>∗</sup> ( _R_ )) = { _x_ 1 _x_ 2 _,x_ 2 _x_ 3 _,x_ 1 _x_ 3} _and F_ ( _s_ ) = { _x_ 1 ∗ _x_ 2 _,x_ 2 ∗ _x_ 3 _,x_ 1 ∗ _x_ 3}

_Let i_ ∈ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>12</sup> ) ∩ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>23</sup> ) _, then we have that Ui_ ( _s_<sup>∗</sup> ( _R_ )) = 1 _and Ui_ ( _s_ ) =<sup><u>1</u></sup> 2<sup>_._</sup> _Therefore, Ui_ ( _s_<sup>∗</sup> ( _R_ )) > _Ui_ ( _s_ ) _, which contradicts Equation (4)._

**_Case 9:_** _F_ ( _s_<sup>∗</sup> ( _R_ )) = { _x_ 1 _x_ 2 _,x_ 2 _x_ 3 _,x_ 3 _x_ 1} _and F_ ( _s_ ) = { _x_ 1 ∗ _x_ 2 _,x_ 2 ∗ _x_ 3 _,x_ 1 ∗ _x_ 3}

_Let i_ ∈ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>12</sup> ) ∩ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )<sup>31</sup> ) _, then we have that Ui_ ( _s_<sup>∗</sup> ( _R_ )) = _θ_ 12 + _θ_ 13 _and Ui_ ( _s_ ) =<sup><u>1</u></sup> 2<sup>_.Letj_∈</sup><sup>_N_1(</sup><sup>_s_∗(</sup><sup>_R_)12) ∩</sup><sup>_N_1(</sup><sup>_s_∗(</sup><sup>_R_)23)</sup><sup>_,thenwehavethatUj_(</sup><sup>_s_∗(</sup><sup>_R_)) =</sup><sup>_θ_12 +</sup><sup>_θ_23</sup> _and Uj_ ( _s_ ) =<sup><u>1</u></sup> 2<sup>_.Let k_∈</sup><sup>_N_1(</sup><sup>_s_∗(</sup><sup>_R_)23)∩</sup><sup>_N_1(</sup><sup>_s_∗(</sup><sup>_R_)31)</sup><sup>_,then we have that Uk_(</sup><sup>_s_∗(</sup><sup>_R_)) =</sup><sup>_θ_23+</sup><sup>_θ_13</sup> _and Uk_ ( _s_ ) =<sup><u>1</u></sup> 2<sup>_.ThisimpliesthatUi_(</sup><sup>_s_∗(</sup><sup>_R_)) +</sup><sup>_Uj_(</sup><sup>_s_∗(</sup><sup>_R_)) +</sup><sup>_Uk_(</sup><sup>_s_∗(</sup><sup>_R_))=2></sup><sup><u>3</u></sup> 2<sup>=</sup><sup>_Ui_(</sup><sup>_s_) +</sup> _Uj_ ( _s_ ) + _Uk_ ( _s_ ) _, which contradicts Equation (4)._ _As a result, we have a contradiction of Equation (4) in any case. Hence, the truthful vote is a Pareto-efficient Nash equilibrium of the associated game_ Γ _R_ = ( _N,X,U_<sup>_dα_</sup> ) _._

**Proof (Theorem 5) :** _Let_ ( _N,X,_ ( _Ri_ ) _i_ ∈ _N ,W,F_ ) _be the ranked-choice election as stated in Theorem 5, and c_ ∈[0 _,_ 1]<sup>_n_</sup> _a profile of voters’ costs._

**_Sufficiency:_** _Assume_ ∣ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ ))∣= ∣ _N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ ))∣ _and ci_ ≤<sup><u>1</u></sup> 2<sup>_foralli_∈</sup><sup>_N,andletus_</sup> _show that s_<sup>∗</sup> ( _R_ ) _is a Nash equilibrium. Since_ ∣ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ ))∣= ∣ _N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ ))∣ _, then we have social incomparability under majority voting. This implies that ui_ ( _s_<sup>∗</sup> ( _R_ )) =<sup><u>1</u></sup> 2<sup>−</sup><sup>_ciforall_</sup> _i_ ∈ _N . If a voter i_ ∈ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )) _deviates from s_<sup>∗</sup> _i_<sup>(</sup><sup>_R_)=1</sup><sup>_toti_∈{0</sup><sup>_,_−1}</sup><sup>_.Ifti_=0</sup><sup>_,then_</sup> _ui_ ( _ti,s_ −<sup>∗</sup> _i_<sup>(</sup><sup>_R_)) = 0 ≤</sup><sup>_ui_(</sup><sup>_s_∗(</sup><sup>_R_)</sup><sup>_,sinceci_≤</sup><sup><u>1</u></sup> 2<sup>_.Ifti_= −1</sup><sup>_,thenui_(</sup><sup>_ti,s_∗</sup> − _i_<sup>(</sup><sup>_R_)) = −</sup><sup>_ci_<</sup><sup>_ui_(</sup><sup>_s_∗(</sup><sup>_R_)</sup><sup>_._</sup> _Therefore, voter i has no incentive to deviate from his truthful vote. Similarly, we show that a voter i_ ∈ _N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ )) _has no incentive to deviate from his truthful vote. This terminates the proof since N_ = _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ )) ∪ _N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ )) _by assumption._

36
**_Necessity :_** _We proceed by contradiction. Assume that s_<sup>∗</sup> ( _R_ ) _is a Nash equilibrium, but at least one of the following conditions fails to hold_ ∣ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ ))∣= ∣ _N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ ))∣ _and ci_ ≤ 2<sup><u>1</u></sup><sup>_foralli_∈</sup><sup>_N.Ifthereexisti_∈</sup><sup>_Nsuchthatci_></sup><sup><u>1</u></sup> 2<sup>_,thenthisimpliesthat_</sup> _ui_ ( _s_<sup>∗</sup> ( _R_ ) = 2<sup><u>1</u>−</sup><sup>_ci_< 0 =</sup><sup>_ui_(0</sup><sup>_,s_</sup> −<sup>∗</sup> _i_<sup>(</sup><sup>_R_))</sup><sup>_.Therefore,votericandeviatefroms_∗</sup> _i_<sup>(</sup><sup>_R_)</sup><sup>_toti_= 0</sup><sup>_,_</sup> _which contradicts the fact that s_<sup>∗</sup> ( _R_ ) _is a Nash equilibrium. If_ ∣ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ ))∣≠∣ _N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ ))∣ _, then let us assume without loss of generality that_ ∣ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ ))∣> ∣ _N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ ))∣ _. It follows that any voter i_ ∈ _N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ )) _can deviate from s_<sup>∗</sup> _i_<sup>(</sup><sup>_R_)</sup><sup>_toti_=0</sup><sup>_,whichcontradictthe_</sup> _fact that s_<sup>∗</sup> ( _R_ ) _is a Nash equilibrium. We find a contradiction in any case, therefore the conditions_ ∣ _N_<sup>1</sup> ( _s_<sup>∗</sup> ( _R_ ))∣= ∣ _N_<sup>−1</sup> ( _s_<sup>∗</sup> ( _R_ ))∣ _and ci_ ≤ 2<sup><u>1</u></sup><sup>_foralli_∈</sup><sup>_Narenecessaryforthe_</sup> _truthful vote to be a Nash equilibrium._

_Furthermore, s_<sup>∗</sup> ( _R_ ) _is not Pareto-efficient since it is Pareto dominated by the strategy profile ti_ = 0 _for all i_ ∈ _N , where every voter decides not to participate in the election. Indeed, we have that ui_ ( _t_ ) =<sup><u>1</u></sup> 2<sup>></sup><sup>_ui_(</sup><sup>_s_∗(</sup><sup>_R_))</sup><sup>_._</sup>

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