1 Introduction
Imagine that you and your friends are choosing a restaurant to go to for dinner. Everybody is asked to name their two most preferred cuisines, and the cuisine named most frequently will be selected (this voting rule is known as 2approval). Your favourite cuisine is Japanese and your second most preferred cuisine is Indian. Indian is quite popular among your friends and you know that if you name it among your favourite two cuisines, it will be selected. On the other hand, you also know that only a few of your friends like Chinese food. Will you vote for Japanese and Chinese to give Japanese cuisine a chance?
This example illustrates that group decisionmaking is a complex process that represents an aggregation of individual preferences. Individual decisionmakers would like to influence the final decision in a way that is beneficial to them, and hence they may be strategic in communicating their individual choices. Moreover, it is essentially impossible to eliminate strategic behavior by changing the voting rule: the groundbreaking result of Gibbard (1973) and Satterthwaite (1975) states that, under any onto and nondictatorial social choice rule, there exist situations where a voter can achieve a better outcome by casting a strategic vote rather than the sincere one, provided that everyone else votes sincerely; in what follows, we will call such voters Gibbard–Satterthwaite (GS) manipulators.
The Gibbard–Satterthwaite theorem alerts us that strategic behavior of voters cannot be ignored, but it does not tell us under which circumstances it actually happens. Of course, if there is just a single GSmanipulator at a given profile, and he^{1}^{1}1We use ‘he’ to refer to voters and ‘she’ to refer to candidates. is fully aware of other voters’ preferences, it is rational for him to manipulate. However, even in this case this voter may prefer to vote truthfully, simply because he assigns a high value to communicating his true preferences; such voters are called ideological. Moreover, if there are two or more GSmanipulators, it is no longer easy for them to make up their mind in favour of manipulation: while the Gibbard–Satterthwaite theorem tells us that each of these voters would benefit from voting strategically assuming that all other voters remain truthful, it does not offer any predictions if several voters may be able to manipulate simultaneously. The issues faced by GSmanipulators in this case are illustrated by the following example.
Example 1.
Suppose four people are to choose among three alternatives by means of 2approval, with ties broken according to the order . Let the profile of sincere preferences be as in Table 1. There are two voters who prefer to to , one voter who prefers to to , and one voter who prefers to to . If everyone votes sincerely, then gets 4 points, gets 3 points and gets 1 point, so is elected. Voters 1 and 2 are Gibbard–Satterthwaite manipulators. Each of them can make the winner by voting , ceteris paribus. Let us consider this game from the first voter’s perspective, assuming that he is strategic; let denote the strategy set of voter , . The strategy set of voter 1 can then be assumed to be (clearly, under 2approval is indistinguishable from , is indistinguishable from , and the two votes that do not rank first are less useful than either and ). Voter 1 has a good reason to believe that voters 3 and 4 will vote sincerely, as voter 3 cannot achieve an outcome that he would prefer to the current outcome and voter 4 is fully satisfied.
voter 1  voter 2  voter 3  voter 4 
Case 1. If voter 1 believes that voter 2 is ideological, then he is analysing the game where , and . In this case he just votes and expects to become the winner.
Case 2. Suppose now that voter 1 believes that voter 2 is also strategic. Now voter 1 has to analyse the game with , and . If either one of the strategic players—voter 1 or voter 2—manipulates and another stays sincere, will be the winner. However, if they both manipulate, their worst alternative will become the winner. Thus, in this case voter 1’s manipulative strategy does not dominate his sincere vote, and if voter 1 is riskaverse, he should refrain from manipulating.
A popular approach (see Section 1.1) is to view voting as a strategic game among the voters, and use various gametheoretic solution concepts to predict the outcomes. The most common such concept is Nash equilibrium, which is defined as a combination of strategies, one for each player, such that each player’s strategy is a best response to other players’ strategies. In these terms, the Gibbard–Satterthwaite theorem says that under every reasonable voting rule there are situations where truthful voting is not a Nash equilibrium. As a further illustration, the game analysed in Example 1 (Case 2) has two Nash equilibria: in the first one, voter 1 manipulates and voter 2 remains truthful, and in the second one the roles are switched.
However, the principle that players can always be expected to choose equilibrium strategies is not universally applicable. Specifically, if players have enough experience with the game in question (or with similar games), both theory and experimental results suggest that players are often able to learn equilibrium strategies (Fudenberg and Levine, 1998). However, it is also wellknown since the early work of Shapley (1964) that learning dynamics may fail to converge to an equilibrium. Moreover, in many applications—and voting is one of them—players’ interactions have only imperfect precedents, or none at all so no learning is possible. If equilibrium is justified in such applications, it must be via strategic thinking of players rather than learning. However, in some games the required reasoning is too complex for such a justification of equilibrium to be behaviourally plausible (Harsanyi and Selten, 1988; Brandenburger, 1992). This is fully applicable to voting, where such reasoning, beyond very simple profiles, is impossible because of the number of voters involved.
In fact, a number of recent experimental and empirical studies suggest that players’ responses in strategic situations often deviate systematically from equilibrium strategies, and are better explained by the structural nonequilibrium level (Nagel, 1995; Stahl and Wilson, 1994) or cognitive hierarchy (CH) models (Camerer et al., 2004); see also a survey by Crawford et al. (2013). In a level model players anchor their beliefs in a nonstrategic initial assessment of others’ likely responses to the game. Nonstrategic players are said to be level0 players. Level 1 players believe that all other players are at level 0, and they give their best response on the basis of this belief. Level 2 players assume that all other players belong to level 1, and, more generally, players at level give their best response assuming that all other players are at level . The cognitive hierarchy model is similar, but with an essential difference: in this model players of level respond to a mixture of types from level to level
. It is frequently assumed that other players’ levels are drawn from a Poisson distribution. Some further approaches based on similar ideas are surveyed by
Wright and LeytonBrown (2010).The aim of our work is to explore the applicability of these models to voting games. We believe that specifics of voting, and, in particular, the heterogeneity of types of voters in real electorates, make the cognitive hierarchy framework more appropriate for our purposes. In more detail, an important feature that distinguishes voting from many other applications of both level and CH models is the role of level0 players. Specifically, level0 (nonstrategic) players are typically assumed to choose their strategy at random, and this type practically does not appear in real games at all. In contrast, in applications to voting it is natural to associate level0 players with ideological voters, who have a significant presence in real elections. For instance, in the famous Florida vote (2000), where Bush won over Gore by just 537 votes, 97,488 Nader supporters voted for Nader—even though in such a close election every strategic voter should have voted either for Gore or for Bush (and an overwhelming majority of Nader supporters preferred Gore to Bush). However, in the level analysis voters of level 2 assume that all other voters have level 1, i.e., level models cannot be used to accommodate ideological voters. We therefore focus on the cognitive hierarchy approach. Moreover, we limit ourselves to considering the first three levels of the hierarchy (i.e., level0, level1, and level2 players), as it seems plausible that very few voters are capable of higherlevel reasoning (see the survey by Crawford et al. (2013) for some evidence in support of this assumption).
Adapting the cognitive hierarchy framework to voting games is not a trivial task. First, it does not make sense to assume that voters’ levels follow a specific distribution. Second, in the standard model of social choice voters’ preferences over alternatives are ordinal rather than cardinal. The combination of these two factors means that, in general, for voters at level 2 or higher their best response may not be welldefined. We therefore choose to focus on strategies that are not weakly dominated according to the voter’s beliefs. We present our formal definitions and the reasoning that justifies them in Section 3.
To develop a better understanding of the resulting model, we instantiate it for a specific family of voting rules, namely, approval with . We develop a classification of level1 strategies under approval and clarify the relationship between level1 reasoning and the predictions of the Gibbard–Satterthwaite theorem (Section 4). We then switch our attention to level2 strategies, and, in particular, to the complexity of computing such strategies. For approval with (i.e., the classic Plurality rule) we describe an efficient algorithm that decides whether a given strategy weakly dominates another strategy; as a corollary of this result, we conclude that under the Plurality rule level2 strategies can be efficiently computed and efficiently recognised (Section 5). We obtain a similar result for 2approval under an additional assumption of minimality (Section 6). Briefly, this assumption means that the level2 player expects all level1 players to manipulate by making as few changes to their votes as possible. However, for larger values of finding level2 strategies becomes computationally challenging: we show that this problem is NPhard for approval with (Section 7). As the problem of finding a level1 strategy under approval is computationally easy for any value of (this follows immediately by combining our characterization of level1 strategies with the classic results of Bartholdi et al. (1989)), this demonstrates that higher levels of voters’ sophistication come with a price tag in terms of algorithmic complexity.
1.1 Related work
There is a substantial body of research in social choice theory and in political science that models nontruthful voting as a strategic interaction, with a strong focus on Plurality voting; this line of work dates back to Farquharson (1969) and includes important contributions by Cain (1978), Feddersen et al. (1990) and Cox (1997), to name a few.
More recently, voting games and their equilibria have also received a considerable amount of attention from computer science researchers, with a variety of approaches used to eliminate counterintuitive Nash equilibria. For instance, some authors assume that voters have a slight preference for abstaining or for voting truthfully when they are not pivotal (Battaglini, 2005; Dutta and Sen, 2012; Desmedt and Elkind, 2010; Thompson et al., 2013; Obraztsova et al., 2013; Elkind et al., 2015b; Obraztsova et al., 2015a). Other works consider refinements of Nash equilibrium, such as subgameperfect Nash equilibrium (Desmedt and Elkind, 2010; Xia and Conitzer, 2010), strong equilibrium (Messner and Polborn, 2007) or tremblinghand equilibrium (Obraztsova et al., 2016), or model the reasoning of voters who have incomplete or imperfect information about each others’ preferences (Myerson and Weber, 1993; Myatt, 2007; Meir et al., 2014). Dominancebased solution concepts have been investigated as well (Moulin, 1979; Dhillon and Lockwood, 2004; Buenrostro et al., 2013; Dellis, 2010; Meir et al., 2014), albeit from a noncomputational perspective. All the aforementioned papers do not impose any restrictions on the voters’ reasoning ability, de facto assuming that they are fully rational. Boundedly rational voters are considered by Grandi et al. (2017); however, their work focuses on strategic interactions among Gibbard–Satterthwaite manipulators, and studies conditions that ensure existence of pure strategy Nash equilibria in the resulting games. In contrast, in this paper we go further and formally define the degree of voters’ rationality by using the cognitive hierarchy approach.
Level models and the cognitive hierarchy framework have been long used to model a variety of strategic interactions; we refer the reader to the survey of Crawford et al. (2013). Nevertheless, to the best of our knowledge, ours is the first attempt to apply these ideas in the context of voting.
A topic closely related to voting games is voting dynamics, where players change their votes one by one in response to the current outcome (Meir et al., 2010; Reijngoud and Endriss, 2012; Reyhaneh and Wilson, 2012; Obraztsova et al., 2015b; Endriss et al., 2016; Lev and Rosenschein, 2016; Koolyk et al., 2017); see also a survey by Meir (2017). However, this line of work assumes the voters to be myopic.
Our work can also be seen as an extension of the model of safe strategic voting proposed by Slinko and White (2014). However, unlike us, Slinko and White focus on a subset of GSmanipulators who (a) all have identical preferences and (b) choose between truthtelling and using a specific manipulative vote, and on the existence of a weakly dominant strategic vote in this setting (such votes are called safe strategic votes). In contrast, our decisionmaker takes into account that manipulators may have diverse preferences and have strategy sets that contain more than one strategic vote. It is therefore not surprising that computing safe strategic votes is easier than finding level2 strategies: Hazon and Elkind (2010) show that safe strategic votes with respect to approval can be computed efficiently for every , whereas we obtain hardness results for .
One of our contributions is a classification of manipulative votes under approval with lexicographic tiebreaking. Peters et al. (2012) propose a similar classification for several approvalbased voting rules. However, they view approval as a nonresolute voting rule, and therefore their results do not apply in our setting.
Paper outline.
The paper is organised as follows. We introduce the basic terminology and definitions in Section 2. Section 3 presents the adaptation of the cognitive hierarchy framework to the setting of voting games. We then focus on the study of approval. Section 4 describes the structure of level1 strategies under approval. In Section 5 we provide an efficient algorithm for identifying level2 strategies with respect to the Plurality rule. Section 6 contains our results for approval, and in Section 7 we present our hardness results for approval with . Section 8 summarises our results and suggests directions for future work.
2 Preliminaries
In this section we introduce the relevant notation and terminology concerning preference aggregation and normalform games.
2.1 Preferences and Voting Rules
We consider voter elections over a candidate set ; in what follows we use the terms candidates and alternatives interchangeably. Let denote the set of all linear orders over . An election is defined by a preference profile , where each , , is a linear order over ; we refer to as the sincere vote, or preferences, of voter . For two candidates we write , if voter ranks above , and say that voter prefers to . For brevity we will sometimes write to represent a vote such that . We denote the top candidate in by . Also, we denote the set of top candidates in by ; note that and for all and .
A (resolute) voting rule is a mapping that, given a profile , outputs a candidate , which we call the winner of the election defined by , or simply the winner at . In this paper we focus on the family of voting rules known as approval. Under approval, , each candidate receives one point from each voter who ranks her in top positions; the approval score of a candidate , denoted by , is the total number of points that she receives. The winner is chosen among the candidates with the highest score according to a fixed tiebreaking linear order on the set of candidates : specifically, the winner is the highestranked candidate with respect to this order among the candidates with the highest score. The approval voting rule is widely used and known as Plurality. We will denote the approval rule (with tiebreaking based on a fixed linear order ) by . We say that a candidate beats candidate at with respect to and the tiebreaking order if or and .
2.2 Strategic Voting
Given a preference profile , and a linear order , we denote by the preference profile obtained from by replacing with ; for readability, we will sometimes omit the parentheses around and write . We will often use this notation when voter submits a strategic vote instead of his sincere vote .
Definition 1.
Consider a profile , a voter , and a voting rule . We say that a linear order is a manipulative vote of voter at with respect to if . We say that manipulates in favour of candidate by submitting a vote if is the winner at . A voter is a Gibbard–Satterthwaite manipulator, or a GSmanipulator, at with respect to if the set of his manipulative votes at with respect to is not empty. We denote the set of all GSmanipulators at by .
Note that a voter may be able to manipulate in favour of several different candidates. Let ; we say that the candidates in are feasible for at with respect to . Note that for all , as this set contains the winner at under truthful voting. We say that two votes and over the same candidate set are equivalent with respect to a voting rule if for every voter and every profile of other voters’ preferences. It is easy to see that and are equivalent with respect to approval if and only if .
2.3 Normalform Games
A normalform game is defined by a set of players , and, for each , a set of strategies and a preference relation defined on the space of strategy profiles, i.e., tuples of the form , where for all ^{2}^{2}2While one usually defines normalform games in terms of utility functions, defining them in terms of preference relations is more appropriate for our setting, as preference profiles only provide ordinal information about the voters’ preferences.. For each pair of strategy profiles and a player , we write if and . A normalform game is viewed as a game of complete and perfect information, which means that all players are fully aware of the structure of the game they are playing.
Given a strategy profile and a strategy , we denote by the strategy profile , which is obtained from by replacing with . We say that a strategy weakly dominates another strategy if for every strategy profile of other players we have and there exists a profile of other players’ strategies such that .
3 The Model
As suggested in Section 1, our goal is to analyse voting as a strategic game and consider it from the perspective of the cognitive hierarchy model. As we reason about voters’ strategic behavior, we consider games where players are voters, their strategies are ballots they can submit, and their preferences over strategy profiles are determined by election outcomes under a given voting rule. We then use the cognitive hierarchy framework to narrow down the players’ strategy sets.
3.1 Cognitive Hierarchy Framework for Voting Games
Recall that, in the general framework of cognitive hierarchy, players at level 0 are typically assumed to choose their action at random. This is because in general normalform games a player who is unable to deliberate about other players’ actions usually has no reason to prefer one strategy over another. In contrast, in the context of voting, there is an obvious focal strategy, namely, truthful voting. Thus, in our model we associate level0 voters with ideological voters, i.e., voters who always vote according to their true preferences.
At the next level of hierarchy are level1 voters. These voters assume that all other voters are ideological (i.e., are at level 0), and choose their vote so as to get the best outcome they consider possible under this assumption. That is, voter votes so as to make his most preferred candidate in the election winner (in particular, if is a singleton, voter votes truthfully). We say that a vote of a voter is a level1 strategy at profile with respect to if for all . Note that a level1 voter that is not a Gibbard–Satterthwaite manipulator has no reason to vote nontruthfully, as he does not expect to be able to change the election outcome according to his tastes; hence we assume that such voters are truthful.
We are now ready to discuss level2 voters. These voters believe that all other voters are at levels 0 or 1 of the cognitive hierarchy. We will further assume that level2 voters are agnostic about other voters’ levels; thus, from their perspective every other voter may turn out to be a level0 voter (which in our setting is equivalent to being sincere) or a level1 voter. Thus, from the point of view of a level2 voter, a voter who is not a GSmanipulator will stick to his truthful vote, whereas a GSmanipulator will either choose his action among level1 strategies or (in case he is actually a level0 voter) vote truthfully. Thus, when selecting his vote strategically, a level2 voter takes into account the possibility that other voters—namely, the GSmanipulators—may be strategic as well.
We further enrich the model by assuming that a level2 voter may be able to identify, for each other voter , a subset of level1 strategies such that always chooses his vote from that subset, i.e., a level2 voter may be able to rule out some of the level1 strategies of other voters. There are several reasons to allow for this possibility. First, the set of all level1 strategies for a given voter can be very large, and a voter may be unable or unwilling to identify all such votes. For example, our level2 voter may know or believe that other voters use a specific algorithm (e.g., that of Bartholdi et al. (1989)) to find their level1 strategies; in this case, his set of strategies for each voter would consist of the truthful vote and the output of the respective algorithm. Also, voters may be known not to choose manipulations that are (weakly) dominated by other manipulations. Finally, voters may prefer not to change their vote beyond what is necessary to make their target candidate the election winner, either because they want their vote to be as close to the true preference order as possible (see the work of Obraztsova and Elkind (2012)), or for fear of unintended consequences of such changes in the complex environment of the game. Thus, a preference profile together with a voting rule define not just a single game, but a family of games, which differ in sets of actions available to GSmanipulators.
3.2 Gibbard–Satterthwaite Games
We will now describe a formal model that will enable us to reason about the decisions faced by a level2 voter. For convenience, we assume that voter is a level2 voter and describe a normalform game that captures his perspective of strategic interaction, i.e., his beliefs about the game he is playing.
Fix a voting rule , let be a profile over a set of candidates , let be the set of GSmanipulators at with respect to , and set . We consider a family of normalform games defined as follows. In each game the set of players is , i.e., voter is a player irrespective of whether he is actually a GSmanipulator. For each player , ’s strategy set consists of his truthful vote and a (possibly empty) subset of his level1 strategies; for voter we have , i.e., can submit an arbitrary ballot. It remains to describe the voters’ preferences over strategy profiles. For a strategy profile , where for , let be the preference profile such that for and for . Then, given two strategy profiles and and a voter , we write if and only if prefers to or . We refer to any such game as a GSgame.
We denote the set of all GSgames for and by . Note that an individual game in is fully determined by the GSmanipulators’ sets of strategies, i.e., (player ’s set of strategies is always the same, namely, ). Thus, in what follows, we write ; when and are clear from the context, we simply write . We refer to a strategy profile in a GSgame as a GSprofile, and we will sometimes identify the GSprofile with the preference profile . We denote the set of all GSprofiles in a game by .
We will now argue that games in reflect the perspective of voter when he is at the second level of the cognitive hierarchy. Fix a game . Note first that, since voter believes that all other voters belong to levels 0 and 1 of the cognitive hierarchy, he expects all voters who are not GSmanipulators to vote truthfully, i.e., he does not need to reason about their strategies at all. This justifies having as our set of players. On the other hand, consider a voter . Voter 1 considers it possible that is a level0 voter, who votes truthfully. Voter 1 also entertains the possibility that is a level1 voter, in which case ’s vote has to be a level1 strategy; as argued above, voter may also be able to rule out some of ’s level1 strategies. Consequently, the set , which, by definition, contains , consists of all strategies that voter considers possible for . Thus, voter ’s view of other voters’ actions is captured by .
We are now ready to discuss level2 strategies. In gametheoretic literature, it is typical to assume that a level2 player is endowed with probabilistic beliefs about other players’ types as well as a utility function describing his payoffs under all possible strategy profiles. Under these conditions, it makes sense to define level2 strategies as those that maximise player 2’s expected payoff. However, in the absence of numerical information, as in the case of voting games, we cannot reason about expected payoffs. We can, however, compare different strategies pointwise, and remove strategies that are weakly dominated by other strategies. On the other hand, if a strategy is not weakly dominated, a level2 player may hold beliefs that make him favour , so no such strategy can be removed from consideration without making additional assumptions about the behavior of players in . This reasoning motivates the following definition of a level2 strategy.
Definition 2.
Given a GSgame , we say that a strategy of player 1 is a level2 strategy if no other strategy of player 1 weakly dominates .
We note that being weakly undominated is not a very demanding property: a strategy can be weakly undominated even if it fares badly in many scenarios. This is illustrated by the following example.
Example 2.
Consider the 4voter profile over given in Table 2. Suppose that the voting rule is Plurality and the tiebreaking rule is . As always, we assume that voter 1 is the level2 voter. Voters 2, 3, and 4 are GSmanipulators; their most preferred feasible candidates are, respectively, , , and . Consider the GSgame where , , . In this game every vote that does not rank first is a level2 strategy for the first player. Indeed, a vote that ranks first is optimal when all other players submit their sincere votes; a vote that ranks first is optimal when players 2 and 3 stay sincere, but player 4 votes for ; and a vote that ranks first is optimal when player 2 votes for , but players 3 and 4 stay sincere. Note, in particular, that, by changing his vote from (his sincere vote) to , player changes the outcome from (his top choice) to (his third choice) when other players vote truthfully; however, this behavior is rational if player 1 expects players 3 and 4 (but not player 2) to vote sincerely.
voter 1  voter 2  voter 3  voter 4 
Example 2 illustrates that level2 strategies are not ‘safe’: there can be circumstances where a level2 strategy results in a worse outcome than sincere voting. Now, a cautious level2 player may prefer to stick to his sincere vote unless he can find a manipulative vote which leads to an outcome that is at least as desirable as the outcome under truthful voting, for any combination of actions of other players that our level2 player considers possible. The following definition, which is motivated by the concept of safe strategic voting (Slinko and White, 2014), describes the set of strategies that even a very cautious level2 player would prefer to sincere voting.
Definition 3.
Given a GSgame , we say that a strategy of player 1 is an improving strategy if weakly dominates player 1’s sincere strategy .
We note that a level2 strategy may fail to be an improving strategy, and conversely, an improving strategy is not necessarily a level2 strategy. For instance, in Example 1 the strategy is a level2 strategy, but not an improving strategy, and none of the level2 strategies in Example 2 is improving. However, it is easy to see that if a player has an improving strategy, he also has an improving strategy that is a level2 strategy. Moreover, an improving strategy exists if and only if sincere voting is not a level2 strategy.
One can also ask if a given strategy weakly dominates all other (nonequivalent) strategies. However, while strategies with this property are highly desirable, from the perspective of a strategic voter it is more important to find out whether his truthful strategy is weakly dominated. Indeed, the main issue faced by a strategic voter is whether to manipulate at all, and if a certain vote can always ensure an outcome that is at least as good, and sometimes better, as that guaranteed by his truthful vote, this is a very strong incentive to use it, even if another nontruthful vote may be better in some situations. This issue is illustrated by Example 3 below, which describes a profile where a player has two incomparable improving strategies.
Example 3.
Let the profile of sincere preferences be as in Table 3, and assume that the voting rule is Plurality and the tiebreaking order is given by . The winner at the sincere profile is . All level1 strategies of voter 2 are equivalent to , whereas all level1 strategies of voter 3 are equivalent to ; voters 4 and 5 are not GSmanipulators. Consider the GSgame where for the set of strategies of player consists of his truthful vote and all of his level1 strategies. Voter 1, who is our level2 player, can manipulate either in favour of or in favour of , by ranking the respective candidate first. Indeed, for player 1 both and weakly dominate truthtelling. However, neither of these strategies weakly dominates the other: is preferable if no other player uses a level1 strategy, whereas is preferable if player 2 uses his level1 strategy, but player 3 votes sincerely.
voter 1  voter 2  voter 3  voter 4  voter 5 
We note that a level2 voter may find it useful to act as a countermanipulator (Pattanaik, 1976; Grandi et al., 2017), i.e., to submit a vote that is not manipulative with respect to the truthful profile, but minimises the damage from someone else’s manipulation.
Example 4.
Let the profile of sincere preferences be as in Table 4, and assume that the voting rule is Plurality and the tiebreaking order is given by . Under truthful voting wins, so voter 6 is the only GSmanipulator: if he changes his vote to then wins, and . Therefore, for voter 1 voting is preferable to voting truthfully: this insincere vote has no impact if voter 6 votes truthfully, but prevents from becoming a winner when voter 6 submits a manipulative vote.
Thus, in this example is an improving strategy, and truthful voting is not a level2 strategy as it is weakly dominated by voting . In contrast, is a level2 strategy, as no other strategy weakly dominates it.
voter 1  voter 2  voter 3  voter 4  voter 5  voter 6 
3.3 Algorithmic Questions
From an algorithmic perspective, perhaps the most natural questions suggested by our framework are how to decide whether a given strategy is a level strategy, or how to compute a level strategy. A related question is whether a given strategy is an improving strategy and whether an improving strategy can be efficiently computed. These questions offer an interesting challenge from an algorithmic perspective: the straightforward algorithm for deciding whether a given strategy is a level2 strategy or an improving strategy relies on considering all combinations of other players’ strategies, and hence has exponential running time. It is therefore natural to ask whether for some voting rules exhaustive choice can be avoided. We explore this question in Sections 5–7; for concreteness, we focus on approval, for various values of .
4 Level1 Strategies Under Approval
The goal of this section is to understand and classify level1 strategies under the
approval voting rule; this will help us reason about level2 strategies in subsequent sections. In what follows, we fix a linear order used for tiebreaking. We start with a simple, but useful lemma.Lemma 1.
Fix . Consider a profile over , let be the approval winner at , and let be an alternative in . Then any manipulative vote by a voter in favour of at falls under one of the following two categories:
 Type 1

Voter increases the score of by 1 without decreasing the score of . In this case , , and the manipulative vote satisfies , . In such cases voter will be referred to as a promoter of .
 Type 2

Voter decreases the score of (and possibly that of some other alternatives) by 1 without increasing the score of . In this case , , and the manipulative vote satisfies , . In such cases voter will be referred to as a demoter of . Manipulations of type 2 only exist for .
Proof.
Suppose that voter manipulates in favour of . If can increase the score of , then . However, must rank higher than (otherwise, this would not be a manipulation). Thus, and therefore voter cannot decrease ’s score. Moreover, if , then would beat under approval in ; thus, .
On the other hand, suppose that cannot increase the score of . This means that and hence is left with reducing the scores of some of ’s competitors including the current winner . For this to be possible, it has to be the case that and . Also, we have , as otherwise would beat under approval in . Finally, as , we can only have if . ∎
The classification in Lemma 1 justifies our terminology: a promoter promotes a new winner and a demoter demotes the old one. Under Plurality, i.e., when , we only have promoters.
Let and be two disjoint sets of candidates. Given a linear order over , we denote by the vote obtained by swapping with for . If the sets and are singletons, i.e., , , we omit the curly braces, and simply write . Clearly, under approval any manipulative vote of voter is equivalent to a vote of the form , where , . We can now state a corollary of Lemma 1, which characterises the possible effects of a manipulative vote under approval.
Corollary 2.
Let be the approval winner at a profile , let , where and . Let , and let be the approval winner at . Then either or but not both.
Proof.
Lemma 1 implies that either the new winner is promoted or the old winner is demoted, but not both. ∎
Consider a manipulative vote of voter at under approval; we say that is minimal if for every other manipulative vote of voter there is a vote that is equivalent to and satisfies . That is, a manipulative vote is minimal if it performs as few swaps as possible. Arguably, minimal manipulative votes are the main tool that a rational voter would use, as they achieve the desired result in the most straightforward way possible.
We now introduce some useful notation. Fix a profile . Let be the approval winner at , and let . Set
and set .
We say that a candidate is competitive at if . The following proposition explains our choice of the term: only competitive candidates can become approval winners as a result of a manipulation.
Proposition 1.
Suppose that some voter can manipulate in favour of a candidate at a profile with respect to approval. Then .
Proof.
Let be the approval winner at ; clearly, . Suppose that voter can manipulate in favour of at by submitting a vote ; let . Set ; then . Note that if , it has to be the case that , since otherwise would beat at . Thus, in this case . Now, suppose that . By Corollary 2 we have either (if was promoted) or (if was demoted). In both cases we have to have , as otherwise would beat at . Therefore, in this case . Finally, note that it cannot be the case that , since in this case by Corollary 2 we would have either , or , , i.e., would beat at . ∎
Suppose that . If , then by we denote the topranked candidate in with respect to ; otherwise, we denote by the topranked candidate in with respect to . Thus, beats all candidates other than at , and would become a winner if it were to gain one point or if were to lose one point. We omit and from the notation when they are clear from the context.
We are now ready to embark on the computational complexity analysis of level2 strategies under approval, for various values of .
5 Plurality
Plurality voting rule is approval with . For this rule we only have manipulators of type 1, and all manipulative votes of voter in favour of candidate are equivalent: in all such votes is placed in the top position.
The main result of this section is that the problem of deciding whether a given strategy of voter 1 weakly dominates another strategy of that voter admits a polynomialtime algorithm. Note that, since under Plurality there are only votes that are pairwise nonequivalent, this means that we can check if a given strategy is a level2 strategy or an improving strategy, or find a level2 strategy or an improving strategy (if it exists) in polynomial time; we formalise this intuition in Corollary 5 at the end of this section.
Fix a preference profile over a candidate set and consider a GSgame , where . Let be the Plurality winner at . As argued above, for each the set consists of and possibly a number of pairwise equivalent manipulative votes; without loss of generality, we can remove all but one manipulative vote, so that for all . We will now explain how, given two votes and , voter can efficiently decide if one of these votes weakly dominates the other.
We will first describe a subroutine that will be used by our algorithm.
Lemma 3.
There is a polynomialtime procedure
that, given a GSgame with , two integers , two distinct candidates , and a partition of candidates in into , , and , decides whether there is a strategy profile in such that

,

, and

for each and each if then .
Proof.
We proceed by reducing our problem to an instance of network flow with capacities and lower bounds, as follows. We construct a source, a sink, a node for each voter and a node for each candidate in . There is an arc from the source to each voter node; the capacity and the lower bound of this arc are set to , i.e., it is required to carry one unit of flow. Also, there is an arc with capacity and lower bound 0 from voter to candidate if and for some or if and . Finally, there is an arc from each candidate to the sink. The capacity of this arc is set to if for some ; the lower bounds for these arcs are . For , both the capacity and the lower bound of the arc to the sink are set to , and for they are both set to . We note that some of the capacities may be negative, in which case there is no valid flow. It is immediate that an integer flow that satisfies all constraints corresponds to a strategy profile in where all candidates have the required scores; it remains to observe that the existence of a valid integer flow can be decided in polynomial time. ∎
We are now ready to describe our algorithm.
Theorem 4.
Given a GSgame and two strategies of player 1 we can decide in polynomial time whether weakly dominates .
Proof.
We will design a polynomialtime procedure that, given two strategies of player 1, decides if there exists a profile of other players’ strategies such that ; by definition, weakly dominates if this procedure returns ‘yes’ for , and ‘no’ for .
Let , . We can assume without loss of generality that , since otherwise and are equivalent with respect to Plurality. Consider an arbitrary profile of other players’ strategies, and let , , , . We note that implies : if beats at , this is also the case at . Similarly, if then also . Now, suppose that and . We claim that in this case . Indeed, suppose for the sake of contradiction that . As , , the argument above shows that . Thus, both and have the same Plurality score at and ; as beats at , this must also be the case at , a contradiction.
Note that if and only if . By the argument in the previous paragraph, this can happen in one of the following three cases: (i) , and ; (ii) , for some , ; (iii) , for some , . (We note that we can merge case (i) into case (ii) or case (iii); we choose not to do so for the sake of clarity of presentation.) We will now explain how to check if there exists a profile that corresponds to any of these three situations.

Case (i): , .
Suppose first that . Then a desired profile exists if and only if there is some value such that and

, for all with , for all with , or

, for all with , and for all with .
Note that and for . Thus, to check if condition (a) is satisfied for some , we set , , and call
Similarly, to determine whether condition (b) is satisfied for some , we set , , and call
The answer is ‘yes’ if one of these calls returns ‘yes’ for some .
For the case the analysis is similar. In this case, we need to decide whether there exists a value of such that and

, for all with , and for all with , or

, for all with , and for all with .
Again, this can be decided by calling the procedure with appropriate parameters; we omit the details.


Case (ii): , for some with . In this case, we go over all candidates with and all values of and call with appropriate parameters.
Specifically, if , we start by setting , and
We then place in if and in otherwise; our treatment of reflects the fact that she gets an extra point at .
If we start by setting , and
We then place in if and in otherwise.
Finally, we call
The answer is ‘yes’ if one of these calls returns ‘yes’ for some and some with .

Case (iii): , for some with . The analysis is similar to the previous case; we omit the details.∎
Theorem 4 immediately implies that natural questions concerning level2 strategies and improving strategies are computationally easy.
Corollary 5.
Given a GSgame and a strategy of player 1 we can decide in polynomial time whether is a level2 strategy or an improving strategy. Moreover, we can decide in polynomial time whether player 1 has a level2 strategy or an improving strategy in .
Proof.
Let . To decide whether is an improving strategy, we use the algorithm described in the proof of Theorem 4 to check whether weakly dominates . Similarly, to decide whether is a level2 strategy, for each we construct a vote with and check whether weakly dominates using the algorithm from the proof of Theorem 4. As every strategy of player is equivalent either to or to one of the votes we constructed, is a level2 strategy if and only if it is not weakly dominated by any of the votes , .
Similarly, to decide whether has a level2 strategy (respectively, an improving strategy), we consider all of his pairwise nonequivalent strategies, and check if any of them is a level2 strategy (respectively, an improving strategy), as described in the previous paragraph. ∎
6 2Approval
In this section, we study the computational complexity of identifying level2 strategies and improving strategies in GSgames under approval. We show that if the level2 player believes that level1 players can only contemplate minimal manipulations, he can efficiently compute his level2 strategies as well as his improving strategies. As argued in Section 4, minimality is a reasonable assumption, as level1 players have no reason to use complex strategies when simple strategies can do the job.
Specifically, we prove that, under the minimality assumption, given two strategies and , the level2 player can decide in polynomial time whether one of these strategies weakly dominates the other; just as in the case of Plurality, this implies that he can check in polynomial time whether a given strategy is a level2 (respectively, improving) strategy or identify all of his level2 (respectively, improving) strategies.
The following observations play a crucial role in our analysis.
Proposition 2.
Consider a GSgame . Let be the 2approval winner at . Then for each player such that it holds that