Robust Voting Equilibria

I first provide two examples that illustrate how the concept of robust equilibrium refines the set of Nash equilibria in the voting game. Generally, every voting profile in which more than a majority of voters support a party is a Nash equilibrium of G(T, θ, q, 0) regardless of voters’ preferences. The example below shows that my solution concept permits such an equilibrium only when it is driven by voters’ preferences.

Example 1 Let n = 101 and $t_{i}=\frac{i-1}{100}$ for each t_i ∈ T. Let ℓ = 3 and $\theta =(0,\frac{3}{8},\frac{9}{16})$ . Let $q=\frac{1}{4}$ . Consider any voting profile $\bar{v}$ in which at least as many as 52 voters vote for party 1. Let ε > 0. Party 1 is the decisive party in the parliament, so the set of acceptable proposals in the bargaining stage is $A(\bar{v})=[-\frac{1}{4},\frac{1}{4}]$ . Then, party 1 would propose zero, and parties 2 and 3 both would propose $\frac{1}{4}$ , i.e., $p(\bar{v})=(0,\frac{1}{4},\frac{1}{4})$ . Then,

Since $b_{1}(\bar{v})\ge 52$ , there must be a voter t such that v_t = 1 and $t>\frac{1}{8}$ . Consider voter t’s deviation by voting for party 2. The voter’s payoff from the deviation is

Thus,

which is negative for every ε > 0 since $t>\frac{1}{8}$ . Thus, the strategies do not constitute a robust voting equilibrium of G(T, θ, q, 0). Similarly, we can show that there is no robust equilibrium in which party 2 receives more than 51 votes.

I now consider the following voting profile $\hat{v}$ :

In the profile, 54 voters support party 3, so the party is a majority party. Then $A(\hat{v})=[\frac{1}{4},\frac{7}{8}]$ , and $p(\hat{v})=(\frac{1}{4},\frac{3}{8},\frac{9}{16})$ . Suppose a voter t with $\hat{v}_{t}=i$ deviates by voting for j. Then, the deviation would decrease the probability of $p_{i}(\hat{v})$ being the outcome by $\frac{\epsilon }{101}$ and increase the probability of $p_{j}(\hat{v})$ being the outcome by the same amount. Thus, such a deviation would be profitable only if the voter prefers j’s proposal to i’s proposal. However, each voter votes for the party whose proposal she likes the most among the three expected proposals in $\hat{v}$ , and, thus, the deviation does not increase the voter’s payoff. As this is true for every positive ε, $\hat{v}$ is a robust equilibrium. Note that party 3 is the unique party that locates on the right of the median voter and that a majority of voters likes the party’s position the most. Thus, in contrast to the Nash equilibria in which party 1 receives more than a majority of votes, $\hat{v}$ seems plausible. $\hfill\Box$

The next example considers robust equilibria with no majority party.

Example 2 Let T be as in Example 1. Let ℓ = 4, $\theta =(0,\frac{1}{4},\frac{1}{2},1)$ , and $q=\frac{1}{8}$ . Consider the following profile v:

Figure 1 illustrates the strategies. The election result is b(v) = (19, 19, 31, 32). Party 3 is the median party, so $A(v)=A_{3}=[\frac{1}{8},\frac{7}{8}]$ . Thus, the proposals are $p(v)=(\frac{1}{8},\frac{1}{4},\frac{1}{2},\frac{7}{8})$ . Consider a voter t who votes for party i in v. If the voter deviated by voting for another party j, then the probability of i’s proposal being the outcome would decrease by $\frac{1}{101}$ and the probability of j’s proposal being the outcome would increase by the same amount. Such a deviation cannot increase voter t’s expected payoff since every voter votes according to her preference over the expected proposals in v. Thus, v is a robust voting equilibrium. $\hfill\Box$

FIGURE 1.

Strategically Sincere Equilibrium: Example 2

In the equilibria of Examples 1 and 2, all voters vote sincerely with respect to their preferences over the equilibrium proposals by parties. I will call such a voting behavior strategically sincere voting.

Definition 4 Given voting profile v = (v_t ) _{t ∈ T} , we say v_t is strategically sincere for t in v if $v_{t}\in \arg \max \lbrace u(p_{i}(v);t)|i\in L\rbrace$ .

Strategically sincere voting is distinguished from naive sincere voting, which might be defined as voting according to voters’ preferences over the party positions θ = (θ₁, . . ., θ _n ). In contrast to that, strategically sincere voting requires that voters should correctly forecast the optimal policy making strategies for parties, which in turn depend on the aggregate election result. As seen in Figure 1, the voters whose ideal points are between $\frac{11}{16}$ and $\frac{3}{4}$ in Example 2 would vote for party 3 if they voted according to their preferences over the original party positions. However, they vote for party 4 in the equilibrium as they understand that party 4 will moderate its policy choice in order to form a majority coalition with party 3. For each voting profile v, let T*(v) be the set of voters who vote strategically sincerely in v. We say a voting profile v is strategically sincere if every voter votes strategically sincerely in it; that is, T*(v) = T. Such a profile seems simple and natural in the sense that, in it, each party represents a single policy and voters’ decisions depend only on their preferences over the policies. I first investigate the conditions for a strategically sincere voting profile to be a robust equilibrium.

Observe that in the equilibria of the two examples, no voter is majority- or median-pivotal. Generally, when a voter is neither majority- nor median-pivotal, a change of the voter’s choice, say from party i to party j, would not alter any party’s proposals and would only transfer a small probability from j’s proposal to i’s proposal in the lottery of the policy outcome. Hence, the voter must vote for a party whose proposal is the best. A lemma follows from this discussion.

Lemma 1 Assume that t ∈ T is neither majority-pivotal nor median-pivotal in v. Then, v_t is a robust best response to v _{− t} if and only if v_t is strategically sincere for t in v.

The lemma is useful to find conditions for strategically sincere equilibria, as it implies the following. If a voting profile is strategically sincere and no voter is majority- or median-pivotal in it, then it must be a robust equilibrium. Conversely, if no voter is majority- or median-pivotal in a robust equilibrium, then the equilibrium must be strategically sincere. Whether or not there is such an equilibrium will depend on the distribution of voter ideal points, the distribution of party positions, and the location of the status quo.

A few more notations are necessary to present our results. For each i, j ∈ L, let $s_{j}^{i}=\min \lbrace \max \lbrace \theta _{j},\underline{x}_{i}(q)\rbrace ,\overline{x}_{i}(q)\rbrace$ . Note that sⁱ _j uniquely maximizes party j’s utility in A_i , and, thus, it would be proposed by party j whenever party i is decisive in the parliament. For each i = 1, . . ., ℓ − 1, let $\overline{y}_{i}=\frac{\theta _{i}+s_{i+1}^{i}}{2}$ , and let $\overline{y}_{\ell }=t_{n}$ . For each i = 2, . . ., ℓ, let $\underline{y}_{i}=\frac{s_{i-1}^{i}+\theta _{i}}{2}$ , and let $\underline{y}_{1}=t_{1}$ . For each i ∈ L, let $X_{i}=\lbrace t\in T|\underline{y}_{i}\le t\le \overline{y}_{i}\rbrace$ . The interpretation of the set X_i is as follows. When party i is the decisive party in the parliament, the party will propose its ideal point θ _i . Thus, sⁱ _{i − 1} and sⁱ _{i + 1} are the two proposals that are adjacent to party i’s proposal. Then, if a voter’s ideal point belongs to X_i , then party i’s proposal is a best proposal for the voter when party i is the decisive party. Thus, X_i is the set of voters whose strategically sincere choices are party i when the party is decisive in the parliament.

In what follows, I present the results of the equilibrium analysis for four different cases that are mutually exclusive and jointly exhaustive. The division of the cases is expressed in terms of properties of X_m , the set of core supporters of party m, loosely speaking. From the above definition, the set is a function of exogenous parameters and can be written as follows: When m ⩾ 2,

and when m = 1, X_m is the set of voter ideal points that are less than or equal to $\frac{\theta _{m}+\min \lbrace \theta _{m+1},\max \lbrace 2\theta _{m}-q,q\rbrace \rbrace }{2}$ . Thus, the distribution of voter ideologies, the distribution of party positions, and the location of the status quo together determine the set. The equilibrium election outcome critically depends on how many and which voters are included in X_m . In general, X_m includes more voters as the positions of parties m − 1 and m + 1 are farther from that of party m, as more voters are located in the vicinity of party m’s position, and as the status quo is farther from party m’s position.

Generalizing the observation in Example 1, my first proposition identifies a necessary and sufficient condition for equilibria in which a party receives more than a majority of votes.

Proposition 1 For every (T, θ, q) ∈ P , the following is true:

1. 1. If |X_m | > M , then there exists a robust equilibrium v of G(T, θ, q, 0) such that b_m (v) > M and v is strategically sincere.

2. 2. If v is a robust equilibrium of G(T, θ, q, 0) and b _k(v)(v) > M , then k(v) = m and |X_m | > M.

Thus, there is a robust equilibrium in which one party receives more than a majority of votes if and only if there is a party such that more than a majority of voters prefer its position to any other party’s expected proposal in the parliament. When the condition is met, party m receives more than M votes in a strategically sincere profile, as all voters in X_m support it. Then, no voter is majority- or median-pivotal in the profile, which, therefore, is guaranteed to be a robust equilibrium by Lemma 1. Conversely, if party i receives more than M votes in equilibrium, then no voter is majority- or median-pivotal. Lemma 1 implies that the equilibrium is strategically sincere. Then, only the voters in |X_i | vote for party i. Hence, X_i must contain more than M voters, and this is possible only for the party that is closest to the median voter, i = m.

Notice that the condition |X_m | > M implies that more than M voters like party m’s position the most among all party positions. Thus, the type of equilibria exist only when one party is the first choice of more than a majority of voters in terms of their sincere preferences. This condition is hardly met by elections in multiparty PR systems, which seems a clear explanation of why we rarely observe a majority party in those systems. However, this most prominent observation in PR elections has not been well accounted for in prior game-theoretic studies on voting. As mentioned earlier, the Nash concept makes many implausible predictions. Moreover, the (iterative) elimination of weakly dominated strategies does not have much bite in multiparty contests. An alternative employed in some studies is strong equilibrium (Baron and Diermeier Reference Baron and Diermeier2001; Indridason Reference Indridason2011). Although it greatly mitigates the problem of indeterminacy, the solution concept still tends to predict election results in which a large majority of voters support one party. Consider, for instance, the voting profile in which all voters vote for party 3 in the setting of Example 2. In it, the policy outcome is party 3’s ideal point which coincides with the median voter ideal point. This is a strong Nash equilibrium in G(T, θ, q, 0) because only a majority coalition could change the outcome and it is impossible to construct a majority of voters all of whom prefer some policy lottery over the median ideal point. However, this equilibrium involves coordination among a large number of voters and seems unlikely to be played.

Applied to my model, the strong Nash concept predicts that the profile in which every voter votes for party m is an equilibrium in many cases. The prediction, of course, is inconsistent with what we observe in PR elections.Footnote ⁴ Why does not such large coordination of votes occur? My solution concept provides one possible reason. Voters may believe that their voting for a party has some chance to increase the party’s influence on policies even when another party holds a majority of seats in the parliament. In general, a majority party gets its own way. However, minority opposition parties may not be totally powerless in policy making, and, moreover, their influence may depend on their seat shares. Voters may believe the possibility of minority influence on policy making because they can be uncertain about the election result or unity of the majority party, or because minority parties may have some institutional sources of power such as committees. If a voting strategy is not robust against an infinitesimal possibility of such a minority influence, it seems unlikely to be chosen. Proposition 1 shows that such a small perturbation is sufficient to remove all implausible Nash equilibria in which one party is supported by more than a majority of voters including those who do not prefer the party.

Next, I discuss strategically sincere equilibria in which no party receives a majority of votes.

Proposition 2 If |X_m | < M − 1 and {t _{M − 1}, t_M , t _{M + 1}}⊆X_m , then there exists a robust voting equilibrium v of G(T, θ, q) such that k(v) = m, b_m (v) < M − 1, and v is strategically sincere.

The proposition finds a sufficient condition for minority equilibria in which party m is the median party and all voters vote strategically sincerely. Suppose the condition in Proposition 2 holds. I consider a strategically sincere profile in which all voters in X_m vote for party m. For those voters who vote for parties on the left of party m, their ideal points must be less than $\underline{y}_{m}$ . But since $t_{M-1}\ge \underline{y}_{m}$ , the sum of the votes for parties on the left of m is less than M − 1. Applying a symmetric argument, we conclude that the same is true for parties on the right of m. This implies that no voter is median-pivotal. Also, since party m receives less than M − 1 votes, no voter is majority-pivotal. Then, Lemma 1 implies that the profile is a robust equilibrium. Notice that Example 2 belongs to this case.

In my view, the conditions in Proposition 2 is highly likely to be satisfied in the real world elections under PR. Note that the set X_m is always contained in the set of voters who rank party m at the top of their preference orderings over the party positions. Thus, the condition |X_m | < M − 1 will be met whenever voters whose first choice is the median party constitute less than 50 percent of the electorate. We seldom find that public opinion polls tell the contrary in multiparty PR systems. The second condition is that the median voter and the two voters adjacent to the median must prefer the median party’s position to any other party’s expected proposal in the parliament. In the model, the condition is equivalent to |θ _m − q| ⩾ 2max{|θ _m − t _{M − 1}|, |θ _m − t _{M + 1}|} and $t_{M+1}\le \frac{\theta _{m}+\theta _{m+1}}{2}$ . In general, it is easily satisfied when the median party locates close to the median voter, and voters are densely populated within a close neighborhood of the median. Suppose, for example, that the median party locates at the median voter ideal point and that there are so numerous voters that the distribution of ideal points are well approximated by a continuous distribution. In that setting, the condition is always met. Although we do not have reliable measures of parties’ expected proposals, the data on ideologies of voters and parties in general show that voters are densely populated around the median (e.g., Powell and Vanberg Reference Powell and Vanberg2000). This suggests that the condition may be satisfied in many cases.

In this most likely scenario, the model provides very intuitive predictions. In the equilibria in Proposition 2, all voters anticipate that party m, the centrist party, will be the median party. Each party, then, is evaluated by the policy it will offer to make a coalition with the median party. They then vote according to their preferences over the compromised policies. Second, as seen in Example 2, each party’s constituency is ideologically connected, and the ideological “order” among voters are preserved in their vote choices. The existence of robust equilibria with such a natural property is generalized as will be shown later. Lastly, our equilibria do not involve a difficult coordination problem among voters. Each voter’s choice in equilibrium remains a best response even when she does not correctly forecast other voters’ strategies as long as she expects the centrist party to be the median party. It is worthwhile to note that the equilibrium with no majority party in Example 2 is not a strong Nash equilibrium. A group of voters on the left of party 3’s constituency could vote for the party to make it a majority party, which would increase all deviators’ payoffs. On the other hand, there is a strong Nash equilibrium in which all voters vote for party 3, as mentioned before. Thus, applied to the setting of Example 2, the strong Nash refinement seems too strong in the sense that it removes a plausible equilibrium and too weak in the sense that it predicts the equilibria where all voters vote for one party.

Proposition 2 helps us understand the aforementioned 1998 Swedish election. The sufficient condition in the proposition was clearly satisfied by the election as opinion polls showed that no party was top-ranked by a majority of the Swedish voters. Also, the election was largely understood as a one-dimensional competition in which the main issue was the “recovery of state finances” implemented the incumbent Social Democratic Party government (Möller Reference Möller1999). We can reasonably assume that the voters expected that no party would receive a majority of votes in the election and important policy programs are decided in parliamentary politics. Although my stylized model is in no way a perfectly accurate description of Swedish parliamentary rules, it still can capture how voters’ choices were shaped by their concerns about parliamentary bargaining. After the election, the minority government by the Social Democratic Party was formed announcing that the Left Party and the Green Party would cooperate with the government. Before the 1998 election, the important austerity measures by the Social Democratic government were mainly supported by the center-right Center Party. Outcome-oriented leftist voters might reasonably anticipate that the Social Democratic government in cooperation with the Left Party would adopt more leftist policies than the Social Democratic government with the support of the Center Party, but it would not adopt the Left Party’s ideal policy. Also, they might believe that an increase in the seat share of the Left Party would increase the likelihood that the Social Democratic Party would turn to the Left Party. Perhaps these expectations made moderate leftist voters vote for the Left Party. Indeed, the Left Party increased its seats dramatically: it obtained 43 seats almost twice as many as what it received in the 1994 election. As a result, the Social Democratic Party turned to them instead of one of the center-right parties. Voters in my simplified model have incentives to evaluate parties in terms of their compromised policies and to increase the seat share of the party that may give the best compromised outcome. This essentially captures the strategic considerations of the Swedish leftist voters described above.Footnote ⁵

I have so far discussed equilibria in which no voter is majority- or median-pivotal. The next result is about the existence of robust equilibria in which some voters are majority-pivotal.

Proposition 3 Assume {t _{M − 1}, t_M , t _{M + 1}}⊆X_m . Then, the following is true.

1. 1. If |X_m | = M , then there exists a robust equilibrium v of G(T, θ, q, 0) such that b_m (v) = M and v is strategically sincere.

2. 2. If |X_m | = M − 1, then there exists a robust voting equilibrium v of G(T, θ, q, 0) such that b_m (v) = M and T*(v) = T∖{t*} for some t* ∈ T.

There is a robust equilibrium in which party m receives the minimum majority of votes when X_m contains {t _{M − 1}, t_M , t _{M + 1}} and the number of voters in the set is either M or M − 1. First, when X_m contains exactly M voters, the equilibrium is strategically sincere. The intuition is the following. Consider a profile, say v, in which all voters vote strategically sincerely and all voters in X_m , including indifferent ones, vote for party m. In it, the voters whose ideal points are less than $\underline{y}_{m}$ vote for some party on the left of party m, and the ones whose ideal points are greater than $\overline{y}_{m}$ vote for some party on the right of party m. Then, since $\underline{y}_{m}\le t_{M-1}<t_{M+1}\le \overline{y}_{m}$ , ∑ ^{m − 1} _{i = 1} b_i (v) < M − 1 and ∑^ℓ _{i = m + 1} b_i (v) < M − 1. This implies that no voter is median-pivotal in v. Moreover, only voters in X_m are majority-pivotal. Then, by Lemma 1, for those who are not in X_m , their strategically sincere choices are robust best responses. Now consider any voter t in X_m and let v′ be the profile that is identical to v except that voter t votes for some other party. Since the voter is not median-pivotal, party m would still be the decisive party in v′. Thus, her deviation would not change the proposals by the parties. In G(T, θ, q, ε) with small positive ε, the policy lotteries induced by v and v′ have the same supports. Note that the deviation transfers the probability close to one-half from party m’s proposal to the other parties’ proposals. However, since voter t already voted for the party whose proposal is the best in v, the deviation is not profitable.

When X_m contains M − 1 voters, there is exactly one voter whose choice is not strategically sincere. Again, consider a voting profile, say $\hat{v}$ , in which all voters vote strategically sincerely and all voters in X_m vote for party m. Now, party m receives M − 1 votes, one vote less than a majority. As in the previous case, no voter is median-pivotal and party m is the median party. Any voter who does not vote for party m could make it a majority party by voting for it. Without loss of generality, assume that the mean of the policy lottery induced by $\hat{v}$ is greater than or equal to θ _m . Let t* be the rightmost voter among those on the left of $\underline{y}_{m}$ . The voter prefers the degenerate lottery on θ _m to the lottery induced by $\hat{v}$ since her preference is strictly concave. Thus, when ε is sufficiently small, she prefers to make party m the majority party. This means that the profile $\hat{v}$ is not a robust equilibrium. However, the profile, say $\bar{v}$ , in which voter t* votes for party m and all else is the same as in $\hat{v}$ is a robust equilibrium. In it, party m is the majority party receiving M votes. Those who do not vote for party m are neither majority-pivotal nor median-pivotal. As they vote strategically sincerely, they play robust best responses. For those who vote for party m except voter t*, the argument in the previous case is applied. Lastly, if voter t* deviates, party m will be the median party, and, thus, no proposal will be changed. Then, the best possible deviation for the voter is to vote strategically sincerely. Note that such deviation from $\bar{v}$ leads to $\hat{v}$ . However, by construction, voter t* prefers $\bar{v}$ to $\hat{v}$ . Thus, $\bar{v}$ is a robust equilibrium.

It should be noted that Proposition 3 only finds a sufficient condition for there to be an equilibrium in which some voters are majority-pivotal. Depending on parameters, there can be a majority-pivotal equilibrium when the conditions in Proposition 1 or Proposition 2 are satisfied. We will later discuss the issue of multiple equilibria generally.

The next result discusses the remaining cases in the parameter space of the model.

Proposition 4 Assume {t _{M − 1}, t_M , t _{M + 1}}⊈X_m . Then, the following is true:

1. 1. If t _{M + 1} ∈ X_m , then there exists a robust voting equilibrium v of G(T, θ, q, 0) such that k(v) = m and v is strategically sincere.

2. 2. If t _{M + 1}∉X_m , then there exists a robust voting equilibrium v of G(T, θ, q, 0) such that either (1) k(v) = m and $T\setminus T^{*}(v)=\lbrace t\in T\:|\: \overline{y}_{m}<t\le t_{M}\rbrace$ ; or (2) k(v) = m + 1 and $T\setminus T^{*}(v)=\lbrace t\in T\:|\: t_{M}\le t<\underline{y}_{m+1}\rbrace$ .

In the equilibria in Proposition 4, some voters are median-pivotal. I first consider the case that t _{M + 1} ∈ X_m but t _{M − 1}∉X_m . In a strategically sincere profile, party m will be the median party, but the parties on the left of it will receive exactly M − 1 votes. Thus, every voter who votes for party m or those on the right of it is median-pivotal. By voting for one of the parties on the left of party m, she could make party m − 1 the median party. Such a deviation would lead to changes in party proposals in general. Specifically, some parties’ proposal would move toward the left. As such, the voter who has the strongest incentive to deviate is the leftmost voter among the median-pivotal voters, the median voter t_M . Essentially, the median compares the policy lottery induced by the strategically sincere profile and the policy lottery which her voting for m − 1 would result in. The fact that t_M ⩾ θ _m guarantees that the median prefers the former to the latter, which will be rigorously proved in the Appendix.

The situation is more complex when t _{M + 1}∉X_m , and the intuition is best illustrated by an example.

Example 3 Let T be as in the previous examples. Let ℓ = 4, $\theta =(0,\frac{3}{8},\frac{3}{4},1)$ , and $q=\frac{1}{4}$ . Define the voting profile $\bar{v}$ by the following:

The election result is $b(\bar{v})=(32,19,37,13)$ . Thus, party 2 is the median party, which implies that $A(v)=[\frac{1}{4},\frac{1}{2}]$ and $p(\bar{v})=(\frac{1}{4},\frac{3}{8},\frac{1}{2},\frac{1}{2})$ . Those voters who vote for parties 3 and 4 are neither majority- nor median-pivotal. Thus, their strategically sincere votes are robust best responses. Any voter who votes for parties 1 or 2 could make party 3 decisive by voting for 3 or 4. Pick a voter t who votes for party 2 in $\bar{v}$ and consider her deviation by voting for party 3. Let $\hat{v}$ denote the profile induced by the deviation. In it, the election result is $b(\hat{v})=(32,18,38,13)$ . Since party 3 is the median party, $A(\hat{v})=[\frac{1}{4},\frac{5}{4}]$ , and, thus, $p(\hat{v})=(\frac{1}{4},\frac{3}{8},\frac{3}{4},1)$ . For every ε ∈ [0, 1),

and

Subtracting (7) from (6), we obtain

which is positive because $t<\frac{1}{2}$ . Similarly, we can show that any deviation for any voter who vote for 1 or 2 is not profitable. Therefore, $\bar{v}$ is a robust voting equilibrium. $\hfill\Box$ .

In the example, party m = 2 because $t_{M}=\frac{1}{2}$ . When party 2 is the median party, the median voter prefers party 3’s proposal, $\frac{1}{2}$ , to party 2’s proposal, $\frac{3}{8}$ . But if the median voted for party 3, party 2 would not be the median party. This implies that there is no strategically sincere equilibrium in this setting. Note that the seven voters whose ideal points are in $(\frac{7}{16},\frac{1}{2}]=(\overline{y}_{m},t_{M}]$ vote for party 2 even though they prefer party 3’s proposal to party 2’s proposal. In a sense, these voters coordinate their votes to make party 2 the median party.

The key intuition in the proof of the proposition is the following. We can construct two profiles $\bar{v}$ and $\hat{v}$ , as in Example 3, so that: (1) Party m is the median party in $\bar{v}$ and party m + 1 is the median party in $\hat{v}$ . (2) The median voter votes for party m in $\bar{v}$ , and she is the rightmost voter among those who do so. (3) The median voter votes for party m + 1 in $\hat{v}$ , and she is the leftmost voter among those who do so. (4) The median’s deviation from $\bar{v}$ by voting for m + 1 leads to a policy lottery that is not better for her than the lottery in $\hat{v}$ . (4) The median’s deviation from $\hat{v}$ by voting for m leads to a policy lottery that is not better for her than the lottery in $\bar{v}$ . Then, either the median weakly prefers $\bar{v}$ to $\hat{v}$ , or $\hat{v}$ to $\bar{v}$ . When the former is true, $\bar{v}$ is an equilibrium (Case 2(1) in Proposition 4); when the latter is true, $\hat{v}$ is an equilibrium (Case 2(2)). In the Appendix, I prove that it is always possible to construct such two profiles.

The previous propositions altogether imply the general existence of robust equilibria. Thus, while my solution concept is able to remove some implausible Nash equilibria, it never has the problem of an empty prediction. Moreover, notice that, in every equilibrium in my examples, the set of voters who support any given party is a “connected” set. Formally, we say a voting profile v is order preserving if, for all t, t′ ∈ T, v_t < v _t′ implies t < t′. In an order-preserving profile, one that is ideologically more rightist than another never votes for a party that is more leftist than the party supported by the latter. The proofs of the previous propositions in fact construct order-preserving equilibria. Thus, we have:

Corollary 1 For every (T, θ, q) ∈ P , there exists a robust voting equilibrium v of G(T, θ, q, 0) such that v is order-preserving.

The predictions in order-preserving robust equilibria substantially differ from those by the behavioral assumption of “sincere (naive) voting.” However, they both predict that “leftist voters tend to support leftist parties and rightist voters tend to support rightist parties.” From eyes of researchers who do not exactly know individual voters’ ideal points, strategic voting in order-preserving equilibria might appear to be sincere voting, which one must keep in mind when examining empirical evidence of the two different hypotheses.

Multiple Equilibria and Policy Prediction

Robust voting equilibria are not generally unique. The multiplicity sometimes is due to the presence of indifferent voters. For instance, suppose X_m contains M voters. By Proposition 3, there is a strategically sincere equilibrium, say v, in which all voters in X_m vote for party m. Now suppose that, among those voters, one voter is exactly located at $\underline{y}_{m}$ and another at $\overline{y}_{m}$ . That is, the former is indifferent between the proposals by parties m − 1 and m, and the latter between those by m and m + 1. Letting them vote for party m − 1 and party m + 1, we have another strategically sincere equilibrium, say v′, in which party m receives less than a majority of votes. The policy outcome in v is the ideal point of party m, the majority party, while that in v′ is a nondegenerate lottery. Thus, the policy predictions by the two equilibria widely differ. However, it may not be a significant problem, as we may rarely encounter such indifferent voters. Hence, I will focus on the generic case in which no voter has two distinct party proposals by assuming the following:

1. A2 For all h, i, j ∈ L and all t ∈ T, if u(s^h _i ; t) = u(s_j ^h ; t) = max{u(s^h _k ; t)|k ∈ L}, then s^h _i = s_j ^h .

There are cases in which multiplicity of equilibria is not driven by voters’ indifference, as is shown by the next example.

Example 4 I reconsider the setting in Example 2 with the quadratic utility function: f(x) = −x ². As is found earlier, the profile v in Example 2 is a robust equilibrium. I now show that the following profile $\bar{v}$ is also a robust equilibrium:

The election result is $b(\bar{v})=(19,32,25,25)$ . Party 2, which is not the closest party to the median voter, is the median party, so $A(\bar{v})=[\frac{1}{8},\frac{3}{8}]$ . Then, $p(\bar{v})=(\frac{1}{8},\frac{1}{4},\frac{3}{8},\frac{3}{8})$ . The voters who vote for 3 or 4 are neither majority- nor median-pivotal. Their choices are strategically sincere and hence robust best responses. The voters who vote for 1 or 2 are median-pivotal. If one of them deviated by voting for 3 or 4, then party 3 would be the median party. Let t be such that $\bar{v}_{t}=i\in \lbrace 1,2\rbrace$ . Consider her deviation by voting for j ∈ {3, 4}. Let $\bar{v}^{\prime }=(j,\bar{v}_{-t})$ . The payoff from $\bar{v}$ is

for every ε ∈ [0, 1]. Note that, after deviation, the proposals by parties 3 and 4 will be $\frac{1}{2}$ and $\frac{7}{8}$ , respectively. Thus,

Then,

which is minimized when $t=\frac{1}{2}$ . Note that

Thus, $\bar{v}$ is a robust voting equilibrium. $\hfill\Box$

I argue that the equilibrium $\bar{v}$ is less plausible than v in the example. Note that v is strategically sincere, and it has some intuitive characteristics as we explained earlier. The profile $\bar{v}$ is not strategically sincere. There, the 19 voters whose ideal points are in $(\frac{5}{16},\frac{1}{2}]$ prefer party 3’s proposal to party 2’s proposal in $\bar{v}$ but vote for party 2. That is, for this equilibrium to be played, these voters need to coordinate to make party 2 the median party, which does not seem easy. Moreover, every voter in this coordinating group is median-pivotal. Thus, even when only one voter in the group fails to follow the coordinated plan and votes for party 3, voting for party 2 will not remain a best response in any of the voters. Thus, they must have a strong common knowledge of the coordination plan, which seems relatively unlikely to occur in a large election. These characteristics of the equilibrium in Example 4 turns out to be shared by every robust equilibrium that is not strategically sincere, as is shown by the next lemma.

Lemma 2 Let v be a robust voting equilibrium of G(T, θ, q). If v_t is not strategically sincere in v , then v_t = k(v).

Thus, all voters who do not vote strategically sincerely in a robust equilibrium support a same party, the decisive party in the equilibrium. That is, the votes that are not strategically sincere can be interpreted as coordinated choices for the purpose of making some party to be decisive. Lemma 1 implies that every voter in the coordinating group is majority- or median-pivotal. If one of them deviated perhaps by voting strategically sincerely, then the decisive party would lose its majority or median status. The coordination is somewhat fragile. By contrast, a complex coordination problem is not present in strategically sincere equilibria. For this reason, I posit that the equilibria that are strategically sincere are more likely to be played than those that are not. The next result shows that, under A2, policy predictions by strategically sincere equilibria are unique. For each voting profile v, let λ ^v denote the lottery over policy outcomes in G(T, θ, q, 0) induced by v.

Proposition 5 Assume A2. For all v, v′ ∈ V(T, θ, q), if v and v′ are both strategically sincere, then λ ^v = λ ^v′.

Thus, if we choose strategic equilibria over other equilibria, a unique probability distribution over policy outcomes is obtained whenever there is a strategically sincere equilibrium. Although strategically sincere equilibria do not always exist, the result is encouraging since they exist in the most likely scenario in the real world election.

Majoritarian Parliament and Strategic Voting

The decision rules in the parliament are ultimately majoritarian. I now examine how the majoritarian nature of the representative body shapes the composition of the parliament under PR. The next result shows the relationship between the median voter and the median party in equilibrium.

Proposition 6 Assume A2. If v is a strategically sincere robust equilibrium, then k(v) = m.

In words, the decisive party is the closest party to the median voter in every strategically sincere equilibrium. Thus, my model in general predicts that the median legislator in the parliament is ideologically close to the median voter in PR systems. Powell (Reference Powell2000) and Powell and Vanberg (Reference Powell and Vanberg2000) find that the congruence between the ideologies of the median voter and the median party is quite high in PR systems, which is consistent with the prediction.Footnote ⁶ Austen-Smith and Banks (Reference Austen-Smith and Banks1988) predict that, in their three party model, one party takes the position of the median voter and the other parties locate symmetrically around the median. Their prediction of the median correspondence is, in a sense, stronger than mine since party positions are endogenous in their model and they predict the exact location of the median party. On the other hand, my result is, in a sense, more comprehensive than theirs since we cover the PR systems with more than three parties.

Klumpp (Reference Klumpp2010) studies a model of legislative elections under the single member district system, in which the policy outcome is determined by Romer and Rosenthal’s (Reference Romer and Rosenthal1978) model. As his model differs from mine only in election rules, a comparison is worthwhile. Two results in Klumpp’s work are related to Proposition 6. First, in his model, the median legislator in equilibrium is the representative of the median district. Second, the electorate in the median district tends to strategically elect a representative who leans toward the status quo more than its median citizen. Thus, there are two possible biases in the relationship between the (national) median voter and the median legislator in Klumpp’s model. First, the median of the median district may not well reflect the national median due to other factors affecting the districting process. Second, the congruence between the median legislator and the district median may not be high because of the strategic delegation. Also, we cannot know whether these two biases work in the same or the opposite direction. Thus, a comprehensive comparison cannot be made without making further assumptions in his model and without endogenizing party positions in my model.

I now turn to the issue of strategic voting. Following the literature on voting behavior, I define strategic voting as voting for a party that is not one’s first choice. Given a profile v, and a voter t ∈ T, we say v_t is sincere if $v_{t}\in \arg \max \lbrace u(\theta _{i};t)|i\in L\rbrace$ . Note that this is different from strategically sincere voting. While strategically sincere voting refers to voting for the party whose expected proposal is the best among the party proposals, sincere voting is voting for the party whose original position is the best among the party ideal points. We say that v_t is strategic if it is not sincere. Given the symmetric single-peaked utility functions, a voter votes strategically if she does not vote for the party ideologically most proximate to her. Our model enables us to examine who votes strategically and how they do so.

There is a type of strategic voting that might occur in equilibrium but seems unlikely to be played in the real world. Consider the equilibrium $\bar{v}$ in Example 3. Given that party 2 is the median party in $\bar{v}$ , Parties 3 and 4 both will propose $\frac{1}{2}$ . Hence, the voters who vote for parties 3 or 4 ( i.e., $t>\frac{1}{2}$ ) are completely indifferent between these two parties. I constructed the profile so that they vote sincerely: among the voters, only those who are ideologically closer to party 4’s position than party 3’s (i.e., $t>\frac{7}{8})$ vote for party 4. Instead, I could construct another profile, say $\hat{v}$ , so that voters with ideal points in $(\frac{3}{4},\frac{7}{8})$ vote for party 4 and everything else is equal to $\bar{v}$ . The new profile is substantively the same as $\bar{v}$ in the sense that the two induce the same lottery over policy outcomes, and, for every deviation from $\hat{v}$ , there is an equivalent deviation from $\bar{v}$ . Thus, $\hat{v}$ is also a robust equilibrium. However, in it, the voters in $(\frac{3}{4},\frac{7}{8})$ vote strategically. In this case, we may predict $\bar{v}$ is more likely to be played than $\hat{v}$ because those voters in $(\frac{3}{4},\frac{7}{8})$ find no positive incentive to vote for their second choices. For this reason, I impose the following condition:

1. C1 For every t ∈ T, if v_t = i and there is j ∈ L∖{i} such that p_i (v) = p_j (v), then u(θ _i ; t) ⩾ u(θ _j ; t).

In words, when two or more parties are expected to offer a same compromised policy in the parliament, then voters vote for the one ideologically more proximate to them when choosing among the parties. When there is a strategically sincere equilibrium, there always is an equilibrium that satisfies C1.

The next result shows the way strategic voting happens in strategically sincere equilibria.

Proposition 7 Assume that v is a strategically sincere equilibrium and satisfies C1. Assume v_t is strategic. Then, there exists i(t) ∈ L such that $\arg \max \lbrace u(\theta _{j};t)|j\in L\rbrace =\lbrace i(t)\rbrace$ . Moreover, either θ _vt < t < θ _i(t) ⩽ θ _m or θ _m ⩽ θ _i(t) < t < θ _vt .

Thus, those who do not vote sincerely are relatively moderate voters. Moreover, when a voter votes strategically, she supports a party that holds a position more extreme than her own position and her first-best party’s position. The intuition is best explained by Example 2. In the equilibrium, the citizens in $(\frac{1}{8},\frac{3}{16})$ vote for party 1 although their first choice is party 2. Also, the voters in $(\frac{11}{16},\frac{3}{4})$ support party 4 although they are closer to party 3’s position. Observe that the incentive here is not a directional or balancing motivation. The moderate right voters’ strategic choices are not affected by the left parties’ expected proposals. Moreover, their strategies will not be altered by the leftist voters’ strategies as long as party 3 remains the median party. The voters support the rightmost party simply because its compromising policy in the postelection bargaining is expected to be more proximate to them than their first choice party’s position is.

As mentioned above, the two parties at both ends of the ideological spectrum in the 1998 Swedish election were supported by voters who were more moderate than them.Footnote ⁷ This observation is consistent with the prediction of my model. Given that the important austerity measures of the outgoing government were supported by the Center party, one reasonably assumes that the status quo in the economic left-right dimension then was slightly more rightist than the center of the voter ideology distribution. Then, relatively moderate rightist voters who disliked both a radical leftist reversal of the reform and an even further rightist reform supported the right-wing Moderate Party, perhaps because they expected that the party’s right-wing policy would anyway be vetoed by the center of the parliament, the Green Party, or the Social Democratic Party. On the other hand, leftist voters who were located on the left of the Social Democratic Party and thus disliked the status quo very much voted for the extreme Left Party because they wanted policies that were more leftist than the Social Democratic Party’s position though not as extreme as the Left Party’s position. Moreover, it should be noted that my equilibrium also predicts sincere voting by centrist voters and far extreme voters as rational decisions by outcome-oriented voters.

The observation that moderate voters support extreme parties is also found by Adams and Merrill (Reference Adams and Merrill1999) and Iversen (Reference Iversen1994).Footnote ⁸ Perhaps these empirical findings refute the sincere proximity voting hypothesis. However, my result suggests that they may be consistent with outcome-oriented proximity voting.

The logic of strategic voting in my model is to some extent similar to that in the model by Kedar (Reference Kedar2005). In both of the models, moderate voters support extreme parties because they expect that the effect of increase in extreme parties’ vote shares will be watered down by postelection bargaining. The difference is that, in her model, the process of “watering down” is exogenously given as a decision-theoretic form of voters’ beliefs. In my model, how much their votes for extreme parties will be watered down depends on strategic choices by parties and other voters.

Lastly, I discuss the issue of extremism under PR. There has been a conventional wisdom that PR promotes representation of extreme minorities. While I do not challenge the claim here, one implication of Proposition 7 is that an increase in vote share of an extreme party does not necessarily correspond to a change of policy outcomes in the direction of the party’s position. The extreme party’s vote share may increase due to an increase in strategic voting by moderate voters, who expect that the party will have to compromise its policy choice in the parliamentary bargaining. That is, the party may increase its support as a consequence of a change in the bargaining environmentFootnote ⁹ which makes voters expect that the parties’ policy influence will be greatly mitigated by centrist parties. In that case, it is not clear in which direction the increase changes the expected policy. On the one hand, the probability that the extreme party is selected as the proposer increases, which will be a force to pull the expected policy toward the extreme party’s position. On the other hand, the increase in the party’s vote share is accompanied with a moderation of its policy proposal, which will be a force in the opposite direction. The overall consequence will vary across different distributions of voters and parties. If the distribution is such that a minor compromise by an extreme party can increase a large number of strategic votes by moderate voters, the increase in the party’s seat share is accompanied by a more extreme expected policy. When the opposite is true, a rise of an extreme party may come together with a moderation of the expected policy.

Figure 2 illustrates a numerical example of the comparative statics. Here, voters are distributed by the standard normal distribution. Four parties locate at − 1, 0, 1, and 3. That is, one party takes the median voter’s position, two moderate parties locate at one standard deviation distant from the median, and an extreme right party takes three standard deviations distant from the median. In this setting, there always is a strategically sincere equilibrium. The graph on the left shows the relationship between the extreme party’s proposal and its vote share. The one on the right draws the relationship between the party’s vote share and the expected policy in equilibrium. In the figure, the expected policy becomes more extreme as the extreme right party’s vote share increases up to about 8 percent. However, if the party’s share exceeds the point, an increase in its seat share is accompanied by a moderation of the expected outcome.

FIGURE 2.

Extreme Party Vote Shares and Expected Policy Outcomes

It should be noted that what I am arguing here is not a causal relationship. Certainly, all else being equal, if more people support an extreme party for reasons that are not captured by the model, it will lead to move the expected policy toward the extreme party’s position.Footnote ¹⁰ The curves in Figure 2 are parametric ones where the independent parameter is the status quo q, which captures the bargaining environment in the parliament. With voters’ and parties’ preferences given, as the bargaining environment changes, we may observe the vote share of an extreme party and the expected policy move as in the figure.Footnote ¹¹ Still, the observational implication is that, in large data where exogenous shocks on preferences and the bargaining environment are presumably random, we may find the positive correlation between vote shares of extreme parties and moderation of final policy outcomes. A lesson is that when we observe a rise of an extreme party, we should separate the effect of the party’s mobilization (i.e., a change in voter preferences) from the effect of the bargaining environment (i.e., the situation in which the centrist outcome anyway is guaranteed). In light of my findings, the latter may not be a serious threat to democracy.