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Incomplete information, proportional representation and strategic voting

Abstract

We introduce incomplete information to a multiparty election under proportional representation: each voter knows her preferences and votes strategically to maximize her payoffs, but is uncertain about the number and the preferences of the other voters. Parties are assumed to be purely office motivated and, hence, the resulting governments are always minimum winning. In this framework, we prove (a) generic existence of equilibria where only two parties receive a positive fraction of the votes and therefore lead to single-party governments and (b) generic inexistence of equilibria that lead to coalition governments. That is, contrary to common wisdom, a proportional rule is found not to promote sincere voting and to be favorable towards single-party governments. The existence of two-party equilibria that lead to single-party governments is robust to parties having ideological concerns.

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Notes

  1. 1.

    While for example, Baron and Diermeier (2001) and Austen-Smith and Banks (1988) focus on some bargaining process where further assumptions regarding the coalition formation process are necessary, Iaryczower and Mattozzi (2013) assume that each party forms a single-party government with a probability equal to its vote share.

  2. 2.

    In a unidimensional policy space and under complete information voters strategically vote only for the two extreme parties. Centrist voters strategically vote only for the extreme parties since such strategic vote moves the implemented policy closer to their ideal point than a sincere vote for a centrist party. The main reason behind such strategic action is that implemented policies are a linear combination of the policies announced by the parties, weighted by parties’ vote shares. With a similar argument, De Sinopoli and Iannantuoni (2008) show that only extreme parties (but maybe more than two) obtain a positive vote share. Matakos et al. (2013) incorporate in such model different levels of the electoral rule disproportionality and show how it affects the number of competing parties. For two-party systems under alternative voting procedures such as the Borda count and approval voting see Dellis (2013) and Dellis et al. (2011).

  3. 3.

    Even though two-party equilibria in PR elections may seem surprising at a first glance, historically, two-party systems contradicting Duverger’s hypothesis were established in Germany, Austria, Ireland, and Australia (Lijphart 1994; Riker 1982). Currently, Malta provides an example of a two-party system in PR elections.

  4. 4.

    Exceptions of countries traditionally governed by minority governments are Denmark and Norway. For a formal analysis of the emergence of minority governments see Kalandrakis (2015), Diermeier and Merlo (2000), Laver and Shepsle (1990) and Austen-Smith and Banks (1990).

  5. 5.

    We explicitly require that Gamson’s Law holds in our setup. Gamson (1961) claimed that each party belonging to a coalition government obtains a share of portfolios proportional to the seats that each partner contributed to the coalition. This law has some empirical support (Warwick and Druckman 2001; Browne and Franklin 1973) and as Laver (1998) claims Gamson’s law is associated with “one of the highest non-trivial R-squared figures in political science”. For recent theoretical advances see for example Le Breton et al. (2008), Carroll and Cox (2007) and Fréchette et al. (2005).

  6. 6.

    In the proofs we use the assumption that \(n\rightarrow \infty \) to simplify math. All the results still hold if we substitute “\(n\rightarrow \infty \)” with “n sufficiently large”.

  7. 7.

    Notice that runoff elections may sustain sincere voting and hence a multiparty equilibrium (Bouton 2013).

  8. 8.

    Under PR elections, and in contrast to FPTP elections, a vote for the last ranked party may not always consist a dominated strategy.

  9. 9.

    Notice that while the calculated expected utilities clearly depend on the numerical values assumed for the valuations voters attach to parties the argument is generalizable and rather depends on the pivotal probabilities.

  10. 10.

    Notice that not in all cases where all three parties receive votes a coalition government has to be formed. For example, let party C be expected to secure an absolute majority and hence form a single-party government. Obviously, in this case the reasoning described before does not apply: types of voters ranking party C first do not expect an \(\{A,B\}\) government and hence need not consider that voting for C is a lost vote.

  11. 11.

    Despite the fact that mixed equilibria with \(v_{C}=\frac{1}{2}\) cannot be ruled out, their possible existence should crucially depend both on distributional assumptions and, perhaps more importantly, on the assumptions regarding voters’ exact cardinal preferences. That is, even if they exist they may in no way represent a robust prediction of the model.

  12. 12.

    The change in the weights of the coalition partners that one can induce by one’s vote is bounded from above by \(\frac{1}{(k+1)/2}=\frac{2}{k+1}\) (when k other voters participate then a coalition government has to receive at least \((k+1)/2\) votes). Hence, the expected change in a coalition’s weights when voting for A is strictly smaller than \(E(\frac{2}{k+1})=2\frac{e^{n}-1 }{ne^{n}}\) while the probability of being pivotal between B and C converges to \(\frac{e^{-n(\sqrt{v_{B}}-\sqrt{v_{C}})^{2}}}{2\sqrt{\pi n\sqrt{ v_{B}v_{C}}}}=\frac{1}{2\sqrt{\pi nv_{B}}}\) (see Myerson 2000). We observe that \(2\frac{e^{n}-1}{ne^{n}}/\frac{1}{2\sqrt{\pi nv_{B}}}=4\frac{(e^{n}-1) \sqrt{\pi v_{B}}}{\sqrt{n}e^{n}}\) and that \(\lim _{n\rightarrow +\infty } \frac{(e^{n}-1)\sqrt{\pi v_{B}}}{\sqrt{n}e^{n}}=0\). That is, the probability that one’s vote is decisive in the choice of A’s coalition partner becomes infinitely larger than the expected utility gain from increasing the weight of one’s prefered party in a given coalition when the electorate becomes arbitrarily large.

  13. 13.

    The description of how a government is formed when no stable coalitions exist is presented here only for completeness. Such description is irrelevant regarding our results since in all cases of interest there always exists at least one stable coalition.

  14. 14.

    The probability with which the coalition government is selected out of a set a stable winning coalitions should obviously depend on parties’ exact vote shares, on their ideology-office holding orientation and on their policy preferences. Since for our analysis an explicit formal structure of all the above is redundant, we refrain from formally defining properties of this probability. The only assumption we impose is that whenever the set of stable winning coalitions includes more than one element, then a non-degenerate positive probability is assigned to each of these elements.

  15. 15.

    Given the numerical values we have attached to the valuations \(u_{t}(J)\) the condition of Lemma 3 is satisfied if more than seven voters participate in the election (\(k>7\)). This numerical condition would vary for different values of \(u_{t}(J)\). However, different thresholds for k do not affect the intuition and results of neither Lemma 3 nor of the following proposition.

  16. 16.

    Where JQ and R are distinct elements of \(\{A,B,C\}\).

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Acknowledgments

We are grateful to the editor and an associate editor for their comments. We also benefited from comments by Martin Gregor, Jan Zapal, and seminar and conference participants in Universitat Autonoma de Barcelona, the IX Meeting of the Spanish Network of Social Choice 2012, and CRETE 2012.

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Correspondence to Dimitrios Xefteris.

Appendix (Proofs)

Appendix (Proofs)

Lemma 1

If party J obtains the majority of the vote (that is \( V_{J}>\frac{1}{2}k\)) then by forming a single-party government \(G=J\) where \( W_{J}^{G}=1\) it obtains payoff \(U_{J}(J)=1\). For any other coalition government \(G\ne J\) then it is straightforward that \(U_{J}(G)<U_{J}(J)\) since it holds that \(W_{J}^{G}<1.\) If no party has a majority of votes then party J maximizes its utility by forming a coalition government with Q if and only if \(V_{Q}\le V_{R}.\) Footnote 16 It is obvious that if all parties tie then any coalition is minimum winning. If at least one party’s number of votes is different from the ones of the other parties then (a) in case all number of votes are different the two bigger parties want to form a coalition with the smallest one and the smallest one wants to form a coalition with the second smallest one (b) if two parties tie and receive less votes than the other party then they wish to form a coalition with each other and (c) if two parties tie and receive more votes than the other party then both want to form a coalition with the smallest party while the smallest party is indifferent. In all three cases only a minimal winning coalition is stable, winning and vote-share consistent. \(\square \)

Proposition 1

A BNE consists of strategies and beliefs. We will prove that if the expected vote share of A is zero and the expected votes shares of B and C are strictly positive then a \(t_{ABC}\) will vote for B (equivalently one can show that a \(t_{ACB}\) voter will vote for C). Consider the first pivotal event \(\hat{V}_{B}=\hat{V}_{C}-1\) as presented in Table 1, where \(\hat{V}_{B}\) and \(\hat{V}_{C}\) denote the number of votes that B and C receive by the rest of the population. If \(\hat{V}_{B}=0\) then the voter is strictly better off voting A. If \(\hat{V}_{B}=1\) then the voter is indifferent between voting A and B. If \(\hat{V}_{B}>1\) then the voter is better off voting B for a large enough number of voters. Notice that \(prob(\hat{V}_{B}=0|\hat{V}_{B}=\hat{V}_{C}-1)\rightarrow 0\), \( prob(\hat{V}_{B}=1|\hat{V}_{B}=\hat{V}_{C}-1)\rightarrow 0\) and \(prob(\hat{V} _{B}>1|\hat{V}_{B}=\hat{V}_{C}-1)\rightarrow 1\) as \(n\rightarrow \infty \) and that utility differences are bounded from above and below (conditional on \(\hat{V}_{B}=\hat{V}_{C}-1\) our voter’s choice will affect her utility by at most \(\frac{1}{2}\) independently of how many voters vote). Therefore, conditional on \(\hat{V}_{B}=\hat{V}_{C}-1\) the voter decides which party to support based on the most probable event in large elections (i.e. \(\hat{V} _{B}>1\)) and votes B (i.e. the expected utility difference by voting B rather than A is equal to \(\frac{1}{2}\)). Exactly the same argument applies for the second pivotal event where \(\hat{V}_{B}=\hat{V}_{C}\). Consider now the third pivotal event \(\hat{V}_{B}=\hat{V}_{C}+1\). If \(\hat{V} _{C}=0\) then the voter is strictly better off voting A. If \(\hat{V}_{C}=1\) then the voter is indifferent between voting A and B. If \(\hat{V}_{B}>1\) then for a large enough number of voters the voter is essentially indifferent between voting A and B. Given the same argument regarding pivotal probabilities, conditional on \(\hat{V}_{B}=\hat{V}_{C}+1\) the expected utility difference by voting B rather than A is negative but converges to zero as \(n\rightarrow \infty \). Notice now that all three pivotal events are equiprobable (Myerson 2000). Given the positive and non-degenerate utility difference from the first two events a \(t_{ABC}\) will vote for B (equivalently a \(t_{ACB}\) voter will vote for C). Hence, beliefs \( \{v_{A}=0,v_{B}=p(t_{BAC})+p(t_{BCA})+p(t_{ABC}),v_{C}=p(t_{CAB})+p(t_{CBA})+p(t_{ACB})\} \) and the strategy profile that is consistent with these beliefs form a BNE. \(\square \)

Proposition 2

Consider that there exists a BNE that leads to a coalition government with a non-degenerate probability. Since we focus on pure strategy equilibria, the beliefs which are part of such a BNE should be consistent with \(0<v_{J}<v_{Q}<v_{R}<1/2\) for any generic distribution of players’ types. Let us assume without loss of generality that \( 0<v_{A}<v_{B}<v_{C}<1/2\). In such case the probability with which the minimal winning coalition \(\{A,B\}\) will be formed after the election converges to one as n becomes arbitrarily large. Let us study the problem that a \(t_{CAB}\) voter faces. Her vote may increase the weight of a party in a coalition government (such a change in the weight is bounded from below by \(\frac{1}{k+1}\)) or/and it may determine the coalition government that will be formed. Conditional on the outcome being such that a vote cannot affect the coalition government that will be formed, a \(t_{CAB}\) voter obviously prefers to vote for party A since the coalition government \(\{A,B\}\) will be formed with infinitely larger probability than any other coalition government. Now, conditional on her vote being pivotal for the determination of the exact coalition (or single-party) government a \(t_{CAB}\) voter could be better off by not voting for party A. Since parties are purely office motivated, all such pivotal events are a subset of the cases in which the difference between the votes of (at least) two parties is not more than one vote plus the case in which a party lacks one vote to form a single-party government. Moreover, the probability of any such pivotal event must not exceed the probability of the most probable tie which is given by \(\frac{ e^{-n(\sqrt{\hat{\psi }}-\sqrt{\hat{\omega }})^{2}}}{2\sqrt{\pi n\sqrt{\hat{ \psi }\hat{\omega }}}}\) (see Myerson 2000) where \((\hat{\psi },\hat{\omega } )=\arg \min \{(\sqrt{\psi }-\sqrt{\omega })^{2}\) such that \(\psi \) and \( \omega \) take distinct values from the set \( \{v_{A},v_{B},v_{C},1-v_{A},1-v_{B},1-v_{C}\}\}\).

Then, to establish that a \(t_{CAB}\) voter would prefer to vote for A it is enough to show that

$$\begin{aligned}&\lim _{n\rightarrow +\infty }\frac{E\left( \frac{1}{k+1}|\frac{k}{2}-1>\hat{V} _{C}>\max \{\hat{V}_{A},\hat{V}_{B}\}+1\right) \times prob\left( \frac{k}{2}-1>\hat{V} _{C}>\max \{\hat{V}_{A},\hat{V}_{B}\}+1\right) }{\frac{e^{-n\left( \sqrt{\hat{\psi }}- \sqrt{\hat{\omega }}\right) ^{2}}}{2\sqrt{\pi n\sqrt{\hat{\psi }\hat{\omega }}}}} =+\infty . \end{aligned}$$

Given that \(E(\frac{1}{k+1}|\frac{k}{2}-1>\hat{V}_{C}>\max \{\hat{V}_{A}, \hat{V}_{B}\}+1)>\frac{1}{2}E(\frac{1}{k+1}|k\ge m)=\frac{1}{2}\frac{\Gamma (1+m)-\Gamma (1+m,n)}{n\Gamma (1+m)}\) where \(\Gamma (z)\) is the Euler gamma function, \(\Gamma (a,z)\) is the incomplete gamma function and m a sufficiently large integer, we can establish the above relationship if the following is true:

$$\begin{aligned} \lim _{n\rightarrow +\infty }\frac{prob\left( \frac{k}{2}-1>\hat{V}_{C}>\max \left\{ \hat{V}_{A},\hat{V}_{B}\right\} +1\right) }{\frac{2\Gamma (1+m)}{\Gamma (1+m)-\Gamma (1+m,n)}\frac{\sqrt{n}e^{-n\left( \sqrt{\hat{\psi }}-\sqrt{\hat{\omega }}\right) ^{2}}}{2 \sqrt{\pi \sqrt{\hat{\psi }\hat{\omega }}}}}=+\infty . \end{aligned}$$

Pure office motivation and \(0<v_{A}<v_{B}<v_{C}<1/2\) suggest that \(prob( \frac{k}{2}-1>\hat{V}_{C}>\max \{\hat{V}_{A},\hat{V}_{B}\}+1)\rightarrow 1\) as \(n\rightarrow +\infty \), m being positive suggests that \( \lim _{n\rightarrow +\infty }\frac{2\Gamma (1+m)}{\Gamma (1+m)-\Gamma (1+m,n)} =2\) and \((\sqrt{\hat{\psi }}-\sqrt{\hat{\omega }})^{2}>0\) suggests that \( \lim _{n\rightarrow +\infty }\frac{\sqrt{n}e^{-n(\sqrt{\hat{\psi }}-\sqrt{\hat{ \omega }})^{2}}}{2\sqrt{\pi \sqrt{\hat{\psi }\hat{\omega }}}}=0\). Hence, the required relationship holds.

This proves that when a strategy profile generates expected vote shares \( 0<v_{A}<v_{B}<v_{C}<1/2\), then a \(t_{CAB}\) voter is strictly better off by voting for A than for C. Similarly one can show that a \(t_{CBA}\) voter is strictly better off by voting for B than for C and that C is not a best response to such expectations for any voter type. That is, beliefs which are consistent with \(0<v_{A}<v_{B}<v_{C}<1/2\) cannot be part of a BNE since they lead to \(\hat{V}_{C}=0\) and, hence, to a voting behavior which is inconsistent with them. \(\square \)

Lemma 2

Similar to Lemma 1 if a party J obtains the majority of the votes then by performing a single-party government \(G=J\) where \( W_{J}^{G}=1\) it maximizes its payoff \(U_{J}(J)=1\). To show that under policy motives the government coalition may be non-minimal let \(\lambda _{A}=\lambda _{B}=\lambda _{C}=\lambda <1\) and \(\tilde{V}_{A}=\tilde{V} _{C}+\varepsilon =\tilde{V}_{B}+2\varepsilon \) where \(\tilde{V} _J=V_J/(V_A+V_B+V_C)\) denotes party’s J vote share. Assume that the policy preferences of party A are compatible with those of a voter of type \( t_{ABC}\) and the policy preferences of party B are compatible with those of a voter of type \(t_{BAC}.\) Then due to both office seeking incentives and ideology proximity party A strictly prefers to form a coalition with B than with C. Now, B strictly prefers to form a coalition with A rather than with C if and only if \(U_{B}(\{A,B\})>U_{B}(\{B,C\}).\) The latter holds if and only if:

$$\begin{aligned}&\lambda \frac{V_{B}}{V_{A}+V_{B}}+(1-\lambda )\left[ \frac{V_{B}}{V_{A}+V_{B}} u_{t_{BAC}}(B)+\frac{V_{A}}{V_{A}+V_{B}}u_{t_{BAC}}(A)\right]>\lambda \frac{V_{B}}{ V_{B}+V_{C}}\\&\qquad +\;(1-\lambda )\left[ \frac{V_{B}}{V_{B}+V_{C}}u_{t_{BAC}}(B)+\frac{V_{C} }{V_{B}+V_{C}}u_{t_{BAC}}(C)\right] \\&\quad \Longrightarrow \lambda \frac{V_{B}}{V_{A}+V_{B}}+(1-\lambda )\left[ \frac{V_{B}}{V_{A}+V_{B}} \right]>\lambda \frac{V_{B}}{V_{B}+V_{C}}\\&\qquad +\;(1-\lambda )\left[ \frac{V_{B}}{V_{B}+V_{C}}- \frac{V_{C}}{V_{B}+V_{C}}\right] \\&\quad \Longrightarrow \frac{V_{B}}{V_{A}+V_{B}}>\frac{V_{B}}{V_{B}+V_{C}}-(1-\lambda )\frac{V_{C} }{V_{B}+V_{C}}\\&\quad \Longrightarrow \frac{\tilde{V}_{B}}{2\tilde{V}_{B}+2\varepsilon }>\frac{\tilde{V}_{B}}{2 \tilde{V}_{B}+\varepsilon }-(1-\lambda )\frac{\tilde{V}_{B}+\varepsilon }{2\tilde{V}_{B}+\varepsilon }\\&\quad \Longrightarrow \frac{\tilde{V}_{B}}{2\tilde{V}_{B}+2\varepsilon }>\frac{\tilde{V}_{B}- \tilde{V}_{B}-\varepsilon +\lambda \tilde{V}_{B}+\lambda \varepsilon }{2 \tilde{V}_{B}+\varepsilon }\\&\quad \Longrightarrow \frac{\tilde{V}_{B}}{2\tilde{V}_{B}+2\varepsilon }>\frac{\lambda \tilde{V} _{B}+\varepsilon (1-\lambda )}{2\tilde{V}_{B}+\varepsilon } \end{aligned}$$

For small values of \(\varepsilon \) (that is for \(\varepsilon \rightarrow 0\)) the above becomes \(\frac{1}{2}>\frac{\lambda }{2}\) and holds for any \( \lambda \in [0,1).\) \(\square \)

Lemma 3

Let \(V_{A}=V_{B}+\eta >V_{C}=1,\) where \(\eta \in \{-1,0,1\}\). Therefore, \(\frac{k}{2}-1\le V_{A}\le \frac{k}{2}+1\). Assume that \(\eta =1\) (the equivalent arguments holds for \(\eta =-1\)). In this case \(V_{A}= \frac{k}{2}\) and the only winning coalition in which C participates is \( \{A,C\}.\) Thus, we are interested in finding for which values of k the \( \{A,C\}\) coalition is also stable. Notice that \(\{A,C\}\) is stable if and only if A prefers to form a coalition with C than with B. It is obvious that \(\frac{V_{A}}{V_{A}+V_{C}}>\frac{V_{A}}{V_{A}+V_{B}}\), that is, if A is a purely office-motivated party it would prefer to form a coalition with C. To establish our result we need to find the condition that guarantees that \(\frac{V_{A}}{V_{A}+V_{C}}u_{t_{ABC}}(A)+\frac{V_{C}}{ V_{A}+V_{C}}u_{t_{ABC}}(C)>\frac{V_{A}}{V_{A}+V_{B}}u_{t_{ABC}}(A)+\frac{ V_{B}}{V_{A}+V_{B}}u_{t_{ABC}}(B)\) as well. The latter inequality implies that even if party A is purely ideology motivated and even if it dislikes the policy of party C the most (party A has preferences compatible with a \(t_{ABC}\) voter), it would still prefer to form a coalition with C rather than with B. We know that \(\frac{V_{A}}{V_{A}+V_{C}}u_{t_{ABC}}(A)+ \frac{V_{C}}{V_{A}+V_{C}}u_{t_{ABC}}(C)=\frac{V_{A}-V_{C}}{V_{A}+V_{C}}\) and that \(\frac{V_{A}}{V_{A}+V_{B}}u_{t_{ABC}}(A)+\frac{V_{B}}{V_{A}+V_{B}} u_{t_{ABC}}(B)=\frac{V_{A}}{V_{A}+V_{B}}.\) We notice that \(\frac{V_{A}-V_{C} }{V_{A}+V_{C}}=\frac{\frac{k}{2}-1}{\frac{k}{2}+1}\) and that \(\frac{V_{A}}{ V_{A}+V_{B}}=\frac{\frac{k}{2}}{k-1}.\) Therefore, the required inequality holds if \(\frac{\frac{k}{2}-1}{\frac{k}{2}+1}>\frac{\frac{k}{2}}{k-1}\) and since k is always a natural number, the later inequality holds if and only if \(k>7.\) We replicate this exercise for \(\eta =0\) and we also find that \( k>7 \) is a sufficient condition for the existence of a stable coalition in which C participates. The only difference of the \(\eta =0\) case compared to the cases in which \(\eta \in \{-1,1\}\) is that there are two winning coalitions in which C participates; \(\{A,C\}\) and \(\{B,C\}.\) Since both A and B received exactly the same votes, party C will choose partner according to its ideological preferences (or it will choose a partner randomly if it is purely office-motivated) and \(k>7\) will guarantee that A (B) strictly prefers to be in a coalition with C rather than with B ( A ). \(\square \)

Proposition 3

A BNE consists of strategies and beliefs. We will prove that if the expected vote share of A is zero and the expected vote shares of the other two parties are positive (formally, \(v_{A}=0\), \(v_{B}>0\) and \(v_{C}>0\) such that \(v_{A}+v_{B}+v_{C}=1\)) then a \(t_{ABC}\) will vote for B (equivalently one can show that a \(t_{ACB}\) voter will vote for C) . That is, we will show that beliefs \( \{v_{A}=0,v_{B}=p(t_{BAC})+p(t_{BCA})+p(t_{ABC}),v_{C}=p(t_{CAB})+p(t_{CBA})+p(t_{ACB})\} \) and the strategy profile which is consistent with these beliefs form a BNE. We know that \(A^{\prime }s\) ideological preferences should be given by one of the strict orders \(\{A\succ B\succ C\}\) or \(\{A\succ C\succ B\}\) (assume without loss of generality that \(A^{\prime }s\) preferences are given by the first linear order). For large polities (\(n\rightarrow \infty \)) the probability that there exists a positive measure of voters with preferences that are compatible with \(A^{\prime }s\) ideological preferences converges to one. Let us investigate the problem that a voter i with such preferences faces in this case. If she expects that \(v_{A}=0\), \(v_{B}>0\) and \(v_{C}>0\) then the only possibility that her vote can define the outcome is when \(\hat{V}_{B}=\hat{V}_{C}+\eta \) where \(\eta \in \{-1,0,1\}\). Notice that a) in such eventualities voting for C is always a strictly dominated strategy and b) \(\eta \in \{-1,1\}\) implies that k is even and \(\eta =0\) implies that k is odd. We detail below the possible pivotal eventualities.

(I) If \(\eta =-1\) then voter i gets utility \(-\frac{1}{2}\) if she votes for B (because B and C will form a coalition in which they will have equal weights). (a) If \(k>7\) and our voter votes for A then both coalitions \(\{A,C\}\) and \(\{B,C\}\) are stable. In this case the coalition \( \{A,C\}\) is formed with probability \(0<<\xi<<1\) and the coalition \(\{B,C\}\) is formed with probability \(1-\xi .\) Voter i gets utility \(-\frac{\hat{V} _{C}-1}{1+\hat{V}_{C}}\) if she votes for A and \(\{A,C\}\) is formed (because the condition of Lemma 3 is satisfied and A and C can form a coalition in which they will have weights \(\frac{1}{1+\hat{V}_{C}}\) and \( \frac{\hat{V}_{C}}{1+\hat{V}_{C}}\) respectively) and utility strictly less than \(-\frac{1}{2}\) if she votes for A and \(\{B,C\}\) is formed. (b) If \(k\le 7\) voter i gets utility of, at most, equal to 0 when the voter votes for A (and government \(\{A,C\}\) is formed if \(\hat{V} _{A}=0,\hat{V}_{B}=0, \hat{V}_{C}=1\)).

(II) If \(\eta =0\) then voter i gets utility 0 if she votes for B (because B will form a single-party government) and if \(k>7\) voter i gets utility \(\frac{1}{1+\hat{V}_{B}}\) if she votes for A. This is because the condition of Lemma 3 is satisfied and, thus, B will prefer to form a coalition with A rather than with C. Due to ideological proximity with B , A will prefer to form a coalition with\(\ B\) and, thus A and B will form a coalition in which they will have weights \(\frac{1}{1+\hat{V}_{B} }\) and \(\frac{\hat{V}_{B}}{1+\hat{V}_{B}}\) respectively. If \(k\le 7\) voter i gets utility of, at most, equal to 1 by voting A (and a single-party government \(\{A\}\) is formed if \(\hat{V}_{A}=0,\hat{V}_{B}=0, \hat{V}_{C}=0 \)).

(III) If \(\eta =1\) then voter i gets utility 0 if she votes for B (because B will form a single-party government) and if \(k>7\) voter i gets utility \(\frac{1}{1+\hat{V}_{B}}\) if she votes for A. This is because the condition of Lemma 3 is satisfied and, thus, B will prefer to form a coalition with A rather than with C. Due to ideological proximity with B , A will prefer to form a coalition with\(\ B\) and, thus A and B will form a coalition in which they will have weights \(\frac{1}{1+\hat{V}_{B} }\) and \(\frac{\hat{V}_{B}}{1+\hat{V}_{B}}\) respectively. If \(k\le 7\) voter i gets utility of, at most, equal to \(\frac{1}{2}\) by voting A (and government \(\{A,B\}\) is formed if \(\hat{V}_{A}=0,\hat{V}_{B}=1, \hat{V} _{C}=0 \)).

Formally, voter i will vote for her second ranked candidate B if the following holds:

$$\begin{aligned} prob(\hat{V}_{B}= & {} \hat{V}_{C}-1\,and\,k>7)\\&\times \left[ -\frac{1}{2}-\xi \left( -E\left( \frac{\hat{V}_{C}-1}{1+\hat{V}_{C}}|\hat{V}_{B}=\hat{V}_{C}-1\,and \,k>7\right) \right) -(1-\xi )\left( -\frac{1}{2}\right) \right] \\&+\;prob\left( \hat{V}_{B} =\hat{V}_{C}-1\,and\,k\le 7\right) \left[ -\frac{1}{2}-0\right] \\&+\;prob\left( \hat{V}_{B} =\hat{V}_{C}\,and\,k>7\right) \left[ 0-E\left( \frac{1}{1+\hat{V }_{B}}|\hat{V}_{B}=\hat{V}_{C}\,and\,k>7\right) \right] \\&+\;prob\left( \hat{V}_{B} =\hat{V}_{C}\,and\,k\le 7\right) [0-1]\\&+\;prob\left( \hat{V}_{B} =\hat{V}_{C}+1\,and\,k>7\right) \\&\times \left[ 0-E\left( \frac{1}{1+ \hat{V}_{B}}|\hat{V}_{B}=\hat{V}_{C}+1\,and\,k>7\right) \right] \\&+\;prob\left( \hat{V}_{B} =\hat{V}_{C}+1\,and\,k\le 7\right) \left[ 0-\frac{1}{2} \right] >0 \end{aligned}$$

We observe that:

$$\begin{aligned} prob(\hat{V}_{B}=\hat{V}_{C}-1\,and\,k\le 7)= & {} prob(\hat{V}_{B}=\hat{V} _{C}-1|k\le 7)\times prob(k\le 7),\\ prob(\hat{V}_{B}=\hat{V}_{C}\,and\,k\le 7)= & {} prob(\hat{V}_{B}=\hat{V} _{C}|k\le 7)\times prob(k\le 7)\, \hbox {and}\\ prob(\hat{V}_{B}=\hat{V}_{C}+1\,and\,k\le 7)= & {} prob(\hat{V}_{B}=\hat{V} _{C}+1|k\le 7)\times prob(k\le 7). \end{aligned}$$

It is obvious that all \(prob(\hat{V}_{B}=\hat{V}_{C}-1|k\le 7),prob(\hat{V} _{B}=\hat{V}_{C}|k\le 7)\) and \(prob(\hat{V}_{B}=\hat{V}_{C}+1|k\le 7)\) take a positive value significantly higher than 0. On the other side we have that \(prob(k\le 7)=\overset{7}{\sum \nolimits _{m=0}}\frac{n^{m}}{m!} e^{-n} \).

Notice that \(prob(\hat{V}_{B}=\hat{V}_{C}-1)=\) \(prob(\hat{V}_{B}= \hat{V}_{C}-1\) and \(k>7)+prob(\hat{V}_{B}=\hat{V}_{C}-1\) and \(k\le 7)\) and \(prob(\hat{V}_{B}=\hat{V}_{C}-1\) and \(k\le 7)=prob(\hat{V}_{B}=\hat{V} _{C}-1|k\le 7)\times prob(k\le 7).\) That is, \(1=\) \(\frac{prob(\hat{V}_{B}= \hat{V}_{C}-1\,and\,k>7)}{prob(\hat{V}_{B}=\hat{V}_{C}-1)}+prob( \hat{V}_{B}=\hat{V}_{C}-1|k\le 7)\times \frac{prob(k\le 7)}{prob(\hat{V} _{B}=\hat{V}_{C}-1)}.\)

By Myerson (2000), we know that \(prob(\hat{V}_{B}=\hat{V}_{C}-1)\simeq \frac{ e^{n(2\sqrt{v_{B}v_{C}}-v_{B}-v_{C})}}{2\sqrt{\pi n\sqrt{v_{B}v_{C}}}}\) for \( n\rightarrow \infty .\) So it must be the case that for \(n\rightarrow \infty , \) \(prob(k\le 7)\) becomes infinitely smaller than \(prob(\hat{V}_{B}=\hat{V} _{C}-1).\) This is due to the fact that \(\overset{7}{\sum \nolimits _{m=0}}\frac{ n^{m}}{m!}e^{-n}<\frac{e^{n(2\sqrt{v_{B}v_{C}}-v_{B}-v_{C})}}{2\sqrt{\pi n \sqrt{v_{B}v_{C}}}}\) is equivalent to \(\frac{2\sqrt{\pi n\sqrt{v_{B}v_{C}}} \overset{7}{\sum \nolimits _{m=0}}\frac{n^{m}}{m!}}{e^{n(2\sqrt{v_{B}v_{C}} -v_{B}-v_{C})+n}}<1\) and \(2\sqrt{\pi n\sqrt{v_{B}v_{C}}}\overset{7}{ \sum \nolimits _{m=0}}\frac{n^{m}}{m!}\) increases in a polynomial rate while \( e^{n(2\sqrt{v_{B}v_{C}}-v_{B}-v_{C})+n}\) increases in an exponential rate (\( n(2\sqrt{v_{B}v_{C}}-v_{B}-v_{C})+n\) is always positive for any \(v_{B}>0\) and \(v_{C}>0\)). That is, \(\frac{2\sqrt{\pi n\sqrt{v_{B}v_{C}}}\overset{7}{ \sum \nolimits _{m=0}}\frac{n^{m}}{m!}}{e^{n(2\sqrt{v_{B}v_{C}}-v_{B}-v_{C})+n}} \rightarrow 0\) and therefore \(\frac{prob(k\le 7)}{prob(\hat{V}_{B}=\hat{V} _{C}-1)}\rightarrow 0.\) This implies that for \(n\rightarrow \infty \) we have \(1=\) \(\frac{prob(\hat{V}_{B}=\hat{V}_{C}-1\,and\,k>7)}{prob(\hat{ V}_{B}=\hat{V}_{C}-1)}+prob(\hat{V}_{B}=\hat{V}_{C}-1|k\le 7)\times 0\) which is equivalent to \(\frac{prob(\hat{V}_{B}=\hat{V}_{C}-1\,and\,k>7)}{prob(\hat{V}_{B}=\hat{V}_{C}-1)}\rightarrow 1\) and to \(prob(\hat{V} _{B}=\hat{V}_{C}-1\) and \(k>7)\rightarrow prob(\hat{V}_{B}=\hat{V}_{C}-1).\)

With the same logic we can demonstrate that \(prob(\hat{V}_{B}=\hat{V}_{C}\) and \(k>7)\rightarrow prob(\hat{V}_{B}=\hat{V}_{C})\) and that \(prob(\hat{V} _{B}=\hat{V}_{C}+1\) and \(k>7)\rightarrow prob(\hat{V}_{B}=\hat{V}_{C}+1).\)

By this last observation and the offset theorem of Myerson (2000) we have that \(\frac{prob(\hat{V}_{B}=\hat{V}_{C}\,and\,k>7)}{prob(\hat{V} _{B}=\hat{V}_{C}-1\,and\,k>7)}\rightarrow \sqrt{\frac{v_{B}}{ v_{C}}}\) and \(\frac{prob(\hat{V}_{B}=\hat{V}_{C}+1\,and\,k>7)}{ prob(\hat{V}_{B}=\hat{V}_{C}-1\,and\,k>7)}\rightarrow \frac{v_{B} }{v_{C}}.\)

All these imply that if we divide our relevant inequality with \(prob(\hat{V} _{B}=\hat{V}_{C}-1\) and \(k>7)\) we should get:

$$\begin{aligned}&\xi \left[ -\frac{1}{2}-\left( -E\left( \frac{\hat{V}_{C}-1}{1+\hat{V}_{C}}|\hat{V} _{B}=\hat{V}_{C}-1\,and\,k>7\right) \right) \right] \\&\quad +\,prob\left( \hat{V}_{B} =\hat{V}_{C}-1|k\le 7\right) \frac{prob(k\le 7)}{prob\left( \hat{V} _{B}=\hat{V}_{C}-1\,and\,k>7\right) }\left[ -\frac{1}{2}-0\right] \\&\quad +\,\sqrt{\frac{v_{B}}{v_{C}}}\left[ 0-E\left( \frac{1}{1+\hat{V}_{B}}|\hat{V}_{B}=\hat{V} _{C}\,and\,k>7\right) \right] \\&\quad +\,prob\left( \hat{V}_{B} =\hat{V}_{C}|k\le 7\right) \frac{prob(k\le 7)}{prob\left( \hat{V} _{B}=\hat{V}_{C}-1\,and\,k>7\right) }[0-1] \\&\quad +\,\frac{v_{B}}{v_{C}}\left[ 0-E\left( \frac{1}{1+\hat{V}_{B}}|\hat{V}_{B}=\hat{V}_{C}+1 \,and\,k>7\right) \right] \\&\quad +\,prob\left( \hat{V}_{B} =\hat{V}_{C}|k\le 7\right) \frac{prob\left( k\le 7\right) }{prob\left( \hat{V} _{B}=\hat{V}_{C}-1\,and\,k>7\right) }\left[ 0-\frac{1}{2}\right] >0 \end{aligned}$$

Since

$$\begin{aligned}&E\left( \frac{\hat{V}_{C}-1}{1+\hat{V}_{C}}|\hat{V}_{B}=\hat{V}_{C}-1\,and\, k>7\right) \\&\quad =1-E\left( \frac{2}{1+\hat{V}_{C}}|\hat{V}_{B}=\hat{V}_{C}-1\,and\, k>7\right) \rightarrow 1 \end{aligned}$$

and

$$\begin{aligned} E\left( \frac{1}{1+\hat{V}_{B}}|\hat{V}_{B}=\hat{V}_{C}\, and\, k>7\right) \rightarrow E\left( \frac{1}{1+\hat{V}_{B}}|\hat{V}_{B}=\hat{V}_{C}+1\, and\, k>7\right) \rightarrow 0 \end{aligned}$$

the above inequality becomes \(\frac{\xi }{2}>0\) and always holds. That is, when \(v_{A}=0\), \(v_{B}>0\) and \(v_{C}>0\) then a \(t_{ABC}\) will vote for B (equivalently a \(t_{ACB}\) voter will vote for C). Therefore, beliefs \(\{v_{A}=0,v_{B}=p(t_{BAC})+p(t_{BCA})+p(t_{ABC}),v_{C}=p(t_{CAB})+p(t_{CBA})+p(t_{ACB})\}\) and the strategy profile which is consistent with these beliefs form a BNE. \(\square \)

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Troumpounis, O., Xefteris, D. Incomplete information, proportional representation and strategic voting. Soc Choice Welf 47, 879–903 (2016). https://doi.org/10.1007/s00355-016-0995-7

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Keywords

  • Vote Share
  • Proportional Representation
  • Winning Coalition
  • Coalition Government
  • Strategic Vote