Strategic Voting and Duverger’s Hypothesis Part 2

6 Conclusions In this paper, we characterized the set of strictly perfect equilibria in three candidate runoff elections. In all equilibria, the front-runner receives the votes of all the voters who prefer her to the runner-up. An equilibrium where all the remaining voters coordinate on the runner-up candidate always exists, that is, similarly to the case of plurality elections, there always exists a Duverger’s Law equilibrium in which only two candidates receive a positive vote share. We also showed that there is at most one Duverger’s Hypothesis equilibrium in which three candidates receive a positive fraction of the votes. The characteristics of that unique Duverger’s Hypothesis equilibrium challenge common beliefs about runoff elections: (i) some voters do not vote for their most preferred candidate (i.e. the sincere voting equilibrium does not exist), (ii) supporters of the front-runner do not vote for a less-preferred candidate in order to "choose" who will face the front-runner in the second round (i.e. there is no push over equilibrium), and (iii) it can lead to the exclusion of the Condorcet Winner from the second round. References [1] Blais, André, Massicotte, Louis, and Dobrzynska, Agnieszka (1997). Direct Presidential Elections: A World Summary. Electoral Studies, 16 (4), pp. 441-455. [2] Bouton, Laurent (2012). A Theory of Strategic Voting in Runoff Elections, American Economic Review, forthcoming. [3] Bouton, Laurent and Castanheira, Micael (2012). One Person, Many Votes: Divided Majority and Information Aggregation. Econometrica, 80, pp. 43-87. [4] Bullock III, Charles and Johnson, Loch (1992). Runoff Elections in the United States. Chapel Hill: University of North Carolina Press. [5] Callander, Steven (2005). Duverger’s Hypothesis, the Run-Off Rule, and Electoral Competition. Political Analysis, 13, pp. 209-232. [6] Cox, Gary (1997). Making Votes Count. Cambridge, UK: Cambridge University Press. [7] De Sinopoli, Francesco and Pimienta, Carlos (2009). Undominated (and) Perfect Equilibria in Poisson Games, Games and Economic Behavior, 66(2), pp. 775-784. 22 [8] Dolez, Bernard and Laurent, Annie (2010). Strategic Voting in a Semi-Presidential System with a Two-Ballot Electoral System. The 2007 French Legislative Election, French Politics, 8 (1), pp. 1 20. [9] Duverger, Maurice (1954). Political Parties. New York: John Wiley & Sons. [10] Engstrom, Richard L. and Engstrom, Richard N. (2008). The Majority Vote Rule and Runoff Primaries in the United States, Electoral Studies, 27 (3), pp. 407-416. [11] Fey, Mark (1997). Stability and Coordination in Duverger’s Law: A Formal Model of Pre-Election Polls and Strategic Voting, American Political Science Review, 91, pp. 135-147. [12] Golder, Matt (2005). Democratic Electoral Systems Around the World, 1946-2000, Electoral Studies, 24, pp.103-121. [13] Golder, Matt and Wantchekon, Leonard (2004). “Africa: Dictatorial and Democratic Electoral Systems since 1946.” In Josep Colomer. ed. Handbook of Electoral System Choice. London: Palgrave. [14] Kawai, Kei and Watanabe, Yasutora (2012). Inferring Strategic Voting, American Economic Review, forthcoming. [15] Martinelli, Cesar (2002). Simple Plurality Versus Plurality Runoff with Privately Informed Voters, Social Choice and Welfare, 19, pp. 901-919. [16] Messner, Matthias and Polborn, Mattias K (2007). Strong and coalition-proof political equilibria under plurality and runoff rule, International Journal of Game Theory, 35, pp. 287-314. [17] Morton, Rebecca and Rietz, Thomas (2006). Majority Requirements and Voter Coordination. New York University Annual Survey of American Law, 63(4), pp. 691-726. [18] Myerson, Roger (1996). Fundamentals of Social Choice Theory, mimeo, University of Chicago. [19] Myerson, Roger (1998). Population Uncertainty and Poisson Games, International Journal of Game Theory, 27, pp. 375-392. [20] Myerson, Roger (2000). Large Poisson Games, Journal of Economic Theory, 94, pp. 7-45. [21] Myerson, Roger (2002). Comparison of Scoring Rules in Poisson Voting Games, Journal of Economic Theory, 103, pp. 219-251. 23 [22] Okada, Akira (1981). On stability of perfect equilibrium points, International Journal of Game Theory, 10, pp. 67-73. [23] Osborne, Martin and Slivinski, Al (1996). A Model of Political Competition with Citizen-Candidates. Quarterly Journal of Economics, 111, pp. 65-96. [24] Palfrey, Thomas and Rosenthal, Howard (1991). Testing Game-Theoretic Models of Free Riding: New Evidence on Probability Bias and Learning, In Laboratory Research in Political Economy, ed. Thomas Palfrey, Ann Arbor: University of Michigan Press. pp. 239-268. [25] Piketty, Thomas (2000). Voting as Communicating. Review of Economic Studies, 67, pp. 169-191 [26] Riker, William (1982). The Two-Party System and Duverger’s Law: An Essay on the History of Political Science. American Political Science Review, 84, pp. 10771101. [27] Saari, Donald (2003). Unsettling Aspects of Voting Theory, Economic Theory, 22, pp. 529-555. [28] Smith, J.H., (1973), Aggregation o f preferences with variable electorates, Econometrica, 41, pp. 1027-1041. [29] Schmidt, Gregory (2007). Back to the Future? The 2006 Peruvian General Election, Electoral Studies, 26, pp. 797-837. [30] Spoon, Jae-Jae (2008). Presidential and Legislative Elections in France, April-June 2007, Electoral Studies, 27, pp. 151-190. Appendices Appendix A1 provides a reminder of some fundamental properties of Poisson games (Myerson 2000 and 2002). Appendix A2 demonstrates the claims made in Section 4. Appendix A1: Large Poisson Games in Runoff Elections A Poisson game Γ ≡ (n, T , F, C, u) is defined by the expected number of voters n ∈ N, the set of types T , a probability measure F defined over T , a set of actions C and a vector of payoffs ut : C × Z (C) → R, each t ∈ T , where Z (C) is the set of all action profiles for the players. 24 Myerson (2000) shows that, in a Poisson game, the probability of an exact profile of vote shares is exponentially decreasing in the expected number of voters, n, and converges to zero at an exponential speed proportional to its magnitude. Furthermore, Myerson (2000, Corollary 1) shows that: Lemma 1. Compare two events E and E 0 with different magnitudes: µ(E) < µ(E 0 ). Then, the probability ratio of the former over the later event goes to zero as n increases: µ(E) < µ(E 0 ) ⇒ Pr (E) → 0 Pr (E 0 ) n→∞ The intuition is that the probabilities of different events do not converge towards zero at the same speed. Hence, unless two events have the same magnitude, their likelihood ratio converges either to zero or to infinity when the electorate grows large. Myerson calls this result the magnitude theorem. Proofs in this paper rely extensively on this property of large Poisson games. We make use of the magnitude theorem to identify the properties of the set of strictly perfect equilibria as n → ∞. As explained in Section 3, there are two types of pivotal events in a majority runoff election: the threshold pivotabilities and the second-rank pivotabilities. As proven in Bouton (2012), the magnitude of a pivotal event piv is larger when the expected outcome of the first round, τ, is close to the conditions necessary for event piv to occur. For instance, the pivotal event pivi/ij is more likely to occur when 1/2 = τi > τj > τk than when 1/2 > τk > τj > τi . Indeed, the occurrence of the pivotal event in the latter case requires a “larger deviation with respect to the expected outcome”. Lemma 2. The magnitudes of the pivot probabilities are: (a) Threshold pivot probability i/ij and ij/i:   p
2 (τj + τk ) τi − 1
µ pivi/ij = µ pivij/i = p  2 2τi √τj τk − 1 if τj τj +τk ≥ 12 ; (2) otherwise (b) Second-rank pivot probability ki/kj and kj/ki: µ pivki/kj

= µ pivkj/ki =  √ √
2   − τi − τj    p √ = 2 2τk τi τj − 1     3 (τi τj τk ) 13 − 1 √ √ if 2 τi τj > τk > τi τj ; √ if τk > 2 τi τj ; √ if τi τj > τk . (3) We are now in position of establishing some preliminary results on the equilibrium behavior of the magnitudes of different pivot probabilities. In particular, Lemma 3 says that the magnitude of pivR/Ri , the event that a single vote being decisive between the front runner winning outright and a second round between the front-runner and the runner-up, is never less than the magnitude of any other first round 25 pivot probability. Also, the magnitude of pivRS/RW , the event that a vote is pivotal in determining which candidate will face the front runner in a second round, is never less than the magnitude of any other second round pivot probability and is strictly larger unless the front runner and the runner-up have the same expected share of votes. Lemma 3. There are three possible rankings of the two largest magnitudes:

(i) µ pivR/Ri ≥ µ pivi/Ri ≥ any other magnitude; or

(ii) µ pivR/Ri ≥ µ pivRS/RW ≥ any other magnitude; or

(iii) µ pivRS/RW > µ pivR/Ri ≥ any other magnitude.
Proof. We first compare µ pivR/Ri with other threshold magnitudes and show that this is the largest threshold magnitude: q q

µ pivR/Ri = 2 (τj + τi ) τR − 1 ≥ 2 (τj + τR ) τi − 1 = µ pivi/Ri and the expression holds with equality only if τj = 0 or τi = τR . Also, trivially q q

µ pivR/Ri = 2 (τj + τi ) τR − 1 > 2 (τi + τR ) τj − 1 = µ pivj/Rj unless τR = τi and q q

µ pivi/Ri = 2 (τj + τR ) τi − 1 > 2 (τi + τR ) τj − 1 = µ pivj/Rj unless τj = τi . We can also show that µ pivR/Ri
= 2 q (τj + τi ) τR − 1 > 2 q
√ 2τR τi τj − 1 = µ pivR/Rj p p √ √ unless τi = τj . Indeed, 2 (τj + τi ) τR − 1 > 2 2τR τi τj − 1 ⇐⇒ τR (τi + τj ) − 2τR τi τj > 0. The   √ LHS of the last inequality can be rewritten as τR τj + τi − 2 τi τj and   √ √ √
2 τR τj + τi − 2 τi τj = τR τi − τj > 0 if τi > τj .


It remains to show that µ pivR/Ri is larger than µ pivi/ij and µ pivj/ij . The first condition is satisfied if τR > τi or τi > τj since µ pivR/Ri 2 q
(τj + τi ) τR − 1 (τj + τi ) τR 2
µ pivi/ij q √ > 2 2τi τj τR − 1 > > 26 √ √ τj τi τi τR . Notice that the first element of the LHS is greater or equal (only if τR = τi = τj = 1/3) than the first element of the RHS since a geometric mean of x, y, . . . is always less than or equal to the arithmetic mean of x, y, . . . , with the equality holding only if x = y = . . . . Also, since τR ≥ τi , the second element is

also greater than or equal to the second element of the RHS. The last case, i.e. µ pivR/Ri > µ pivj/ij follows a very similar argument. Furthermore, q q

√ µ pivi/Ri = 2 (τj + τR ) τi − 1 > 2 2τi τj τR − 1 = µ pivi/ij

unless τj = τR , and similarly µ pivi/Ri > µ pivj/ij .
We now compare µ pivRS/RW with other second-rank magnitudes and we show that this is the √ √ largest second-rank magnitude. First, notice that for τR ≥ τi ≥ τj , τR ≥ τi τj , τj ≤ τR τi , and

√ √ τi > 2 τR τj ⇒ τR > 2 τi τj . Hence, to prove that µ pivRS/RW > µ pivjR/ji it is sufficient to show 1 √ √ √
2 two conditions: 1) if τR < 2 τi τj , then it is sufficient to show that − τi − τj > 3 (τi τj τR ) 3 − 1. The inequality can be rewritten as (using τR + τi + τj = 1) 1 √ τi + τj − 2 τi τj < τi + τj + τR − 3 (τi τj τR ) 3 and therefore as √
1 τR + 2 τi τj √ τi τj 2 τR 3 . > 3 The RHS and the LHS are, respectively, the weighted geometric and arithmetic means of τR with weights 2 and 1. It follows that they are equal if and only if τR = τi = τj = 1 3, √ τi τj and otherwise, the inequality holds. 1 p √ √ 2) if τR ≥ 2 τi τj , then it is sufficient to show that 2 2τR τi τj − 1 ≥ 3 (τR τi τj ) 3 − 1. Taking logs and simplifying, we get 3 ln τi τj − 2 ln τR ln 2 − ln 3 ≥ 2 12   √ 3 !12   2  ≥ ln τi τj ln  3 τR2  √ 3 6 √ 2 τi τj . Notice that ≈ .7023 < 2. Hence, since τR > 2 τi τj , we 3

have shown that µ pivRS/RW > µ pivjR/ji .

√ To show that µ pivRS/RW ≥ µ piviR/ij , we divide the analysis into three cases. 1) if τi < τR τj ,



then µ piviR/ij = µ pivjR/ji and we have just shown that µ pivRS/RW ≥ µ pivjR/ji with equality which simplifies to τR ≥  √ 3 6 2 3 √ holding only if τR = τi = τj = 31 . 27

√ 2) if τi > 2 τR τj , then µ pivRS/RW ≥ µ piviR/ij if and only if q √ 2 2τR τi τj − 1 √ τR τi τj √ τR ≥ ≥ ≥ q √ 2 2τi τR τj − 1 √ τi τR τj √ τi which is trivially true for all τR ≥ τi , with equality only if τR = τi .

√ √ √ 3) if τR τj < τi < 2 τR τj and τR < 2 τi τj , then µ pivRS/RW ≥ µ piviR/ij if and only if − √ τi − √
2 τj √ τi ≥ ≤ − √ √ τR − √
2 τj τR which is trivially true for all τR ≥ τi , with equality only if τR = τi .


√ Last, we compare µ pivR/Ri and µ pivi/Ri with µ pivRS/RW . Notice that if τR ≥ 2 τi τj , then


µ pivRS/RW = µ pivR/Rj ≤ µ pivR/Ri with the last inequality holding with strict sign unless τi = τj .

√ Otherwise, if τR ≤ 2 τi τj , then there exist two regions of ∆ (C) such that µ pivi/Ri > µ pivRS/RW

and µ pivi/Ri > µ pivRS/RW , respectively.

√ Furthermore, if τR ≥ 2 τi τj , µ pivi/Ri ≥ µ pivRS/RW ⇒ Using τi ≥ √ q 2 (τj + τR ) τi − 1 ≥ (τj + τR ) τi ≥ q √ 2 2τR τi τj − 1 √ 2τR τi τj . τi τj , we have (τj + τR ) τi τj √ ≥ 2τR τi ≥ 2τR τi τj ≥ τR √ with equality holding only if τR = τS = τW = 1/3. Otherwise, if τR ≥ 2 τi τj , there exist two regions of



∆ (C) such that µ pivi/Ri > µ pivRS/RW and µ pivi/Ri > µ pivRS/RW , respectively. Appendix A2: Proofs of Section 4 Proof. [Proof of Proposition 1] In a Duverger’s law equilibrium there exists j ∈ {W, S} : τj (σ ∗ , F ) = 0.
That is, we need σt∗ (j) = 0 for all but at most a measure zero of voter types. This implies µ pivR/Ri =

µ pivi/Ri > any other magnitude, i 6= j. We can construct Gt (c, nτ ) , ∀c ∈ C and divide by Pr pivR/Ri and take the limit when n → ∞. To prove that there exists a Duverger’s law equilibrium, it is sufficient to show that for all t ∈ T , there exist c 6= j : limn→∞ Gt (c, nτ ) ≥ limn→∞ Gt (j, nτ ). 28 Let
Pr pivS/Ri
> 0. φ = lim Pr pivR/Ri Then, we have lim n→∞ Gt (j, nτ )
Pr pivR/Ri = U (R, i | t) − U (R | t) + φ [U (R, i | t) − U (i | t)] = lim n→∞ Gt (i, nτ )
Pr pivR/Ri = U (R, i | t) − U (R | t) + φ [U (i | t) − U (R, i | t)] = lim n→∞ Gt (R, nτ )
Pr pivR/Ri (1 + φ) U (R, i | t) − U (R | t) − φU (i | t) ; (1 − φ) U (R, i | t) − U (R | t) + φU (i | t) ; = U (R | t) − U (R, i | t) + φ [U (R, i | t) − U (i | t)] = (φ − 1) U (R, i | t) + U (R | t) − φU (i | t) . Notice that U (R, i | t) is a strict convex combination of U (R | t) and U (i | t). It is easy to show that limn→∞ Gt (R, nτ ) ≥ limn→∞ Gt (j, nτ ) if U (R | t) ≥ U (i | t) and limn→∞ Gt (i, nτ ) ≥ limn→∞ Gt (j, nτ ) otherwise. Hence, σt∗ (R) = 1 is a best response if U (R | t) > U (i | t), σt∗ (i) = 1 is a best response if U (i | t) > U (R | t), and σt∗ (j) = 0 is a best response for all t ∈ T , since for all t ∈ T : U (i | t) = U (R | t), Rt (τ (σ ∗ , F )) (c) = [0, 1] , ∀c ∈ C. Notice also that t ∈ T : U (i | a, b) = U (R | a, b) has measure zero for all F ∈ F. Hence, this is a strict equilibrium for all but a measure zero of voters. We imposed no restrictions on F , hence for any F ∈ F, if σt∗ (j) = 0, for all t ∈ T but at most t ∈ T : U (i | t) = U (R | t), then τj (σ ∗ , F ) = 0. To show that a Duverger’s law equilibrium is strictly perfect, consider any τ ∈ ∆ (C) : |τ − τ (σ ∗ , F )| <  for some  > 0 and τj (σ ∗ , F ) = 0. Notice that all magnitude formulae are continuous. For  sufficiently

small, the order of the magnitudes is µ pivR/Ri > µ pivi/Ri >any other magnitude, i 6= j. In which case, lim Gt (j, nτ )
Pr pivR/RS = U (R, i | t) − U (R | t) lim Gt (i, nτ )
Pr pivR/RS = U (R, i | t) − U (R | t) n→∞ n→∞ 29 lim n→∞ Gt (R, nτ )
Pr pivR/RS = U (R | t) − U (R, i | t) . Trivially, σ ∗ is a best response. Proof. [Proof of Proposition 2] The starting point of the proof is to consider a tuple (σ ∗ , F ) such that



µ pivR/Ri > µ pivRS/RW ≥ µ pivR/Rj and µ pivRS/RW > any other magnitude (that is, i is the runner up candidate). We then divide the rest of the proof is into three parts. First, we show that all voters with U (R | t) > U (i | t) vote for candidate R. Second, we analyze the behavior of voters with U (R | t) < U (i | t) . We show that those with U (R, S | t) ≥ U (R, W | t) vote for S, whereas the others vote for W. Finally, we prove that the equilibrium is strictly perfect. First, we can construct, lim Gt (j, nτ )
= U (R, i | t) − U (R | t) Pr pivR/Ri lim Gt (i, nτ )
= U (R, i | t) − U (R | t) Pr pivR/Ri n→∞ n→∞ Gt (R, nτ )
= U (R | t) − U (R, i | t) n→∞ Pr pivR/Ri lim and conclude that all the voters with U ( R| t) > U ( i| t) vote for candidate R.


Second, divide the expected gains by Pr pivRS/RW and assume µ pivRS/RW > µ pivR/Rj . Then " lim n→∞ Gt (S, nτ ) Gt (W, nτ )
−
Pr pivRS/RW Pr pivRS/RW # = U (R, S | t) − U (R, W | t) − U (R, W | t) + U (S, W | t) = 2 [U (R, S | t) − U (R, W | t)] . Hence, a voter of type t (with U ( R| t) < U ( i| t)) votes for S only if U (R, S | t) ≥ U (R, W | t) ⇐⇒ Pr(R | RS)U (R | t) + (1 − Pr(R | RS)) U (S | t) ≥ ≥ Pr(R | RW )U (R | t) + (1 − Pr(R | RW )) U (W | t) (4) with any mixed strategy allowed for the measure zero of voters’ types with U (R, S | t) = U (R, W | t). Otherwise she votes for W . Notice that the condition in 4 is independent of which candidate, S or W , 30 is expected to receive more votes. Furthermore, since Pr(R | RS) < Pr(R | RW ), a type t indifferent

between S and W must be a W ’s supporter. If instead µ pivRS/RW = µ pivR/Rj , then " lim n→∞ Gt (W, nτ ) Gt (S, nτ )
−
Pr pivRS/RW Pr pivRS/RW # = U (R, S | t) − U (R, W | t) − U (R, W | t) + U (S, W | t) + +φ0 [U (R, j | t) − U (R | t) − U (R, j | t) + U (R | t)] = with 2 [U (R, S | t) − U (R, W | t)]
Pr pivR/Rj
> 0. φ = Pr pivRS/RW 0 Hence, the condition for voting for S or W is not changed. To show that this equilibrium is strictly perfect, consider any τ ∈ ∆ (C) : |τ − τ (σ ∗ , F )| <  for some



 > 0 and τj (σ ∗ , F ) : τ : µ pivR/Ri > µ pivRS/RW ≥ µ pivR/Rj and µ pivRS/RW > any other magnitude. Notice that all magnitude formulae are continuous. For  sufficiently small, the order of the magnitudes is unchanged. In which case, σ ∗ is a best response. Proof. [Proof of Proposition 3] Propositions 1 and 2 characterize the set of equilibria when the order of magnitudes is as in points 1 and 2 in lemma 3. Together, Lemmata 1 and 3 imply that no other strictly
perfect equilibrium σ ∗ can exist unless τ (σ ∗ , F ) implies point 3 in lemma 3, i.e. when µ pivRS/RW is the (strictly) largest magnitude. Notice that this implies τR (σ ∗ , F ) > 0 (indeed, by definition, R is the front runner). We can construct, lim Gt (S, nτ )
Pr pivRS/RW = U (R, S | t) − U (R, W | t) lim Gt (W, nτ )
Pr pivRS/RW = U (R, W | t) − U (R, S | t) n→∞ n→∞ lim n→∞ Gt (R, nτ )
Pr pivRS/RW = 0. Hence, voting for R is a best response for a measure zero of voter types, those with t ∈ T : U (R, S | t) =
U (R, W | t). Hence, τR (σ ∗ , F ) = 0, contradicting the assumption that µ pivRS/RW is the largest magnitude. 31