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Probabilistic votingSettingsReturn to the discrete settings with three policies in Section 4.2.1. A Condorcet winning lottery does not exist when voters have convex utility functions, as in the case of multiple policy dimensions. One method of finding an equilibrium when there are multiple policy dimensions is to introduce probabilistic voting, in which candidates are uncertain about voters' preferences. The voters’ preference relations on policies and utilities are the same as those in (4.1). However, suppose that the members of group m have L ~ R. such that the utilities from L and R are both v.^{5} A continuum of voters is distributed to each group according to a probability mass function f:G—>[0,l/2), with /(»/) = / and /(/) = /(r) = (ly)/2, where ye[0,1/2); that is a symmetric distribution. The parameter у represents the degree of centralization of the voter distribution. Two candidates 1 and 2 simultaneously determine the weight to allocate to each policy, @_{(} = ,Qf^{1}. j e AX before the election, where/ = 1 or 2. The value of Of e[0,l] is the weight assigned to policy xe X, where 9 +0/^{w} + 6? = . Note that @_{(} is not a mixed strategy on X, because the policy is chosen after the election, while in a mixed strategy, a policy is chosen before the election. The model also supposes that voters believe that the probability that policy x will be implemented after an election is the same as the weight on x; therefore, candidates can affect voters’ beliefs by allocating weights, as Callander and Wilson (2008) modeled. Candidate i obtains the share of votes given by where ©__{(} is the lottery chosen by / ’s opponent, and 0_', is the weight on policy a in the opponent’s promise. Suppose that an officemotivated candidate i maximizes П(0,,©_,). The function n: R —> [0,1] is strictly increasing (n'{t) > 0), satisfying 7t(t) + 7t(t) = 1 (thus, лг(0) = 1 / 2), and is strictly concave (tr"(t) < 0) for all t e [0,°°). Since n{t) + n{t) = 1, n'{t) = rr'(t) for all / e [0,°°). Here, 2, OfUg (a) is the expected utility of a voter in group g when candi date i wins the election. In addition, ^ Gfu_{g} (a)  / ®i^{u}g ( ^{v}) is the difference in the expected utility of a voter in group g between the promise of candidate / and that of his/her opponent. If this is positive (negative), candidate Гs lottery gives a higher (lower) expected utility than that of his/her opponent. In the deterministic model, 7t(l)= 1 when / > 0, and rt(t) = 0 when / < 0. However, in the case of probabilistic voting, even if t > 0, n(t) e (1/2,1]. One interpretation of this is that voters make decisions based not only on candidates’ policies but also on other factors, and therefore, their voting behavior is probabilistic. Equilibrium with convergenceThere exist multiple equilibria in this game. In order to clarify a situation where both candidates choose an ambiguous promise, I use the following corollary to show equilibria where both candidates choose the same lottery (0, = ©_,■), that is, both candidates converge. Corollary 4.4 i If v <1/2 and
an equilibrium with ©, = ©, must satisfy df = Off = 0. (ii) Otherwise, an equilibrium with 0, = 0_, must satisfy 6j^{xt} = 6^ = 1. Proof: See Appendix 4.A.3. When voters have convex utility functions (v< 1/2), and the proportion of median voters (y) is sufficiently small, an ambiguous lottery such as ©, =©,■ = (1/2,0,1/2) can be an equilibrium. Note that there exist many equilibria with ©, = ©_, and = Off = 0 such that 0, = 0__{(}=(1/3,0,2/3). Otherwise, both candidates converge to the median policy M. When voters have concave or linear utility functions (v >1/2), there are no conflicts of interest regarding the degree of ambiguity because all voters (weakly) prefer the lower degree of ambiguity. Therefore, the candidates should converge on M with certainty in equilibrium. This situation is the same as that of the Downsian model, and the median voter becomes critical in deciding the winner. On the other hand, when voters have convex utility functions, conflicts of interest among the voters on the degree of ambiguity do exist, because the voters in groups / and r (extreme voters) prefer a higher degree of ambiguity, whereas those in group m (median voters) still prefer a less ambiguous policy. If the proportion of median voters у is sufficiently high (у > (1 — 2v) / (3 — 4v)), candidates need to consider the median voters’ interests, and thus, they converge on the median policy. However, if у is low(y < (l2v)/(34v)), candidates care more about the extreme voters than they do about median voters, and thus, choose an ambiguous policy. In many extensions of the Downsian model of electoral competitions, the candidate who wins the support of the median voter is the winner. However, when (i) voters have convex utility functions and (ii) the proportion of median voters are small, a candidate cannot win even if he/she gets the support of the median voter. Rather, candidates must ignore the interests of the median voter to win the election. Equilibrium with divergenceThere also exist other equilibria with divergence, that is, Denote
I then have the following proposition.^{6} Proposition 4.5 Suppose v < 1 / 2. A strategy profile with Of^{4} = Off = 0 with Of  в^{4} = 9 (hence 0*  0^ = 0) for all i is a Nash equilibrium when у < у. Proof: See Appendix 4.A.3. As in Corollary 4.4, when the degree of political centralization у is sufficiently small, political ambiguity can emerge. Note that when voters have concave or linear utility functions, both candidates choosing M definitively is a unique equilibrium. Corollary 4.6 If v >1/2, 6^{VI} = Of^{1} = 1 is a unique equilibrium. Proof: See Appendix 4.A.3. The probabilistic voting model adopted here is based on that of Kamada and Kojirna (2014), who suppose that candidates can choose only a single policy (not a lottery). They show that with convex utility functions of voters and a polarized voter distribution, perfectly divergent candidates result in a unique equilibrium. Here, perfect divergence means that, without exception, the left candidate chooses a left policy, while the right candidate chooses a right policy, that is, 0, =(1,0,0) and 0_, =(0,0,1). On the other hand, the model of this chapter allows candidates to choose a lottery instead of a single policy, which increases the number of equilibria. Thus, ambiguity can arise in the form of equilibrium strategies in the context of convex voter utilities. In some equilibria, candidates choose the same ambiguous lottery, so policy divergence does not occur. On the other hand, perfectly divergent equilibrium shown by Kamada and Kojirna (2014) also exists in this model. Therefore, this model demonstrates that a probability voting model with convex utilities is useful to show not only political polarization but also political ambiguity, and candidates may choose partially divergent policies: they combine policy divergence and political ambiguity (i.e., вf > 0 and Qf > 0 with 0, Ф ©_,). In addition, the following corollary is obtained. Corollary 4.7 As j increases, у increases. Therefore, (i) it is less likely to lead to an equilibrium with more divergence, that is, higher в and (ii) if voters are more sensitive to differences between candidates, candidates tend to be converged to the same promise (i.e., ©, = Because n"{t) < 0 for all t e [0,~), a policy with more divergence has lower 7Г'(б/), which decreases y. Thus, the condition у According to Proposition 4.5, both policy divergence and political ambiguity can occur when voters have convex preferences with low y. However, Corollary 4.6 indicates that political ambiguity is more likely to occur compared to policy divergence. 
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