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Electoral Promises with Vague WordsIntroductionPolitical parties and candidates usually prefer making ambiguous promises, a practice referred to as “political ambiguity.” Prior studies usually interpret political ambiguity as a lottery: a probability distribution on single policies. In accordance with these past studies, this chapter extends the standard Downsian model with fully office motivated candidates to allow a candidate to choose a lottery, rather than a single policy.^{1} Then, this chapter identifies the conditions under which candidates choose ambiguous promises in equilibrium. Past studies indicate many possible reasons that political ambiguity occurs (Subsection 1.3.5), but this chapter analyzes convex voter preference as a possible explanation (Zeckhauser, 1969; Shepsle, 1972; Aragones and Postlewaite, 2002). To simplify the analysis, this chapter considers completely binding platforms, that is, candidates announce a lottery as a platform before an election, and the winner implements the policy according to the probability distribution of the announced platform after he/she wins the election. This chapter shows that candidates choose ambiguous promises in equilibrium when (i) voters have convex utility functions, (ii) candidates are uncertain about voters’ preferences, and (iii) the distribution of voters’ preferred policies are polarized. Therefore, for political ambiguity to be considered as an equilibrium phenomenon with convex utility functions, voters must be polarized and voting must be probabilistic. Through this chapter, I do not intend to say that the convexity of utility functions is the only reason political ambiguity emerges; many reasonable mechanisms have been suggested. However, while prior studies recognize the convexity of a voter’s utility function as one reason for the emergence of ambiguity, none shows it as an equilibrium phenomenon without any restrictions on a candidate’s strategy. Thus, one of the main contributions of this chapter is to explicitly show additional conditions (probabilistic voting and polarized voters) in which candidates choose a vague promise in equilibrium, given voters’ convex utility functions while past studies did not establish the existence of equilibria in which candidates announce ambiguous promises with convex utility functions. Deterministic votingDiscrete spaceFirst, this subsection presents the implications of the deterministic model as a benchmark. Let me denote X as the set of policies and define g(.v,y) as the majority margin for x,y g A; the number of voters who prefer .v to у minus the number of voters who prefer у to x, where x and у are single policies. A policy x is the Condorcet winner when g(.v,y) > 0 for all у g X (Black, 1948). Let us denote AX as the set of probability distributions over X. g(p,q) is defined as the majority margin for lotteries p.q g AX. I call a Condorcet winner on AX a Condorcet winning lottery, which is defined as follows:^{2} Definition 4.1 A Condorcet winning lottery is a lottery p, such that g(p,q)> 0 for all q g AX. Suppose three policies, X = {L.M.R}. where an element of AX is (Pl.Pm.Pr) g AX. and p_{x}> 0 is the probability that ag X occurs, where p_{L} + p_{M} + p_{R} = 1. In addition, suppose there is a population of voters from mass one, divided into three discrete groups, /, m, and r, and the proportion of voters in each group is less than 1/2; that is, no group constitutes a majority. Denote the set of groups as G = {/,»;,/ }, and its element as g g G. Suppose that members of each group have the following preference relations:
These preference relations satisfy singlepeakedness. Furthermore, the median point is M. which is the Condorcet winner in Black (1948).^{3} These preference relations of voters in group g can be represented by the Von NeumannMorgenstern utility function u_{g} : X —>{0,v,l}, with v e (0,1) The function assigns the value one to the most preferred alternative, v to the secondbest alternative, and zero to the worst alternative. A voter has a concave utility function if v> 1/2, a linear utility function if v = 1 / 2, and a convex utility function if v < 1 /2. Note that if a member of group m has L~ R, the utilities of both L and R are v = u_{m}(L) = u_{m}(R). 1 refer to the lottery (pi, Pm,Pr) = (0,1,0) simply as M. Assume that voters vote sincerely; they vote for the most preferred lottery among the alternatives. Assume also that voters choose to abstain when they are indifferent. If the two lotteries receive the same number of votes, the winner is chosen with an equal probability (50% each). Then, the following proposition is attained. Note that the result does not change even if voters choose a lottery with an equal probability when they are indifferent.^{4} Proposition 4.2 A Condorcet winning lottery is M when v > 1 / 2, and does not exist when v < 1 / 2. Proof: See Appendix 4.А.1. The rationale is as follows. Policy M cannot be the Condorcet winning lottery if voters have convex utility functions (i.e., v < 1 / 2). If M is chosen, the utilities of voters in groups/,m, and r are v, 1, and v, respectively. On the other hand, if lottery q with {Pl,PMiPr) = { 1 /2, 0, 1 /2) is chosen, the utilities of voters in /, /;/, and r are 1/2, v / 2, and 1 / 2, respectively. Thus, if v < 1 / 2, the voters in / and r prefer cp to M, and M is defeated by cp in a pairwise election. Moreover, cp cannot be a Condorcet winning lottery. If lottery cj2 with (pl,Pm,Pr) = (2/3, 1/3, 0) is chosen, the utilities of voters in /, m, and r are (2 + v)/3, (l + 2v)/ 3, and v / 3, respectively. Thus, the voters in / and m prefer <72 to cp. However, <72 is also defeated by 1/2, M is not defeated by <71 (or any other lotteries). Such a preference cycle usually occurs when a policy space has multiple dimensions. Supposing that candidates can choose a lottery instead of a single policy, the space of lotteries has two dimensions, since pi and рм should be identified (and p_{R} is determined by Pr =  Pl~ Pm), even though the dimension of a single policy is one. However, if voters have concave utility functions, all voters prefer the least risk, that is, to make a certain choice. If they have linear utility functions, voters in / and r are indifferent, but voters in m still prefer M to q because their utility is maximized when M is chosen. Consequently, if voters have concave or linear utility functions, they all (w'eakly) prefer a less ambiguous choice, in which case the space is considered one dimensional [pi = 1, рм = к or рц = l). Therefore, a Condorcet winning lottery exists if v > 1 /2. On the other hand, if voters have convex utility functions, conflicts of interest will arise among them: voters in group m prefer M to q because their utility is maximized when M is definitively chosen. However, voters in groups / and r prefer q to M because q is riskier. Thus, both the position of a lottery and its degree of ambiguity matter, and this multidimensional space induces the nonexistence of a Condorcet winning lottery. Continuous spaceWhen voters have concave or linear utility, the median policy is still the Condorcet winning lottery within a continuous policy space. Suppose that the policy space is continuous, fcR. There is a continuum of voters, and each voter / has a singlepeaked preference with his/ her ideal point (the most preferred policy) x, e X. Voter /’s preference can be represented by the function n, (^x,) where X is the implemented policy. Since a voter has a singlepeaked preference, it satisfies u'i(d) > 0 for d > 0. Voter’s ideal points are distributed on X, and the distribution function is continuous and strictly increasing, meaning that there exists a unique median policy denoted by x,„. In a lottery /у, the implemented policies are distributed on [x/,x/], and the probability distribution is #/(.)• The upper bound of the lottery is Xj and the lower bound is xj, that is, X/
Denote also as the expected value of the implemented policy of a lottery p/. Then, the following proposition is obtained. Proposition 4.3 Suppose iij(d) > 0 and u'fd) > 0 for d > 0. A Condorcet winning lottery is the median policy (a lottery which gives a probability one to x_{m}), and it is unique. Proof: See Appendix 4.A.2. In words, when voters are either risk averse or risk neutral, the Condorcet winning lottery is still the median policy. However, it may not be true when voters are risk loving. 
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