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Electoral Promises as a SignalIntroductionThis chapter analyzes the signaling role of campaign promises by extending the model with partially binding platforms discussed in Chapter 2.^{1} In particular, I introduce asymmetric information by assuming that candidate policy preferences are private information. One candidate’s ideal policy is to the left of the median policy, while that of the other candidate is to the right. Each candidate is one of two types  moderate or extreme  and the moderate type’s ideal policy is closer to the median policy than that of the extreme type. A candidate knows his/her own type, but voters and the opponent do not. In the remainder of this chapter, I refer to an extreme type as “he” and a moderate type as “she.” Note that both candidates have the same cost of betrayal and the degree of policy motivation (A = /2 = 1), so only policy preferences are possibly asymmetric. Most studies on the twocandidate model of political competition show a politician’s convergence with the ideal policy of the median voter. However, in realworld elections, politicians are frequently polarized. This chapter provides one possible reason why politicians are polarized and shows that an extreme candidate has a higher probability of winning than does a moderate candidate, although the extreme candidate will implement a more extreme policy. The important reason for the extreme candidate’s higher probability of winning is that he has a stronger incentive to prevent an opponent from winning because his ideal policy is further from the opponent’s policy than is that of a moderate candidate. The model in this chapter describes this incentive for an extreme candidate by introducing partially binding platforms and uncertainty about a candidate’s preference. The modelSettingThe policy space is R.^{2} There is a continuum of voters, and their ideal policies are distributed on some interval of R. The distribution function is continuous and strictly increasing, so there is a unique median voter’s ideal policy, x_{m}. There are two candidates, L and R, and each candidate is one of two types: moderate or extreme. Let лf* and л/ denote the respective ideal policies for the moderate and extreme types, where /' = L or R, and xf < xf^{!} After the types of candidates are decided, each candidate announces a platform, denoted by zf e R, where / = L or R and t = M or E. On the basis of these platforms, voters decide on a winner according to a majority voting rule. After an election, the winning candidate chooses an implemented policy, denoted by z', where / = L or R and t = M or E. As in Chapter 2, the policy to be implemented will lie somewhere between the platform policy and the ideal policy, as shown in Figure 1.1 in Chapter 1. If the implemented policy is different from the candidate’s ideal policy, all candidates — both winner and loser — experience disutility. This disutility is represented by , where / = L or R,t = M or E, and z is the policy implemented by the winner. Assume that г/(.) satisfies г/(0) = 0, м'(0) = 0, u'[d)> 0, and u"[d)> 0 when d> 0. Unlike in Chapter 2, I do not consider linear utility here. If the implemented policy is not the same as the platform, the winning candidate needs to pay a cost of betrayal. The function describing the cost of betrayal is Assume that c(.) satisfies c(0) = 0, c'(0) = 0, c'(0, and c’(d)> 0 when d> 0. The loser does not pay a cost. After an election, the winning candidate chooses a policy that maximizes мЦ^лфсАг' zj That is, the winner chooses Z! (=i) = argmax*  «(*  */)  f ( ='  *)’ I also assume that c'(d)l c[d) strictly decreases as d increases. That is. Assumption 1 introduced in Chapter 2 holds. I also suppose the following assumption, which is similar to Assumption 2 in Chapter 2. Assumption 3 u'{d) / «(and u"(d) / u'(d) strictly decrease as cl increases. This assumption means that the relative marginal disutility decreases as^A? increases, and the ArrowPratt measure of absolute risk aversion is decreasing in ^a'. For example, if the function is a monomial, this assumption holds, and many polynomial functions satisfy the assumptions as well. Upon observing a platform, the utility of voter n when candidate i of type t wins is w^ % (r,)A„j. Assume that w(.) satisfies w'(r/)>0 when d>0. Let /?,(/ jr) denote the voters’ revised beliefs that candidate i is of type t upon observing platform z. The expected utility of voter n when the winner is candidate i, who promises z,, is p,(M  z,) «(x!“ {=i)x„)(Pi(M  z,))m(z,) — a„j. Voters vote sincerely, which means that they vote for the most preferred candidate, and weakly dominated strategies are ruled out. Assume that all voters and the opponent have the same beliefs about a candidate’s type. Let n (z,zj j denote the probability of candidate i of type t winning against opponent j of type s, given z/ and zj. Let F’ (.) denote the distribution function of the mixed strategy chosen by candidate i of type /. The expected utility of candidate / of type t who promises z/ is where i,j = L,R and t = M,E. The first term indicates when the candidate defeats each type of opponent. The second term indicates when the candidate loses to each type of opponent. To simplify, I do not introduce a benefit from holding office (i.e., b = 0) in a candidate’s utility. A benefit from holding office would not change my results significantly when the benefit is small. Here, candidates approach the median policy more closely, but the main characteristics of the equilibria do not change. Therefore, I simply assume that the benefit from holding office is zero. However, if the benefits from holding office are great, candidates’ implemented policies converge to the median policy regardless of type like as Lemma 2.4 in Chapter 2. This is what Huang (2010) shows by introducing sufficiently high benefits from holding office, which compensate for all disutility resulting from the policy and the cost of betrayal. In summary, the timing of events is as follows. Note that this chapter does not consider candidates’ decisions to run.
In what follows, I concentrate on a symmetric, purestrategy, perfect Bayesian equilibrium consisting of strategies and beliefs. Policy implemented by the winnerFollowing an election, the winning candidate implements a policy that maximizes his/her utility following a win, m( X a,'I)  с (I r' x (=,')) like as Lemma 2.1 in Chapter 2. Lemma 3.1 The implemented policy % () satisfies u'^xj () —v/) = c'(r  x ()), for x () g (x',r) when r > x'_{h} andx () e (z,xf) when : < xj. When the platform differs from the ideal policy, the implemented policy must differ from the platform or the ideal policy since it is decided by the winner (who no longer cares about the loser’s platform, but cares about the cost of betrayal) after an election. If voters know the candidate’s type (ideal policy), they can also know the future implemented policy by observing the platform. However, with asymmetric information, they may not know the candidate’s type. Here, the median voter x_{m} is pivotal. Thus, if the candidate is more attractive to the median voter than is the opponent, this candidate is certain to win. 
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