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Winner of an asymmetric election

From Proposition 2.9, if candidate i has a greater incentive to approach xm (i.e., y/j(zj, zj) is greater than ijf j(zj, z,) in the event of a tie with ipj{zj, Zj) + b = 0), then, in equilibrium, candidate / always wins. This implies that to find the winner of an asymmetric election, it is sufficient to compare candidates’ degrees of incentive to win. This can be given as follows.

In order to prove that i wins against j, assume that the policies each candidate would implement are initially fixed at symmetric positions (i.e., |Xi (-/) ~xm = Xj (zy)_ |)■ This implies that the electoral outcome (a tie) is fixed by fixing the policies to be implemented. Note that because voters have complete information, they can correctly guess the policy each candidate would implement, making the positions of these policies critical to the electoral outcome. Suppose also that two candidates are initially symmetric (i.e., they have symmetric cost and disutility functions, and their ideal policies are equidistant from the median policy) and indifferent between winning and losing, that is, VP, (Zj, zj)+b = '¥j(zj, Zj)+b = 0 . Then, differentiate 'Vj{zj, z, ) by the parameter of a candidate’s characteristic (such as xj, A,-, or pj). Now, suppose j’s parameter value is higher than that of/. If Ч'Д-у, decreases with this parameter value, it means that vF/(-/, -,) is lower than '¥i(=j, =j) in a tie with j(zj, z,)+b = 0. Hence, / is certain to win, according to Proposition 2.9.

In the following subsections, I use the above method to show the asymmetric electoral outcomes for asymmetric ideal policies, asymmetric costs of betrayal, and asymmetric policy motivations. Although I only consider these basic characteristics, this model could be used to derive more implications by adding other characteristics (e.g., competence and valence) or other players (e.g., special interest groups and media).

Asymmetric ideal policies

Assume that xr - xm xm - xL; that is, the candidate’s ideal policy is asymmetric. The cost and disutility functions are the same for both candidates. Suppose also the following assumption.

Assumption 2

u"(d) / u'{d) is non-increasing in d.

This assumption means that the Arrow-Pratt measure of absolute risk aversion is non-increasing in Xi{=i)~лу|. If the function is monomial, this assumption holds and will be satisfied by many polynomial functions.

Corollary 2.10

Suppose Assumptions l and2, and that two candidates run. Furthermore, suppose that candidate i is more moderate (i.e., |.v,- - л„,| < |лу -л„ф, but that the candidates are symmetric in all other respects. Then, in equilibrium, we have the following: (i) when u"(d) > 0, for any d > 0, candidate i wins with certainty and the expected utility from winning is higher than the expected utility from losing; and (ii) when u"(d) = 0 .for alld > 0, the result is either a tie or candidate i wins with certainty.

Proof: See Appendix 2.A.9.

A more moderate candidate, whose ideal policy is closer to the median policy, will not severely betray his/her platform after an election. On the other hand, in order to implement the same policy, a more extreme candidate will pay a higher cost of betrayal because he/she will betray the platform more severely. Hence, his/her degree of incentive to win decreases to avoid paying such a high cost of betrayal. As a result, the more moderate candidate wins.

When the candidates’ utility functions are linear, they tie in most cases. When a candidate has a linear utility function, the policy he/ she will implement is not affected by the ideal policy, x,. Therefore, the situation is the same for both candidates and they have the same probability of winning. However, if the moderate candidate’s ideal policy is very close to xm, the moderate candidate does not have an incentive to approach xm from his/her ideal policy, and the extreme candidate does not have an incentive to win even though the moderate candidate announces a,. Thus, the moderate candidate announces his/her ideal policy as the platform and then implements it after he/she wins. This case is similar to Corollary 2.7, but the moderate candidate wins with certainty, as his/her ideal policy is closer to the median policy.

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