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Candidates with asymmetric characteristics

Equilibrium

This section shows that the model of partially binding platforms can predict the winner when candidates have asymmetric characteristics (such as having asymmetric ideal policies, A, and p). This section also shows the basic method for deriving a winner.

I denote

That is, *P ,•(=,■, zA refers to the difference between the utility of candidate / when candidate / wins ((|ж<(-/)-л:||)-Л'с(|-г - A/(-/)|)) and the utility of / when the opponent, j, wins i-u^Xj(=j)-ignoring the fixed values, b and k. When the candidates tie, a candidate will want to make ensure its win by approaching the median policy if 4/, (r,-, =j)+b > 0 but will want to lose if 4*, :j^+b< 0. The candidate with the higher 4х ,■ (=h r,) has the greater incentive to win. Therefore, 4х, refers to the degree of incentive to win.

1 also denote

when b < АсЦг,- (л„,) - a„,J). That is, candidate / is indifferent between winning and losing when the opponent’s policy is equidistant from xm as the own policy, and this distance is d,. When b > Acl|z, (An,)-Am|j, as a candidate has an incentive to commit to implement xm, 'P, =j)+b > 0 for all symmetric pairs of Xi(=i)

and Xj{=j) sj (л'т))+^-0). In this case, suppose that

dj = 0. From Corollary 2.3 and Proposition 2.6, (i) ;y)+/>>0

if Xi(=i)-xm = Xj{=j)-xm>di; (ii) 4/,(r„ - /)+6<0 if

Xi{=i)~ xm = xj[=j)-xm< di (and dj> 0); and (iii) the value of с/, is uniquely determined. In words, if the distance between Xi(=i) and x,„ (Xj(=j) and x,„) is longer than i has an incentive to win while / does not have such an incentive if this distance is shorter than d,. Then, suppose d, < dj, that is, candidate / has an incentive to commit to a more moderate policy in the event of a tie with Ч1 j (ry, r,-)+/> = 0. In this situation, the following proposition shows that candidate / announces a platform such that the policy he/she will choose to implement is slightly closer to the median policy than that of j, ensuring that in equilibrium, / will win.

One technical issue is that equilibrium may not exist in a deterministic model with a continuous policy space. Suppose L wins with certainty; that is, L commits to <|^лл)-л„,|, which is a

more moderate policy than that of R. In this case, L prefers to move to a more extreme policy such that L would still win against R, but the policy L would implement would be closer to his/her ideal policy. Note that such a policy exists because the policy space is continuous.

By contrast, if a discrete policy space is introduced in the above case, L may not be able to find such a policy. Suppose we have a grid of evenly spaced policies. The distance between sequential policies is e > 0. The other settings remain the same. Note that the purpose of introducing a discrete policy space is to ensure equilibrium, not to show new implications from a discrete case. Thus, assume that e is a very small value so that the situation is almost the same as that of a continuous policy space.'’ In the following, I assume such a discrete policy space.

Proposition 2.9

Consider a case of discrete policy space. Suppose d, < dj. Then, there exists an equilibrium, and the pair of platforms {r,-, r ; } is an equilibrium strategy if and only if

where Xi(=t) « closer to x,„ than Xj{=j) ЬУ that is> Xl(=l) = xm ~{Xr{=r)~x„,) + c if i = L, and Xr(=r) = + (*m - Xl{=l))-* if

i = R. In equilibrium, i is certain to win, and hence, there is no equilibrium in which both candidates have the same probability of winning. Proof: See Appendix 2.A.8.

The intuition is as follows. Suppose that candidate i has a greater incentive to approach x„, than opponent j does when they tie with 'V j{zj, r,)+/b = 0 (i.e., dj Then, a pair of platforms exists, {r,, ry}, such that the policies each will choose to implement are equidistant from the median policy ||д;,(-,)-лт| = |^у(гу)-л„,|| and candidate j has an incentive to lose |i//, (r,-, =j) + b < Oj, while candidate / has an incentive to win ^-yj + />>oj. In equilibrium, candidate / announces such a platform, while candidate j announces a slightly more extreme platform than that of / (bye) and so chooses to lose. Figure 2.1 shows the policies to be implemented, given such platforms.

Candidates Having Asymmetric Characteristics

Figure 2.1 Candidates Having Asymmetric Characteristics.

Suppose di < dR. Then, candidate L has an incentive to win :r)

+b > 0), while candidate R does not have it (IPr(:r, zi)+b < 0) when both candidates’ symmetric policies are within the bold area. Candidate L announces a platform such that his/her policy is within the bold area, and R loses.

Note that in this equilibrium, i wins, and j loses with certainty. There does not exist any equilibrium where a candidate’s probability of winning is less than 1 and more than 0. Note also that as the policy implemented by the winner is only slightly (

Equilibrium satisfies У, (г,-,-y) + /> > 0 and Т'Дгу, r,)+/><0, or 'Vi =j)+b > 0 and 'Ey (zy, z,•)+/>< 0. As a result, multiple equilibria exist. Denote zj- as the most extreme platform of /, and =i as the most moderate platform of / among all possible equilibrium platforms. More precisely, zj satisfies 4#y(zy, zj)+6 = 0 where Xi(=i)-xm = Xj[=j)-xm, and rsatisfies 'P/(s, =/)+/> = 0 where Xi{=j)~ Xm = Xj{=j)~ xm if b < Лс(|=, (а,„)-л,„|). If b> А,с(|г((л-т)-л-т|), Zj = r,-(xm), which is the platform committing to implement the median policy. Any platform between 2} and zj can be an equilibrium strategy of the winner i. Figure 2.1 also shows the positions of Xi (zj) and X; (£, )•

 
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