# Candidates with symmetric characteristics

This subsection analyzes two candidates who have symmetric cost and disutility functions and whose ideal policies are equidistant from the median policy, xm -xL = xR-xm.

## Platforms

First, the policies candidates choose to implement never overlap, and they also never choose a policy that is more extreme than their own ideal policy.

Lemma 2.5

In equilibrium, the pair of platforms, {zR, zR}, satisfies xl m < Xr (=r) =£ xR. where xL < xm < xR.

Proof: See Appendix 2.A.4.

However, there is a possibility that candidates’ platforms may encroach on the opponent’s side of the policy space (i.e., zR < хгп L), which I do allow for. See Subsection 2.3.3 for more details.

When candidate i wins, the utility of i is -м(|^,(с,)-луЬ -Яс(|г,- Xi{=i)) + b. When opponent j wins, the utility of / is

-м(|^у(гу)-А-,|). In equilibrium, these two utilities must be the same. Proposition 2.6

Suppose u"(d) > 0, for any d > 0. Suppose also that two symmetric candidates choose to run. The pair of platforms, {zL, zR}, is an equilibrium strategy if and only if

for i, j = L, R andi Ф j. Such an equilibrium strategy exists, and is symmetric and unique.

Proof: See Appendix 2.A.5

The main idea of the proof is as follows. When two candidates will tie, if-wfl*,- (r,) - a,-|) - Ac(jz(- - *,■(=,-)|) + b > -u^Xj (=j) - -v,|), each candidate prefers to be certain of winning because his/her utility will be higher than when the opponent wins. If a candidate approaches xm, he/ she is certain of winning. Therefore, the candidate will deviate in this direction. If -u(xi (=,-) - */|)- Acflz,- - Xt{=t)|) + й < ~u(Xj (=jj~ */|)> the candidate would actually prefer the opponent to win. In this case, the candidate deviates away from x,„ and so is certain to lose. I assume b < Ас-(|г,(лш)-л„,|), and hence, Xi(=t) and Xj[=j) should diverge to satisfy equation (2.2).

By contrast, if the disutility function is linear, and xr-xl is quite small, a candidate does not mind if the opponent wins because the opponent’s ideal policy is similar to his/her own ideal policy. Therefore, the candidates may prefer to stay with their ideal policies.

Corollary 2.7

Consider that u’[d) = 0,for all d> 0. Then, if

the candidates choose r, = Xi (-/) = *, in equilibrium. Otherwise, the candidates choose {:i, zR}, which satisfies (2.2).

Proof: See Appendix 2.A.6.

## Comparative statistics: cost of betrayal

Suppose the following assumption.

Assumption 1

c'(d) / c(d) strictly decreases with d, and goes to infinity as d goes to zero.

This assumption means that the relative marginal cost decreases as | =j - x increases. For example, if the function is monomial, this assumption holds and will be satisfied by many polynomial functions. Therefore, this assumption is quite weak.

This subsection shows the comparative statistics of the relative importance of betrayal, Я. To commit to implementing the same policy, a candidate needs to pay a larger cost of betrayal when Я decreases.

Proposition 2.8

Suppose Assumption 1 holds. Suppose also that two symmetric candidates choose to run. Then, the realized cost of betrayal, Яс(|г,- - Xi (-/ )|), decreases as Я increases, given the policy to be implemented. The realized cost of betrayal goes to zero as Я goes to infinity, and the candidates' policies and platforms converge to xm.

Proof: See Appendix 2.A.7.

Note that with complete information, voters can correctly guess the policy a candidate will implement by observing the announced platform. Thus, to win the election, the position of the policy that will be implemented is more important than the position of the platform. This is why 1 investigate the realized cost of betrayal given the policy to be implemented (i.e., the electoral outcome).

When Я increases, a candidate does not want to betray the platform. Therefore, and c(|-/ - 2h'(ri)|) decrease, and the decrease in czi - Xi (-/)|) is faster than the increase in Я. As a result, Яс(|;,- — Xi (—i)|) decreases with Я. When Яс(|г,--£,(-,)|) goes to zero, b>H =i(xm) xm|j as b > 0. From Lemma 2.4, both candidates will implement xm. Therefore, if Я reaches infinity, the two candidates converge to the median policy, as in the case of completely binding platforms. However, when Я < ~, they prefer to diverge. As Я goes to zero, the policy the candidates would choose to implement converges to their respective ideal policies.3 Therefore, completely binding and nonbinding platforms are extreme cases of partially binding platforms.

## Position of the platforms and a probabilistic model

In my model, there is no overlap between polices to be implemented; that is, xm - Xr{:r) >n equilibrium, from Lemma 2.5. However, there is a possibility that the platforms are further from the candidate’s ideal policy than the median policy. In other words, platforms may encroach on the opponent’s policy space; that is, zr < x,„ < zi. This could happen when г/(|лГ/(-v„,)- jc,|) - (.v,,,) - Ay|) + > -Яс*(|а,„ -

Xi(xm)|)- If this equation holds, the candidates have an incentive to compromise more when their platforms are the same as xm. Fortunately, this point should not be a serious problem for the following two reasons.

First, for simplification, my model assumes that candidates know every decision-relevant fact about voter preferences. If candidates are uncertain about voter preferences - that is, a probabilistic voting model is considered - the above situation does not hold in many cases. That candidates have a greater divergence of policies in a probabilistic model is well known (Calvert, 1985). Thus, the platform can enter the candidate’s own side in a probabilistic voting model.

Second, the platform may encroach on the opponent’s policy space. There are two main parties in Japan: the Liberal Democratic Party of Japan (LDP), which supports increased public work to sustain rural areas, and the Democratic Party of Japan (DPJ), which supports economic reforms and the reduction of government debt. In 2001, Prime Minister Junichiro Koizumi, a member of the LDP, promised to implement radical economic reforms that were also suggested by the DPJ, including a reduction in government works and debt. Thus, Koizumi and the LDP promised policies also advocated by the DPJ (Mulgan, 2002, pp. 56-57). Moreover, in the 2007 Upper House election, the LDP and Prime Minister Shinzo Abe promised continued implementation of Koizumi's economic reforms, while the DPJ promised some policies to recover and support rural areas.4 This was a complete reversal of the original stance of the parties. Some media also indicated that Hillary Clinton seemed more conservative than John McCain in the 2008 US presidential preliminary elections. My model can explain both cases in which the platforms encroach or do not encroach on the opponent’s side.

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