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Decisions to runThis section analyzes candidates’ decisions to run. There are two potential candidates in the district, and they decide whether to run for office. There are two possible cases: (1) the two potential candidates are symmetric and (2) they are asymmetric (about ideal policies, A, and p). In a twocandidate competition, when one candidate deviates by withdrawing, the remaining candidate who runs can announce his/ her own ideal policy as his/her platform and will implement it after an election because he/she no longer has a rival. Symmetric twocandidate equilibriumSuppose that two potential candidates are symmetric. As in Section 2.3, suppose pj = pj = 1 and A, = Ay = A. Suppose also u"(d) > 0 for any d > 0. Denote and r, as the unique equilibrium pair of platforms when two symmetric candidates run; that is, equation (2.2) holds. For candidate i, the utility when he/she runs is (1/2) к. The utility of / when he/she does not run is м(лу  Ayj. Because condition ^{<2} i ^{h}T/.^{,2)}Ь(к'(^{г}'Т^{х}))^{Лс}(1''*(^{;}))"(кЛ1)^{д:}') +/>Jnl^y(ryjA_{l}J Therefore, both candidates do not deviate by withdrawing if
for /, j = L, R and i Ф j. if (2.6) does not hold, an equilibrium where only one candidate runs exists. In such an equilibrium, one candidate announces his/her ideal policy as a platform and implements it after an election. Note that as the payoff for all players is if no one runs, this candidate never deviates by not running. Therefore, if к is sufficiently low and/or л, лу is sufficiently large, a symmetric two candidate equilibrium exists. Corollary 2.13 Suppose u"{d)> 0 for any d>0. A symmetric twocandidate equilib rium exists if two potential candidates satisfy equation (2.6). Otherwise, one potential candidate runs and wins. Asymmetric twocandidate equilibriumIn general, potential candidates may not be symmetric, and hence, I suppose two asymmetric potential candidates. Suppose candidate j is the loser (with a more extreme ideal policy, lower relative importance of betrayal, or more policy motivation) and candidate / is the winner. Furthermore, suppose / announces z_{h} as introduced in Subsection 2.4.1. For the loser, j, the utility when he/she runs ispj_{U}^_{j} (=j.)_ _{A}j The utility of j when he/she does not run is ДуЩлу  xyV It is always > н(лу  Ayj. Thus, the loser j does not deviate by withdrawing, if
For the winner i, the utility when he/she runs is piu(_{Xi} (=,)*, )  V( =tXi(=i)) + b k The utility of / when he/she does not run is ДЩл,  xjj• Thus, the winner / does not deviate by not running, if
As I have shown, in any case (Corollaries 2.102.12), the winner’s expected utility is higher than or the same as the loser’s expected utility, and hence,Дн(*,• (=})  лу)  V(=t ~ Xt(=t) ) + ^{b}^ ~Pj^{u}[Xi {=t) ~ */) However, if candidates have asymmetric policy motivations, Д < /Т, which means ДиЦл,  xA < ДупЦлу  Ayj. When Ду  Д is sufficiently small and b is high, (2.8) holds if (2.7) holds. This implies that when the loser j does not deviate, the winner / also does not deviate. When Pj  Д is sufficiently large and b is small, (2.7) holds if (2.8) holds. This means that when the winner i does not deviate, the loser j also does not deviate. Note that a sufficiently large Д/  Д means that Д is small. This means that candidate / cannot get sufficient benefits even if he/she is certain to run and win, and hence (2.8) becomes a critical condition in this case. In either case, if the cost of running is sufficiently small, then both asymmetric candidates do not have an incentive to deviate by withdrawing. If (2.7) or (2.8) is not satisfied, an equilibrium exists in which only / or j runs. If (2.7) is satisfied with inequality, but (2.8) is not satisfied, only candidate j runs. Similarly, if (2.8) is satisfied with inequality, but (2.7) is not satisfied, only candidate / runs.^{9} If neither inequality is satisfied, either of the two potential candidates runs.^{10} In either case, one candidate announces his/her ideal policy and implements it. Therefore, if к is sufficiently low and/or лу Xj is sufficiently large, an asymmetric, twocandidate equilibrium exists. Note that candidate /”s platform is the most extreme platform among all possible equilibrium platforms. Thus, if 5} satisfies (2.7) and (2.8), other possible equilibrium platforms also satisfy them. Proposition 2.14 An asymmetric twocandidate equilibrium exists if two potential candidates satisfy equations (2.7) and (2.8). In such an equilibrium, the loser j runs in order to induce the winner i to approach j’s ideal policy even though j loses the election. If j deviates by not running, i will implement his/her ideal policy л,. On the other hand, i will approach the median policy (and hence, j’s ideal policy) more closely if j runs. Therefore, j runs to induce i to approach xj, even though it is certain j will lose. In the models of nonbinding platforms, the winner will implement his/her ideal policy after an election, which means that the loser’s decision to run does not affect the winner’s policy. Thus, the loser does not have any reason to run. In the models of completely binding platforms, as both candidates have an equal probability (50%) of winning, an explicit loser does not exist. Thus, the setting of a partially binding platform is important to derive such strategic behaviors. Application: the Social Democratic Party of JapanIn Japan, the LDP was established in 1955, and it was the party in power right from 1955 to 1993. Sartori (1976) uses it as one example of the predominantparty system: the major party consistently forms the government by winning a majority, even though there is a completely competitive electoral system. The largest opposition party in Japan, the Social Democratic Party of Japan (SDPJ), was also established in 1955. However, in 1960, the conservative faction left the SDPJ and established a new third party. After this event, the SDPJ was unable to win a majority, marking the beginning of the LDP’s predominantparty system. Except in the 1958 election, the SDPJ ran candidates for less than 50% of the legislative seats in all elections, with the number of SDPJ candidates decreasing over time. The SDPJ’s policy position shifted further to the left (socialism) after the conservative faction left, and the SDPJ remained adamant about its stance on policy issues, not wishing to compromise its core beliefs to appeal to the voter base. The SDPJ was not able to form the central government until 1994, but it was able to change or reject bills proposed by the LDP through negotiations or by using their limited veto power, such as boycotts (Ihori and Doi, 1998).^{11} The above strategies and incentives of the SDPJ match the model in this section: The SDPJ gave up trying to gain the majority of seats (i.e., accepted defeat) even though it even though it could have pacified voters by adopting a softer stance (i.e., approached the median policy). This result may have occurred because the SDPJ might have been content with only modifying the LDP's policies to bring them closer to SDPJ’s ideal policies, instead of trying to be the main party of Japan. 
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