We extend our previous work on two-party election competition [Lin, Lu & Chen 2021] to the setting of three or more parties. An election campaign among two or more parties is viewed as a game of two or more players. Each of them has its own candidates as the pure strategies to play. People, as voters, comprise supporters for each party, and a candidate brings utility for the the supporters of each party. Each player nominates exactly one of its candidates to compete against the other party's. A candidate is assumed to win the election with higher odds if it brings more utility for all the people. The payoff of each player is the expected utility its supporters get. The game is egoistic if every candidate benefits her party's supporters more than any candidate from the competing party does. In this work, we first argue that the election game always has a pure Nash equilibrium when the winner is chosen by the hardmax function, while there exist game instances in the three-party election game such that no pure Nash equilibrium exists even the game is egoistic. Next, we propose two sufficient conditions for the egoistic election game to have a pure Nash equilibrium. Based on these conditions, we propose a fixed-parameter tractable algorithm to compute a pure Nash equilibrium of the egoistic election game. Finally, perhaps surprisingly, we show that the price of anarchy of the egoistic election game is upper bounded by the number of parties. Our findings suggest that the election becomes unpredictable when more than two parties are involved and, moreover, the social welfare deteriorates with the number of participating parties in terms of possibly increasing price of anarchy. This work alternatively explains why the two-party system is prevalent in democratic countries.

翻译：我们将先前关于两党选举竞争的研究[Lin, Lu & Chen 2021]扩展至三党或更多党派的情境。两个及以上政党参与的竞选活动被视作两个及以上玩家的博弈。每个玩家都拥有自己的候选人作为纯策略。作为选民，民众是各党派的支持者，而每位候选人会为各党派支持者带来效用。每个玩家恰好提名一名候选人，与其他党派的候选人展开竞争。若某候选人为全体民众带来更高效用，则其获胜的几率更大。每位玩家的收益是其支持者获得的期望效用。若每位候选人给本党支持者带来的效用高于任何竞争党派候选人，则该博弈是利己的。本文首先论证：当胜选者由hardmax函数决定时，选举博弈始终存在纯纳什均衡；但在三党选举博弈中，即使博弈是利己的，也可能不存在纯纳什均衡。其次，我们提出利己选举博弈存在纯纳什均衡的两个充分条件。基于这些条件，我们设计了一种固定参数可追踪算法来计算利己选举博弈的纯纳什均衡。最后，令人意外的是，我们证明利己选举博弈的无政府状态价格上限为党派数量。研究结果表明：当超过两个政党参与时，选举变得不可预测；更重要的是，随着参与政党数量增加，社会福利会因无政府状态价格可能上升而恶化。本文从替代视角解释了为何两党制在民主国家中普遍存在。