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. \emph{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：我们将在两党选举竞争[Lin, Lu & Chen 2021]的前期工作扩展到三党或更多党派的情形。两党或多党竞选活动被视为两个或多个参与者的博弈。每个党派拥有自己的候选人作为纯策略。作为选民，民众构成各党派的支持者群体，而候选人会为各党派支持者带来效用。每个党派恰好提名一位候选人与对手竞争。*候选人若为全体民众带来更高效用，则被认为有更高胜选概率。*每个参与者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所有竞争党派候选人，则该博弈是利己性的。本研究首先证明：当胜选函数采用硬最大值函数时，选举博弈始终存在纯纳什均衡；然而在三党选举博弈中，即使博弈是利己性的，也可能存在无纯纳什均衡的实例。其次，我们提出利己性选举博弈存在纯纳什均衡的两个充分条件，并据此提出一个固定参数可解算法来计算利己性选举博弈的纯纳什均衡。最后，令人意外的是，我们证明利己性选举博弈的无政府代价受党派数量的上界约束。研究结果表明：当涉及两个以上党派时选举变得不可预测，且社会福利随参与党派数量增加而恶化（表现为无政府代价可能上升）。该研究从另一角度解释了为何两党制在民主国家广泛存在。