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]扩展至三党或更多政党的场景。两个及以上政党参与的竞选活动被视为一个多玩家博弈。每个政党拥有各自的候选人作为纯策略。选民作为投票者构成各政党的支持者群体，每位候选人会为各政党支持者带来效用。每个政党提名其一名候选人与其他政党候选人竞争。假设若候选人为全体选民带来更高效用，则其获胜概率更大。每个政党的收益是其支持者获得的期望效用。若某政党所有候选人为本党支持者带来的效用均高于其他政党的任何候选人，则称该博弈为利己博弈。本研究首先论证：当采用hardmax函数选择获胜者时，选举博弈始终存在纯纳什均衡；但在三党选举博弈中，即使博弈为利己性质，仍存在无纯纳什均衡的博弈实例。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，我们设计了一个固定参数可解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代价上界等于政党数量。研究结果表明：当涉及两个以上政党时，选举结果变得不可预测；此外，随着参与政党数量的增加，社会福利可能因无政府代价上升而恶化。本研究从另一角度解释了为何两党制在民主国家普遍存在。