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工作[Lin, Lu & Chen 2021]拓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一个两个及以上玩家的博弈。每个政党拥有自己的候选人作为纯策略。作为选民，人们构成了各政党的支持者群体，而每位候选人为每个政党的支持者带来效用。每个政党提名恰好一位候选人与其他政党的候选人竞争。假设候选人若能给全体人民带来更多效用，其获胜概率就更高。每个玩家的收益是其支持者获得的期望效用。若每位候选人为本党支持者带来的效用超过任何竞争党派候选人所带来的，则该博弈被定义为利己博弈。在本工作中，我们首先论证当胜者由硬最大值函数决定时，选举博弈始终存在纯纳什均衡；而在三党选举博弈中，即使博弈是利己的，也存在没有任何纯纳什均衡的实例。其次，我们提出了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，或许令人惊讶的是，我们证明了利己选举博弈的“无政府价格”以政党数量为上界。我们的研究结果表明，当超过两个政党参与时选举变得不可预测，而且，随着参与政党数量的增加，社会福利可能因无政府价格上升而恶化。本工作从另一角度解释了为何两党制在民主国家中普遍存在。