In this paper, we propose a simple and intuitive model to investigate the efficiency of the two-party election system, especially regarding the nomination process. Each of the two parties has its own candidates, and each of them brings utilities for the people including the supporters and non-supporters. In an election, each party nominates exactly one of its candidates to compete against the other party's. The candidate wins the election with higher odds if he or she brings more utility for all the people. We model such competition as a "two-party election game" such that each party is a player with two or more pure strategies corresponding to its potential candidates, and the payoff of each party is a mixed utility from a selected pair of competing candidates. By looking into the three models, namely, the linear link, Bradley-Terry, and the softmax models, which differ in how to formulate a candidate's winning odds against the competing candidate, we show that the two-party election game may neither have any pure Nash equilibrium nor a bounded price of anarchy. Nevertheless, by considering the conventional "egoism", which states that any candidate benefits his/her party's supporters more than any candidate from the competing party does, we prove that the two-party election game in both the linear link model and the softmax model always has pure Nash equilibria, and furthermore, the price of anarchy is constantly bounded.

中文翻译：

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两党选举游戏有多好？

在本文中，我们提出了一个简单直观的模型来研究两党选举制度的效率，尤其是在提名过程方面。两党各有自己的候选人，各自为包括支持者和非支持者在内的人民带来公用事业。在选举中，每一方都只提名一名候选人与另一方的候选人竞争。如果他或她为所有人带来更多效用，候选人赢得了更高的赔率。我们将这种竞争建模为“两党选举博弈”，这样每一方都是一个具有两个或多个与其潜在候选人相对应的纯策略的参与者，每一方的回报是来自选定的一对竞争候选人的混合效用。通过研究三个模型，即线性链接，Bradley-Terry 和 softmax 模型在如何制定候选人对竞争候选人的获胜几率方面有所不同，我们表明两党选举博弈可能既没有任何纯纳什均衡，也没有无政府状态的有界价格。尽管如此，通过考虑传统的“利己主义”，即任何候选人都比来自竞争党的任何候选人都更有利于他/她的政党的支持者，我们证明了线性链接模型和 softmax 模型中的两党选举博弈总是有纯纳什均衡，而且，无政府状态的代价是不断有界的。我们表明，两党选举博弈可能既没有任何纯纳什均衡，也没有无政府状态的有界价格。尽管如此，通过考虑传统的“利己主义”，即任何候选人都比来自竞争党的任何候选人都更有利于他/她的政党的支持者，我们证明了线性链接模型和 softmax 模型中的两党选举博弈总是有纯纳什均衡，而且，无政府状态的代价是不断有界的。我们表明，两党选举博弈可能既没有任何纯纳什均衡，也没有无政府状态的有界价格。尽管如此，通过考虑传统的“利己主义”，即任何候选人都比来自竞争党的任何候选人都更有利于他/她的政党的支持者，我们证明了线性链接模型和 softmax 模型中的两党选举博弈总是有纯纳什均衡，而且，无政府状态的代价是不断有界的。