  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.
