In this chapter we formulate mathematical definitions of a game and an election and give examples. The examples of games will indicate some of the categories we will study and will also serve to introduce the two main devices we will use to represent games: the game tree and the payoff matrix. For elections, we introduce the notions of true preferences and insincere voting. We consider the famous example of the 2000 presidential election in the state of Florida as an illustration of the connection between games and elections.

We begin by thinking a bit about how to define the notion of a game. The term has broad meanings. What properties should we use to limit the scope? We can probably agree that a game should involve an act of competition with roles for skill, ingenuity, and maybe luck. Should a game be fun? Should it be fair? Fun is, of course, a matter of taste. Fairness also seems difficult to ensure. A grandmaster can play a game of Chess against a novice, but surely this this is not a “fair” game. However, allowing a weaker player to have a handicap destroys the intrinsic fairness of a game but may be the only way to create a good match.

We define a game in a manner that is quite inclusive, making no commitment either way as to the questions of fun, fairness, or other such qualities. We focus instead on what a game produces. This is the outcome, by which we mean some particular, well-defined final status such as “win,” “loss,” or “tie.” Whatever the method or means of play, when the dust settles and the game is over, we expect to have a well-defined final result.

Further, we often want to know the score of the game or the winnings for each player. We will refer to these as the payoffs. Precisely, the payoffs are numeric quantities assigned to each player for each particular outcome. We can think of the payoff to a player as either points or a dollar amount.