The Level Sets of Typical Games

Abstract

W hat is a noncooperative game? You probably know the Rock-PaperScissors (RPS) game. Let’s play: ready, set, go! What did you choose? I chose rock. If you chose scissors, then I win, because rock crushes scissors. If you chose paper, then you win, because paper covers rock. If you also chose rock, then it’s a draw. This is an example of a two-player game in which each player has the same three pure strategies: rock, paper, and scissors. In order for this game to have any significance, we ought to define a payoff function. For example, we could say the winner receives $1, which the loser must pay to the winner, so the loser’s payoff is −$1. If it’s a draw, then neither of us receives anything, so the payoffs are both 0. This is an example of what we’ll call a discrete game. A more general notion is a continuous game, also called a mixed game, which we will simply call a game. Rather than just thinking about playing the game once, we think of the game being repeated an arbitrary or possibly infinite number of times. Instead of deciding upon one of rock, paper, or scissors, we decide upon a probability distribution which is a list of three numbers corresponding to the probabilities of drawing rock, paper, or scissors. The sum of these three numbers is 1, because we assume that we must draw something. One example is (1/3,1/3,1/3), which means the probabilities of drawing rock, paper, or scissors are equal to 1/3. If you only want to draw rock, then your probability distribution would be (1,0,0). We can use these to compute our expected payoffs; these are known as expected values in probability theory. The expected payoff is the sum of the probabilities of each possible outcome multiplied with the payoff