N-Person Nonzero Sum Games and Games with a Continuum of Strategies
The race is not always to the swift nor the battle to the strong, but that’s the way to bet.
—Damon Runyon, More than Somewhat
5.1 The Basics
In previous chapters, the games all assumed that the players each had a finite or countable number of pure strategies they could use. A major generalization of this is to consider games in which the players have many more strategies. For example, in bidding games the amount of money a player could bid for an item tells us that the players can choose any strategy that is a positive real number. The dueling games we considered earlier are discrete versions of games of timing in which the time to act is the strategy. In this chapter, we consider some games of this type with N players and various strategy sets.
If there are N players in a game, we assume that each player has her or his own payoff function depending on her or his choice of strategy and the choices of the other players. Suppose that the strategies must take values in sets Qi, i = 1, …, N and the payoffs are real-valued functions:
Here is a formal definition of a pure Nash equilibrium point, keeping in mind that each player wants to maximize their own payoff.
Definition 5.1.1 A collection of strategies q* = (q1*, …, qn*) Q1 × · · · × QN is a pure Nash equilibrium for the game with payoff functions ...