CO7212 - Game Theory in Computer Science - University of Leicester

Question 1. Solving a zero-sum game

Consider the following 3 × 2 zero-sum game. Player I is the Max player, player II the min player.

Part (a) Draw the payoff diagram showing the payoff to player I for each of player I's pure strategies in response to player II's mixed strategies (1 -q, q)T for 0 ≤ q ≤ 1 (where q is the probability for playing r).

Part (b) Find all Nash equilibria (in pure and mixed strategies) of the game.

Part (c) List the max-min strategies of the Max player, and the min-max strategies of the min player.

Part (d) What is the value of the game?

Question 2. A parameterized zero-sum game

Consider the following 2 3 zero-sum game, where x is a parameter, an arbitrary real number. The payoffs are payoffs to player I. (Note that x appears in two places; the number x is given and cannot be influenced by either player.)

Find, depending on x, all Nash equilibria of this game in pure or mixed strategies, and the corresponding equilibrium payoff for player I. For which x is the game degenerate? [Hint: You will have to make case distinctions for different values of x. Draw goal post diagrams!]

Question 3. Weak domination in zero-sum games

Let G be a zero-sum game, and let a and b be two pure strategies of player II (the min player) such that a weakly dominates b. Let S be a pure strategy of player I (the Max player).

Part (a) Prove that if (S, b) is a Nash equilibrium of G, then (S, a) is also a Nash equilibrium of G.

Part (b) Give an example showing that the statement from (a) does not hold for arbitrary bimatrix games.