Reference no: EM132476970

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.

Max\Min	l	r
T	-1	3
M	4	-1
B	1	1

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.)

Max\Min	l	m	r
T	3	2	x
B	0	4	4x

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.