IA168 — Problem set 1
Problem 1 [5 points]
Consider a zero-sum two-player strategic-form game with pure strategies only, where each player has
exactly four strategies, called A1, B1, C1, D1, and A2, B2, C2, D2, respectively. Define the utility function of this
game so that for both i ∈ {1, 2}, all of the following conditions are satisfied:
• the strategy Ai of player i is strictly dominated;
• the strategy Bi of player i is never-best-response, but not strictly dominated;
• the strategy Ci of player i is not never-best-response;
• (D1, D2) is the only Nash equilibrium of the game.
Problem 2 [7 points]
Consider a two-player strategic-form game with mixed strategies, where each player has exactly two pure
strategies, called A1, B1, and A2, B2, respectively. The utility functions are defined by the following table:
A2 B2
A1 (a, 4) (−a, 2)
B1 (3, 1) (1, 3)
In dependence on the parameter a ∈ R, find all Nash equilibria of this game, and for each of them, decide
whether it is Pareto-optimal.
Problem 3 [8 points]
Prove or disprove the following two propositions: In every strategic-form game with pure strategies only, it
holds that:
a) every rationalizable equilibrium is a Nash equilibrium;
b) every Nash equilibrium is a rationalizable equilibrium.