IA168 — Problem set 1
Throughout this problem set, we consider only two-player strategic-form games. Except for Problem 4, we
consider only pure strategies.
Definitions. A strategy profile s ∈ S Pareto-dominates a strategy profile s ∈ S iff ui(s) ≥ ui(s ) for all
players i ∈ N and ui(s) > ui(s ) for some player i ∈ N.
A strategy profile s ∈ S is Pareto-optimal iff it is not Pareto-dominated by any other strategy profile.
Problem 1 [2 points]
Consider a game where Player 1 selects a positive even integer less than 7 and, at the same time, Player 2
selects a positive odd integer less than 7. The payoff of the player with the higher selected number is the absolute
difference of the selected numbers and the payoff of the other player is the sum of the selected numbers. Give
a formal description of this game as a two-player strategic-form game (i.e., according to the definition from the
lectures).
Problem 2 [4 points]
Consider a zero-sum game, 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 3 [4 points]
Find a game with exactly 2 Nash equilibria and exactly 2 Pareto-optimal strategy profiles such that:
a) both of the Nash equilibria are Pareto-optimal;
b) exactly one of the Nash equilibria is Pareto-optimal;
c) neither of the Nash equilibria is Pareto-optimal.
Problem 4 [5 points]
Consider a 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. Justify your reasoning.
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Problem 5 [5 points]
Prove or disprove:
a) IESDS creates no new Nash equilibria in any game with finitely many strategies.
b) IESDS creates no new Nash equilibria in any game.
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