Problem set 1

Strategic Decision Theory Problem Set 1 Due 5:00 PM on Thursday, September 14th. Please submit a PDF, hand-written or typed, to the Brightspace page. Remark This is to get you started. You can work on the first two problems right away. The third might be easier after the next lecture. You may collaborate with a small study group. If so, please indicate who you collaborated with and write up your answers individually so that it is clear that you understand the solution and it is not simply a copy of someone else's solution . Problem 1 - Dominant Strategies 1. Give an example of a game in which no strategy is strictly dominant, but some strategy is strictly dominated. 2. Recall our definition of "iterative elimination". In that context prove that S k i ̸ = ϕ for k = 1 , 2 , . . . and that the procedure terminates in a finite number of steps. You may confine attention to the simpler pure strategy definition of dominance. 3. Consider a game with a Nash Equilibrium strategy profile (one strategy for each player) s . Now consider Γ ′ , a modified version of Γ, in which a strictly dominated strategy has been deleted for each player. Is s a feasible strategy profile of Γ ′ ? Is it a Nash Equilibrium of Γ ′ ? What about the converse: is any Nash Equilibrium strategy of the game Γ ′ also a Nash Equilibrium of Γ? 4. A strategy s ′ i weakly dominates s ′′ i if s ′ i does at least as well as s ′′ i always, and strictly better sometimes. Write down this casual definition formally. 5. It is well known that the final result of iterative elimination of weakly dominated strategies may depend on the order of the elimination. Please show this by constructing a simple example. (Hint: The simplest example is a 3 × 2 game in which a player with three strategies has two weakly dominated strategies.) Problem 2 - Common Knowledge (Please justify your answer as clearly as you can.) Many centuries ago in a land far far away there was a village of 100 married couples, who were all perfect logicians but had somewhat peculiar social customs. Every evening the men of the village would have a meeting, in a great circle around a campfire, and each would talk about his wife. If when the meeting began a man had any reason to hope that his wife had been always faithful to him, then he would praise her virtue to all of the assembled men. On the other hand, if at any time before the current meeting he had ever gotten proof that his wife had been unfaithful, then he would moan and wail. Furthermore, if a wife had been unfaithful, then she and the lover would immediately inform all of the other men in the village except her husband. All of these traditions were common knowledge among the people of this village. 1

In fact, every wife had been unfaithful to her husband. Thus, every husband knew of every infidelity except for that of his own wife, whom he praised every evening. The situation endured for many years, until a travelling holy man visited the village. After sitting through a session around the campfire and hearing every man praise his wife, the holy man stood up in the center of the circle of husbands and said in a loud voice, "A wife in this village has been unfaithful." After that the meeting adjourned and the holy man departed. What happens the next evening? Does any man moan and wail? If not, does life in this village continue as before for ever after? Problem 3 - Mixed Strategy Equilibrium Let G be a finite normal form game and ( s ∗ 1 , s ∗ 2 , . . . , s ∗ n ) be a pure strategy equilibrium of G . Let G m be the mixed extension of G and p ∗ i be the mixed strategy that assigns probability 1 to the pure strategy s i . Prove that ( p ∗ 1 , p ∗ 2 , . . . , p ∗ n ) is an equilibrium of G m . This implies that our definition of mixed strategy equilibrium generalizes our definition of pure strategy equilibrium and a pure strategy equilibrium of G (if it exists) has a natural analog in the mixed extension. 2