14.122 Problem Set #3 1. Prove the following theorem (known as the Debreu-Fan-Glicksburg Theorem), whichestablishes sufficient conditions for the existence of a pure strategy Nash equilibrium incertain games with continuous action spaces.Theorem 1 Let G be a normal form game. Suppose that the strategy sets Siare nonempty,compact, convex subsets of Euclidean space and that the payoff functions uiare continuousin s and quasiconcave in si. Then G has a pure strategy Nash equilibrium.(Hint: the proof is very similar to that of Nash’s Theorem.)2. Let G be a two player normal form game with strategy spaces A1and A2. A mixedstrategy Nash equilibrium σ∗is said to be totally mixed if σi∗(ai) > 0 for all ai∈ Ai. Showthat if G has more than one totally mixed Nash equilibrium then it must have infinitelymany.Can you give an example of a game which has more than one but finitely mixed strategyNash equilibria (which are not pure strategy NE)?Let G be the game shown below where a drunk student is trying to get from KendallSquare to Harvard Square, but can not recall each time the doors of the subway open howmany stops (if any) have passed.1 4.3. (a) Write out a complete description of the set of nodes, information sets, action,successor and payoff functions, etc. in the game below. (b) Write out the normal representation of the game and find all of the pure strategyNash equilibria. AA ---- -- ---- --:.:.:. . . . . . . . .does this idea not extend to games of imperfect recall in the way in which you might haveexpected? What might the proper generalization be? (Feel free to not think about thisgame much if you think it confuses you and will make you forget the basic point that payoffsare linear in mixing probabilitites.)2 (a) A game is said to have perfect recall if whenever x and x are in the same informationset neither is a predecessor of the other, and whenever x∈ h(x), x is a predecessor of xand the same player i moves at x and x there exists a node ˆx (possibly equal to x) suchthat ˆ x is a predecessor of xand the action takenx is in the same information set as x,ˆat x along the path to x is the same as the action taken at ˆx along the path to x. (SeeFudenberg-Tirole p. 81.) Show formally that the game above does not have perfect recall.(b) How many pure strategies does player 1 have in this game. What is the set of mixed(behavior) strategies available to him?(c) What payoff does player 1 receive from each of his pure strategies? What payoffdoes player 1 receive from mixing and getting off with probability p? What is the optimumchoice of p?(d) I’ve repeated many times in this class that players utility functions are a linearfunction of the probabilities with which they mix over their pure strategies strategies.