1- Game Theory Concepts, True-False: For each of the following statements, state whether it is true or false. If it is false, provide a counter-example: (a) In any game, any strategy of any player is a best response to some beliefs this player has about the strategies of his opponents. (b) In any game, any player can have at most one strictly dominant strategy. (c) In any game, any player can have at most one strictly dominated strategy. ((1) Every game is dominance solvable. (e) In every dominance solvable game, every player is better off when players play the rationalizable action prole, than if they play any other action prole. 2- Chicken: (based loosely on \"Rebel without a Cause\"): Jim and Buzz are racing towards each other in their stolen sports cars at lmph. Each of them can either stay course, or chicken out and swerve. If both keep going straight they will both die: utility _12. If Jim keeps going straight and Buzz chickens out Jim will gain the admiration of Judy for a utility of 55 and Buzz looses in social prestige for a total utility of 0, and vice versa. If both chicken out the loss in prestige is not as great: utility 1- Buzz Straight Chicken Jim Straight _I2,_I2'5, O Chicken| 0 , 5 1,1 (a) Does either player have a strictly dominant or strictly dominated strategy? (b) What are the rationalizable strategies for the players? (c) Is this game dominance solvable? (d) Jim believes that Buzz will go straight with probabilityiO and chicken out with probability 1 _P. What is his best response as a function of P? 3- Simpler Beauty Contest: Consider the version of the Beauty Contest game from the class, in which the players are trying to guess the average guess minus one. More precisely, the utilities now are: \"5(3,,s,-)=_|3,_(3_1)|,where5= Sj/#I E