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. (0) In any game, any player can have at most one strictly dominated strategy. ((1) A player can have at most one strictly dominant strategy. (e) A player can have at most one weakly dominant strategy. 2- Chicken: (based loosely on \"Rebel without a Cause\"): Jim and Buzz are racing towards each other in their stolen sports cars at 100mph. 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 5; 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 Stmz'ght Chicken Jim Straight _I2,_I2 5, 0 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? ((1) Jim believes that Buzz will go straight with probability P 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)|,where3= Sj/#I E (a) Which strategies are strictly dominated in this game, and by which alternative strategies? And so, what is the set of strategies 313,1