Please help solve Question 1.

EXAMPLE 4.2.3 (Chicken). Two drivers speed headon toward each other and a collision is bound to occur unless one of them chickens out and swerves at the last minute. If both swerve, everything is OK (in this case, they both get a payoff of 1). If one chickens out and swerves, but the other does not, then it is a great success for the player with iron nerves (yielding a payoff of 2) and a great disgrace for the chicken (a penalty of 1). If both players have iron nerves, disaster strikes (and both incur a large penalty M). player II _ Swerve (S) Drive (D) 33 Swerve (S) (1, 1) (1, 2) 3 Drive (D) (2, 1) (M, M) 3:, There are two pure Nash equilibria in this game, (S,D) and (13,3): if one player knows with certainty that the other will drive on (respectively, swerve), that player is better off swerving (respectively, driving on). To determine the mixed equilibria, suppose that player I plays 3 with probabil ity a: and D with probability 1 m. This presents player II with expected payoffs of :1:+(1.a';)o(1),i.e.,2x1ifhe plays 3, and 293+(1z1c)' (M) = (M+2):z:M FIGURE 4.5. The game of Chicken. if he plays D. We seek an equilibrium where player II has positive probability on each of S and D. Thus, 2x-1 = (M +2)x -M; i.e., a=1- M The resulting payoff for player II is 2x - 1 =1 -2/M.1. [10 pts] Consider the game Chicken (Example 4.2.3), modied as follows: there is a probability p (between 0 and 1) such that, when a player Swerves, the move is changed to Drive with probe.- bility 33.1 Write the matrix for the modied game, and show how the effect of increasing the value of M (the loss in the case of both players selecting Drive) changes the equilibrium strategies and expected values from the original version. 2. [10 pts] Consider the game Location-sensitive Pollution (Example 4.3.9). Since it satises the denition of a symmetric game, it must have a symmetric equilibrium strategy. Find such a strat- egy and prove that when all players play it, they are in Nash equilibrium. 3. [10 pts] Find an example of a two-player game that has at least one pure Nash equilibrium, but where it is possible for best-resPonse dynamics to cycle (unlike in a potential game). Argue why your example is correct. 4. [10 pts] Find the symmetric Nash equilibrium of the game of Hawks and Doves, as below. dove hawk dove (1,1) (0,3) hawk (3,0) (21,4) Table I: The payoff matrix for Hawks and Doves 5. [10 pts] Consider the following congestion game with four destinations: A, B, C, and D. - The road between A and B costs 3 for one driver to use and 6 per driver if two drivers use it. - The road between A and D costs 1 for one driver to use and 5 per driver if two drivers use it. - The road between B and C costs 1 for one driver to use and 2 per driver if two drivers use it. - The road between C and D costs 2 for one driver to use and 4 per driver if two drivers use it. All these roads can be taken in either direction. No roads connect A to C or B to D directly. Suppose Alice wants to get fmm A to C and Bob wants to get from B to D. Write this game between Alice and Bob in matrix form and nd all the pure Nash equilibria.2 1This corresponds to the situation from Footloose that we watched in Class, where Kevin Bacon's character tries to swerve, but his shoelace randomly gets stuck, so is strategy is switched to drive. 2Since this is a potential game, we know at least one such equilibrium exists