2. Please answer question 114.2. I also give exercise 114.1 to help solve 114.2

player chooses both B and S with positive pntbahlllty (so that each of the four outcomes (B, B). (8.5), (S, B), and (5,5) occun with punitive pniblbllity). . EXERCISE 114.1 (Mixed strategy equilibria of HawkDoor) Consider the two-player game with vNM preferences in which the players' preferences over deterministic action profiles are the same as in HawkDove (Exercise 31.2) and their preference- over lotteries satisfy the following two conditions. Each player is indifferent be- tween (1) the outcome (Passive, Passive) and (ii) the lottery that assigns probability 5 to (Aggressive, Aggressive) and probability to the outcome in which she is aggres- sive and the other player is passive; each player is indifferent also between (i) the outcome in which she is passive and the other player is aggressive and (ii) the lottery that assigns probability g; to the outcome (Aggressive, Aggressive) and prob- ability ; to the outcome (Passive, Passive). Find payoffs whose expected values represent these preferences (take each player's payoff to (Aggressive, Aggressive) to be 0 and each player's payoff to the outcome in which she is passive and the other player is aggressive to be 1). Find the mixed strategy Nash equilibrium of the resulting strategic game. Both Matching Pennies and EDS have finitely many mixed strategy Nash equi- libria: the players' best response functions intersect at a finite number of points (one for Matching Pennies, three for 805). One of the games in the next exercise has a continuum of mixed strategy Nash equilibria because segments of the players' best response functions coincide. O EXERCISE 114.2 (Games with mixed strategy equilibria) Find all the mixed strategy Nash equilibria of the strategic games in Figure 114.1. 0 EXERCISE 114.3 (A coordination game) Two people can perform a task if, and only if, they both exert effort. They are both better off if they both exert effort and perform the task than if neither exers effort (and nothing is accomplished); the worst outcome for each person is that she exerts effort and the other person does not (in which case again nothing is accomplished). Specifically, the players' prefer- ences are represented by the expected value of the payoff functions in Figure 115.1, where c is a positive number less than 1 that can be interpreted as the cost of ex- erting effort. Find all the mixed strategy Nash equilibria of this game. How do the equilibria change as 5 increases? Explain the reasons for the changes. . EXERCISE 1144 (Swimming with sharks) You and a friend are spending two days at the beach; you both enjoy swimming. Each of you believes that with probabil- L R L R r 6,0 0,6 1 o,1'o,2 33,2 6,0 3 2,2(o,i Figure 114.1 Two strategic games with vNM preferences. No effort Eim mum 1 ity It the water is infested will surely be attacked. Each of you of a payoff function that lulgm beach, and 1 to a day's worth 0 mer is attacked by sharks on I! will surely be attacked the next at least one of you swims on th that the water is shark-free. If n tains the belief that the probabil the second clay swims if m: + thus receiving an expected payt a strategic game in which you I ming on your first day at the be day, you (and your friend, if Iht in which case you stay out of t she goes swimming) swim und swim on the second clay. Thu! is 7r(c+0) + (1 n)(l +1) action. Find the mixed strategy 7r). Does the existence of a fri swimming on the first day? ( ' sometimes said to face the some 4.3.4 A usellcharacterlzatlon o! The method we have used no i libria of a game involves comtr method is useful in simple gs i now present a characterization able in the study of general go check whether a mixed strategy procedure (described in Section The key point is an observati games: a player '5 expected payo of her expected payoffs to her p pure strategy is the probability strategy. This property holds for