Pset1 (8)

GTSB Problem Set 1 Moshe Hoffman and Erez Yoeli October 2, 2015 If you are working with a partner, you and your partner may turn in a single copy of the problem set. Please show your work and acknowledge any additional resources consulted. 1 The Coordination Game First consider the payoff matrix in Fig. 1, which represents a special case of the coordination game. L L R R 2 0 1 1 Figure 1: A Special Case of the Coordination Game (a) What is the set of strategies for each player? (b) Find both pure Nash equilibria 12 of this game. Is one equilibrium preferred by both players? Now consider the general version of the coordination game is presented in Fig. 2. (c) Show that ( L, L ) and ( R, R ) are the only pure Nash equilibria of this game. (c) Now suppose c > a , but still assume d > b . What are all the pure Nash of this game? 1 For the definition of a Pure Nash equilibrium, check the Game Theory handout, Section 2. 2 Note that pure Nash equilibrium does not require that the players play the same strategy. When the two players do play the same strategy and it is Nash, this is called a symmetric Nash equilibrium . 1

L L R R a b c d a > c , d > b Figure 2: The Coordination Game 2 Hawk-Dove (a) Two animals are fighting over some resource worth v > 0. Each can be a dove (passive) or a hawk (aggressive). If one is aggressive and the other passive, the aggressive opponent gets the resource. If both are aggressive, both animals incur cost c > 0, and each obtains the resource with equal likelihood, where c > v . If both are passive, each obtains the resource with equal likelihood. Write down the set of strategies. Then, in the form of a payoff matrix, write the payoffs each player gets from each strategy profile. Finally, find all pure Nash equilibria. (b) Imagine that there is an imbalance of power between the two animals, and that, if both play hawk, player 1 wins with probably q such that 0 . 5 < q < 1. Assume both animals still incur the cost c of fighting. Write down the payoff matrix of this new game and find all pure Nash equilibria. Does the prediction of the model depend on the likelihood one animal wins? (c) Now, instead, imagine that the imbalance of power causes the costs of fighting to be unequally distributed between the two animals. For example, one animal might incur a cost of - c 8 and the other animal might incur a cost of - 7 c 8 . Write down the payoff matrix of this new game and find all pure Nash equilibria. Does the prediction of the model change this time? (d) Let’s go back to assuming that animals are equally likely to obtain the object and that they pay the same cost c of fighting (i.e. the assumptions in question a. Suppose that before the players play Hawk-Dove, nature randomly assigns one of the players to arrive at the resource first. Suppose, for now, that the players always know who arrived first. Only then do the players play, simultaneously choosing their strategies, as usual. In this setting, a strategy must specify how players will play if they arrive first and how they will play if they arrive second. There are four possible strategies: play hawk always (we can shorten this to hawk), play dove always 2

still costly: fighting costs both animals c > 0 per unit of time. The payoffs of the game are thus: u 1 ( t 1 , t 2 ) =      - ct 1 if t 1 < t 2 1 2 v - ct 1 if t 1 = t 2 v - ct 2 if t 1 > t 2 u 2 ( t 1 , t 2 ) =      - ct 2 if t 2 < t 1 1 2 v - ct 2 if t 2 = t 1 v - ct 1 if t 2 > t 1 For this problem we will consider that each player can choose a pure strategy, but not a mixed strategy, i.e. players are forbidden to randomize. (a) Let t 1 = 0 and t 2 = v c . Consider what happens when you change the values of t 1 and t 2 according to the above payoffs above (ie., let t 1 be 0 < t 1 < t 2 , t 1 = t 2 , etc.). Is the strategy profile ( t 1 = 0, t 2 = v c ) a Nash Equilibrium? Why? Note that the strategy pair where player 2 devotes 0 and player 1 devotes v c follows the same argument. How does this compare to the Hawk-Dove game we are used to? (b) Using a similar technique, argue that any strategy pair where player 1 devotes t 1 > 0 but t 1 < v c is not a Nash equilibrium. (Note that the same will be true for strategy pairs in which t 2 > 0 but t 2 < v c , by a symmetry argument.) (c) Argue that the strategy pair t 1 = t 2 = 0 is not a Nash equilibrium. Similarly, argue that the strategy pair t 1 = t 2 = v c is not a Nash equilibrium. (d) Are any strategy pairs where player 1 devotes t 1 > v c and player 2 devotes 0 < t 2 < t 1 Nash equilibria? (Note that the same will be true for strategy pairs in which t 2 > v c and 0 < t 1 < t 2 .) (e) What about strategy pairs where t 1 > v c and t 2 = 0? (Note that the same will be true for strategy pairs in which t 2 > v c and t 1 = 0) (f) What’s left? Are there any Nash equilibria other than the two identified in question (a)? If so, how do you interpret the additional equilibria you found? How does the War of Attrition compare to Hawk-Dove? (g) Describe, in general, a method for finding the Nash equilibria of a continuous game. Explain all major steps and justify each. 4 More on the Coordination Game: Mixed Strategies, Risk-Dominance and Mixed Nash (a) Consider the special case of the coordination game presented in Fig. 1. If player 2 is playing L with probability . 1, does player 1 prefer to play L or R ? What about . 9? At what probability 4

is player 1 indifferent between playing L and R ? This number is called the risk-dominance of the coordination game. It will play an important role later in our class. (b) Now consider the general version of the game presented in Fig 2. The parameter p = d - b a - c + d - b is the risk-dominance of the game. If player 2 is playing L with probability < p , does player 1 prefer to play L or R ? What about a probability > p ? Show that player 1 is indifferent between L and R when player 2 plays L with probability p . (c) Based on you answer to b, argue that it is a mixed Nash for both players to play L with probability p , that is, that neither player can benefit by deviating. 3 5 Mixed Nash Equilibria of Hawk-Dove Let’s take another look at the Hawk-Dove game, and find its mixed Nash equilibria. (a) Assume the same problem setup as in question 2, part (a). Describe a mixed strategy. For any mixed strategy, what are the player’s payoffs? (b) Find the mixed Nash equilibrium of this game. (c) Now let’s change the payoffs. Say that the payoff matrix looks as follows: Hawk Dove Hawk a, a b, c Dove c, b d, d ! where a < c and b > d . Find a mixed Nash equilibrium using the same process as above. 6 Population Game: Fisher’s Explanation for 50-50 Sex Ratios Suppose the population size is large and fixed; it doesn’t change from generation to generation. Let f represent the proportion of females in the population, and 1 - f represent the proportion of males. Assume that each offspring must have one male parent and one female parent. Further assume: • The cost of having a male and female offspring is the same • No inbreeding (a) Suppose the proportion of females is . 7. 3 If you a need more of a reminder, check out the Mixed Nash section of the Game Theory Handout or section 3.1 of Osborne’s textbook. 5