Question

1. Assuming all players use pure strategies, descrbe the best response strategy for each player and determine the pure strategy Nash Equilibrium for each of the following games. a) Coordination Game: In such games the two players benefit from coordinating their actions. The "Prisoner's Dlemma" game we studied in class was an example of such coordination game. Here is another example known as "Battle of the Sexes. In this game a husband and wife trying to decide how they should spend Friday evening. The husband prefers to go to a football game, whereas the wife would prefer to go to an opera. Both would however like to spend the evening together rather than apart. However, they cannot communicate with each other (let's say they left their cell phone at home!) Wife Football Game Opera Football Game Husband 2,1 Opera 0,0 Note: The first payoff in each cell indicates the Husband's payoff. 0.0 1.2 b) Zero-Sum Game: In these games the payoff to one player is the loss of the other player. Consider for example how teams score points from penalty kicks in soccer. Team A is making the penalty kick and Team B is defending the goal. If Team A makes the penalty wins the game. On the other hand if Team A does not make the goal with the penalty kick, Team B wins. Team A can kick left or right. Similarly Team B can defend left or right Team B Defend Left Defend Right Kick Left -1,1 Team A Kick Right Note: The first payoff in each cell indicates the Team A's payoff. c) Games of Coexistence: This is known as the "Hawk-Dove game. Such games are used in evolutionary biology to predict the evolutionary path of a given species. Consider two grizzly bears that come across a piece of food. They have to decide whether they will fight each other for that piece of food or if they will share it. Fighting is the hawkish strategy where one will win and one will lose. Sharing is a dovish strategy. But this works only when both are being dovish. If one is dovish while the other is hawkish, the hawk gets all the food while the dove gets none. Hawk Dove Hawk -2.-2 4,0 0,4 2.2 Note: The first payoff in each cell indicates the Row player's payoff. Dove

Answer 1

a Th husband chooses football crame, then best response for life to choose Football game as 100 ehen wife chooses Football name, then best luspaine for Husband is Football game as 270. so (Football Crame, fadetball crame) is a nash equilibrium. - When Husband chooses Opera, wlife will choose Opra as axo, if culife chooses cepura then husband will choose opua too as to So, (opea, Opua) is a rash equilibrium. b) If Team A chooses kick left best rupane for leam B is to kick defend to left as 17-). celhen ream B Defendshf, Team A. Defend left team A will kick Right as 17-1 when heam A kick Right then team b will Defend Right as it. When team B Defend Right team A will kick left as 17-1 So, there is no pure strategy hash equilibrium. bear 22 * Ih bear 1 chooses will beanal Hawk, then bear 2 -2,-2 / 4,0 choose Dove, as @ 10,4 2,2 o7-2. If bear 2 chooses Dave bear 1 will choose bo Hauck as 4 > . so (Hauck, Deane) is a nash equilibrium. When bear 1 chooses cleave then bear 2 will choose Hauck as 472. Bear 2 choose Hank then, bear 1 will choose Dove as o>-2, so (Dou, Hawk) is a pure shakegy rash equem