# Quiz 11: Game Theory and Asymmetric Information

A game is the strategic interaction between two or more players in the process of economic decision making. Each agent who makes the decision in a game is called the player. Each player has a set of choices to decide upon, these choices are called the strategies. The strategies are tied to certain outcomes called payoffs. The Nash equilibrium of the game occurs when both the player takes their best strategies in response to the strategies of the other players. The prisoner's dilemma is a situation where the players choses a Nash equilibrium that is not associated with the highest payoff of the players. The move that the player choses no matter what the other player do is called the dominant strategy. The move that a player never plays no matter what the other player do is called the dominated strategy. The equilibrium where each player plays their dominant strategies is called the dominant strategy equilibrium. a. If each player chooses their dominant strategy, the game has dominant strategy equilibrium. This can be Nash equilibrium is the dominant strategy equilibrium if the optimum move of each player given the moves of other player is their dominant strategy. This is given by the "prisoner's dilemma game". The table below gives the game between gray and white over the choice to confess or don't confess. . Figure To solve the Nash equilibrium the best response of each player is selected given the moves of the other players. If White chooses to "don't confess", it will be best for Gray to "confess" that will get him 0 year in jail than to "don't confess" which will get him 1 years in jail. On the other hand, if White chooses to "confess", it will be best for Gray to "confess" that will get him 10 years in jail than to "don't confess" which will get him 15 years in jail. Hence, "confess" is the dominant strategy of Gray. This means no matter what strategy White choose, Gray will choose "confess" Similarly, if Gray chooses to "don't confess", it will be best for White to "confess" that will get him 0 year in jail than to "don't confess" which will get him 1 year in jail. On the other hand, if Gray chooses to "confess", it will be best for White to "confess" that will get him 10 years in jail than to "don't confess" which will get him 15 years in jail. Hence, "confess" is the dominant strategy of White. This means no matter what strategy Gray choose, White will choose "confess". Hence, "confess" is the dominant strategy for both players and the Nash equilibrium occurs where both "confess" and serve 10 years in jail. Here it will be best for both to choose "don't confess" as it will give them a much less time in jail. But given the strategy of the opponent they reached an equilibrium which is not associated with the highest payoff of the players. This is the prisoner's dilemma. Here each player plays their dominant strategy that is the best response given the moves of the other player. Therefore, the equilibrium here is the dominant strategy equilibrium as well as the Nash equilibrium. b. Nash equilibrium on the other hand may not be dominant strategy equilibrium. This is given by the game below between white and gray : The table below gives the profits for White and Gray. Figure 2 In this game, if White chooses to "I", it will be best for Gray to choose "A" that will get him 5 units of profit than to "B" which will get him 4. On the other hand, if White chooses to play "II", it will be best for Gray to "A" that will get him I unit of profit than to "B" which will get him 0. Hence, "A" is the dominant strategy of Gray. This means no matter what strategy White choose, Gray will choose "A" Similarly, if Gray chooses to "A", it will be best for White to choose "I" that will get him 2 units of profit than to "II" which will get him 1. On the other hand, if Gray chooses "B", it will be best for White to choose "II" that will get him 5 units of profit than to choose "I" which will get him 4. Hence, there is no dominant strategy of White. This means the white will act according to the action of gray. Now gray will always chose A. Hence the equilibrium of the game occurs at white choosing "I" and gray choosing A. The profits will be [2, 5]. This is Nash equilibrium. As white does not have a dominant strategy, the equilibrium is not dominant strategy equilibrium. c. If a player has a dominant strategy and the other players do not, the player with no dominant strategy can take advantage of the situation to evoke threat and promises to alter the equilibrium of a game. In the game in figure 2, white can threat Gray to declaring to choose II, if gray chooses A. The white can also promise to choose I if gray chooses B. In this case, gray is facing a threat of losing 4 units of profit if white choose II in case gray's strategy A. On the other hand, gray will lose 1 unit of profit by choosing B as white choose I. In this situation gray will chose B and gray will stick to his promise and choose I. The equilibrium strategies of White and Gray are I and B respectively. The threat in this case, proves to be profit maximizing for white. This is a credible threat and it is rational for white to declare a credible threat to maximize the outcome of the game. d. Each player's dominant strategy is one that the player plays no matter what the opponent choses to do. Therefore, each player has only one dominant strategy and there can be only one dominant strategy equilibrium in a game. On the other hand, a game can have more than one Nash equilibrium. This is explained by the game below: Figure 3 In this game, if White chooses "I", it will be best for Gray to choose "B" that will get him 5 units of profit than to "A" which will get him -3(loss). On the other hand, if White chooses "II", it will be best for Gray to choose "A" that will get him 10 units of profit than to "B" which will get him -2. Hence, there is no dominant strategy for Gray. Similarly, if Gray chooses to play "A"; it will be best for White to choose "II" that will get him 5 units of profit than to "I" which will get him -3. On the other hand, if Gray chooses "B", it will be best for White to choose "I" that will get him 10 units of profit than to choose "II" which will get him -2. Hence, there is no dominant strategy of White. This means the white will act according to the action of gray. Now gray does not have a dominant strategy either. Hence, the equilibrium of the game occurs according to simultaneous choice of each player. If white chooses "I"; gray will choose "A". On the other hand, if white chooses "II"; gray will choose "B". Therefore, there will be two equilibria at IA and IIB. Both of them are Nash equilibrium.

A game is the strategic interaction between two or more players in the process of economic decision making. Each agent who makes the decision in a game is called the player. Each player has a set of choices to decide upon, these choices are called the strategies. The strategies are tied to certain outcomes called payoffs. The one shot game is one that is played for only one period. A repeated game is played for more than one period. The table below gives the Prisoner's dilemma game for Gray and White. Figure 1 To solve the Nash equilibrium the best response of each player is selected given the moves of the other players. If White chooses to "don't confess", it will be best for Gray to "confess" that will get him 0 year in jail than to "don't confess" which will get him 1 years in jail. On the other hand, if White chooses to "confess", it will be best for Gray to "confess" that will get him 10 years in jail than to "don't confess" which will get him 15 years in jail. Hence, "confess" is the dominant strategy of Gray. This means no matter what strategy White choose, Gray will choose "confess" Similarly, if Gray chooses to "don't confess", it will be best for White to "confess" that will get him 0 year in jail than to "don't confess" which will get him 1 year in jail. On the other hand, if Gray chooses to "confess", it will be best for White to "confess" that will get him 10 years in jail than to "don't confess" which will get him 15 years in jail. Hence, "confess" is the dominant strategy of White. This means no matter what strategy Gray choose, White will choose "confess". Hence, "confess" is the dominant strategy for both players and the Nash equilibrium occurs where both "confess" and serve 10 years in jail. Here it will be best for both to choose "don't confess" as it will give them a much less time in jail. But given the strategy of the opponent they reached an equilibrium which is not associated with the highest payoff of the players. This is the prisoner's dilemma. The prisoner's dilemma is a one shot game. That is the prisoner's dilemma arises when a game is played for one period. If the game is repeated for more than one period, the players tend to learn from their past mistakes and their rival. There is also some opportunity that the players will engage in some bargaining and arrive at a solution which is mutually beneficial. In that case the game will not remain a prisoner's dilemma. Therefore, the prisoner's dilemma is more of a problem for one shot game.

A game is the strategic interaction between two or more players in the process of economic decision making. Each agent who makes the decision in a game is called the player. Each player has a set of choices to decide upon, these choices are called the strategies. The strategies are tied to certain outcomes called payoffs. The Nash equilibrium of the game occurs when both the player takes their best strategies in response to the strategies of the other players. The prisoner's dilemma is a situation where the players choses a Nash equilibrium that is not associated with the highest payoff of the players. The move that the player choses no matter what the other player do is called the dominant strategy. The move that a player never plays no matter what the other player do is called the dominated strategy. The equilibrium where each player plays their dominant strategies is called the dominant strategy equilibrium. a. In the game the stores will have higher profits if they are both keep their shops closed on Sundays. This is because; if they are both open on Sunday the market demand on Sunday is equally divided between them. This will divide the profit of the stores. The profit will be reduced. b. The table below gives the payoff matrix for white and gray for choices of "open" and "close". Figure c. In this game, if White chooses to "Open", it will be best for Gray to choose "Open" that will get him $20,000 profit than to "close" which will get him $17000. On the other hand, if White chooses to play "close", it will be best for Gray to "open" that will get him $25,000 of profit than to "close" which will get him $21,000. Hence, "Open" is the dominant strategy of Gray. This means no matter what strategy White choose, Gray will choose "Open". Similarly, if Gray chooses to "Open", it will be best for White to choose "Open" that will get him $20,000 of profit than to "close" which will get him $17,000. On the other hand, if Gray chooses "close", it will be best for White to choose "open" that will get him $25,000 of profit than to choose "close" which will get him $21,000. Hence, "open" is dominant strategy of White. This means no matter what strategy gray choose, white will choose "Open". Now, gray will always chose "open". Therefore, both the players have dominant strategies. The equilibrium of the game occurs at white choosing "open" by both white and gray. The profits will be [$20,000, $20,000]. As both have a dominant strategy, the equilibrium is dominant strategy equilibrium. At equilibrium as they both remain open on Sunday; the equilibrium is not joint profit maximizing. The profit will be maximize if they both remain close on Sundays and earn $21,000 of profit each. d. The equilibrium will be stable in the long run, because if any player deviates from this strategy, he will end up losing profit from $20,000 to $17,000. To change the equilibrium to mutually beneficial one, one of the stores can promise to remain close on Sundays if the other remains close as well. This will give a mutual beneficial equilibrium with each store earning $21,000 profit. But there is always an incentive to cheat on the other player to earn higher profit in the next period. After that the equilibrium will again move to the dominant strategy equilibrium. e. In this game, it will be best for both to choose "close" on Sundays as it will give them a much higher profit. But given the dominant strategy they reached an equilibrium which is not associated with the highest payoff of the players. This is the prisoner's dilemma.