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1. Define Dominant Strategy and Dominant equilibrium in
2. Do we depict the equilibrium of a game in pay-offs or strategies in Game theory?
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- 7.Consider the attached game depicted in Normal Form. Find one Nash equilibrium and verify that it is a Nash equilibrium. Note: 1. Remember to express the Nash equilibrium in the form of a strategy profile. 2. Use the definition of Nash equilibrium to verify that the strategy profile you find is really a Nash equilibrium.8) Consider the following game: There are 10 players. Each player is asked to write a number between 0 and 100. The player whose choice is closest to half of the average is the winner. Describe the game and discuss what would you expect to be the average number if the game were played in class with your colleagues. Do you expect them to play the Nash equilibrium strategy? Explain.
- 7. Which of the following is not true of the “Battle of the Sexes" Game? a. The predicted outcome demonstrates how independent and rational play by players can quickly reach an agreement or a convergence of expectations concerning which strategy will be played by each player in equilibrium. b. The Battle of the Sexes Game can be used to depict a variety of situations where the players' interests do not precisely coincide, except for the fact that they both want to avoid the outcomes that would result from choosing different actions from one another. c. Players do not have a dominant strategy but playing “Boxing" say, is a best response to the other playing "Boxing" even if it doesn't give the player the payoff associated with his or her preferred alternative. d. Played only once, the outcome where each goes to their preferred venue cannot be Nash Equilibrium.In this game, which of the following claim is wrong: X с D с D 2 3 Veuillez choisir une réponse. a. 2 0 0 1 3 1 This game is similar to the prisoner dilemma. b. Players do not always play the same best response. C. A the Nash equilibrium, the gains for each player is smaller than for other outcomes. d. D is a dominant strategy for player 1.2
- 1. The table below shows a prisoner's dilemma in normal form. Players 1 and 2 each choose between D and C. D C D 2,2 1,10 10, 1 5,5 Answer the following questions. Remember to explain your reasoning. (a) If the game is played only once, what is the equilibrium in dominant strategies? (b) If the game is played three times in a row, what action does each player choose in every round? Suppose that the game is repeated indefinitely. The players do not discount the future. How- ever, the game ends with probability p € (0, 1) after each round. Assume that each player uses the following grim-trigger strategy. Each player chooses C in the first round. Each player chooses C in the current round if both players chose C in every previous round. Each player chooses D in the current round if either player chose D in some previous round. (c) If p = 1, is it optimal for each player to use the grim-trigger strategy above given that the other player uses the grim-trigger strategy above? [Hint: compare…Player 1 Up Down [Select] Left 2,4 6,5 Player 2 In the above game, player 1 would play [Select] Right 1,0 4,2 , player 2 would play resulting in a Nash Equilibrium of [Select]4
- 1. Static and Dynamic Game. Consider the following 2-by-2 game: C D 1A (1,10) (1,1) B (2, a) (0,1) (a) erwise, find the pure-strategy Nash equilibria, if any of this game. (There is no need to look for equilibria in mixed strategies) Assume for now that a = 2. Using dominant strategies or oth- (b) Now, consider the dynamic game in which player 1 moves before player 2, and the payoffs remain unchanged. i.: possible strategies for each player? Recall that a strategy profile for a player not at the initial node of the game tree must specify an action for the player at every node. Draw the game tree for this dynamic game. What are the ii "Find the backward-induction solution(s) to this game. Calculate the equilibrium payoffs for each player. Com- pared to the simultaneous game, is there a first-mover advantage or a ii. second-mover advantage? (c) Now, assume that a = 0 in the original static game. i. Find all the Nash equilibria in pure and mixed strategies. Denoting p the probability…3. Determine the Nash Equilibrium for the following normal form of the game. A) MARY B) UP DOWN PLAYER 1 HARLEY UP (10, 10) (4,8) LEFT (3,0) UP MIDDLE (1, 1) DOWN (0,1) DOWN (8,4) (5,5) PLAYER 2 CENTER RIGHT (2,1) (1,1) (4,2) (0,0) (5,0) (0,1)12. What is the subgame-perfect outcome from a finitely repeated game? a. Iterated Elimination Outcome played in each stage b. Backwards Induction Outcome played in each stage Nash Equilibrium played in each stage d. Nash Equilibrium in stage 1 only
