Alice and Bob are playing a repeated game in which a certain stage game is played twice in succession. All behaviour in the first period is observed by both players before the second period commences. In the stage game, Alice has 4 pure strategies and Bob has 3 pure strategies. How many pure strategies does Bob have in the repeated game? (If necessary, use a calculator to compute this number.) 1728 531441 12 1594323 67108864 16777216 20736 3. 13
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- FINITE REPETITION Alice and Bob are playing a repeated game in which a certain stage game is played twice in succession. All behaviour in the first period is observed by both players before the second period commences. In the stage game, Alice has 2 pure strategies and Bob has 3 pure strategies. How many pure strategies does Bob have in the repeated game? (If necessary, use a calculator to compute this number.) 6. 36 216 128 64 3 729 2187An electronics retailer wants to send codes for $10 off a pair of headphones to the subscribers of its e-mail list. The coupon code will have two letters followed by three digits. The letters A , B , C , and D and the digits 6 and 9 will not be used. So, there are 22 letters and 8 digits that will be used. Assume that the letters and digits can be repeated. How many such coupon codes can be generated? couponcodesFind the perfect Bayesian equilibria in the following two-person game: B. Chance 1/4 U 3/4 2 L \R (:) (?) (3) (6) () (:) (:) (*) 8
- If Firm 1 chooses to release the console in October with probability of 0.692 or December with a probability of 0.308, then Firm 2 is indifferent between choosing a release date. If Firm 2 released the console in October with probability of 0.50 or December with a probability of 0.50, then Firm 1 is indifferent between choosing a release date Suppose now that instead of choosing the release date at the same time, the firms choose sequentially (but still in advance). Firm A chooses its release date first, then firm B observes that date and chooses its own date. Thepayoffs are otherwise the same as above. Represent the game tree corresponding to this dynamic game.FINITE REPETITION Alice and Bob are playing a repeated game in which a certain stage game is played twice in succession. All behaviour in the first period is observed by both players before the second period commences. In the stage game, Alice has 3 pure strategies and Bob has 2 pure strategies. How many pure strategies does Bob have in the repeated game? (If necessary, use a calculator to compute this number.) 2 6 2187 3 36 64 7 729 128 216 Please do fast ASAP fastn people guess an integer between 1 and 100, and the winner is the player whose guess is closest to the mean of the guesses + 1 (ties broken randomly). Which of the following is an equilibrium: a) All announce 1. b) All announce 50. c) All announce 75. d) All announce 100
- (b) Consider the simultaneous-move game below with two players, 1 and 2. Each player has two pure strategies. If a player plays both strategies with strictly positive probability, we call it a strictly mixed strategy for that player. Show that there is no Nash equilibrium in which both 1 and 2 play a strictly mixed strategy. Player 2 b₁ b₂ Player 1 a₁ 3,0 0,1 a2 2,1 2,1A bunch of cookies should be divided among students by a teacher. Each student writes secretly on a sheet of paper his name and the amount of the cookies (his bid) he wants to get (other students cannot see this number). Then the teacher sorts these bids in ascending order and gives cookies to the students starting from the bid with the smallest amount. If cookies are finished at some point the rest of the students get nothing. If there are several equal bids that cannot be satisfied simultaneously with the current amount of cookies, cookies are equally divided among the students who named these bids. If there are extra cookies, the teacher keeps them. You should assume that cookies are perfectly divisible. Find a Nash Equilibrium in pure strategies. Explain why the set of strategies you propose is indeed a Nash Equilibrium in pure strategies. Explain why there are no other Nash equilibria.a W 3,5 3,4 8,4 0,0 3,3 8,9 y 0,1 5,9 9,8 Describe a strategy for player 1 that dominates x. O (1/3.0. 2/3) 1.0,0) O01.1) to
- (ii) A mixed strategy profile (p, q) is one in which p = (p,P2.... P) is the mixed strategy of player 1, and q- (g1, q2,..q4) is the mixed strategy of player 2. Show that if p, >0 in a Nash equilibrium profile (p*, q*), the player 2 must also play i with strictly positive probability q'; > 0. (State clearly any theorem you use to show this. You are not required to justify the theorem.) %3DPlayer 2 Y1 Y2 Y3 X1 1,4 8,4 7,4 6,2 4,5 Player 1 X2 3,5 3,5 X3 5,3 2,2R (3,2) (0,0) (2,4) (5,4) (0,0) In the extensive form representation of the game between Player 1 and Player 2, Player 1 moves first and chooses L or R. If Player 1 chooses R the game ends, if Player 1 chooses L then Player 1 and 2 play a simultaneous move game. The game has strategy Subgame Perfect Nash Equilibria (SPNE). The maximum payoff Player 2 gets in a SPNE is pure strategy Nash equilibria and pure (Please, enter only numerical answers like: 1, 2, 3, ...)