In the lecture we have considered the situation of constant discount rates (u;= Six) where 8₁ = 8₂ = 8 and time does not matter much, i.e. 8 → 1. Now consider a situation where time does not matter much for one player. Time is still relevant for the other player. If player 1 is very patient we write 8₁ - 1. For player 2 we still have 8₂ < 1 a fixed number. What is the subgame perfect equilibrium of this game?
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where x1 + x2 = 1 and y1 + y2 = 1
(Topic: Bargaining game with alternating offers)
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- Each of Player 1 and Player 2 chooses an integer from the set {1, 2, ..., K}. If they choose the same integer, P1 gets +1 and P2 gets -1; if they choose different integers, P1 gets -1 and P2 gets +1. (a) Show that it is a NE for each player to choose every integer in {1, 2, ..., K} with equal probability, K1 . (b) Show that there are no NE besides the one you found in (a).Normal Form: Which one of the following descriptions below is CORRECT according to this Normal Form shown? 1.) If Player 2 believes that Player 1 randomly choose H or L with same probability, then Player 2's expected for choosing HC is 2. 2.) If Player 1 has 20% chance to play H and 80% chance to play L, Player 2 has 40% chance to play HD and 60% chance to play LC; then Player 1's expected payoff is 2. 3.) If Player 2 randomly play one of its 4 strategies without any preference, then Player 1's expected payoff for playing L is 2.5. 4.) If Player 2 believes that Player 1 has no probability to play L, then Player 2 would prefer to choose HC or HD.Consider the following 3×3 two player normal form game that is being repeated infinite number of times. The discounting factor for player 1 is δ1 and the discounting factor for player 2 is δ2. left center right up (10 ,40) (32 ,75) (65 ,58) middle (55 ,63) (21 ,45) (23 ,83) down (14 ,76) (16 ,65) (37 ,42) a. Find the total discounted utility for player 2 if player 1 decides to play middle all the time and player 2 decides to play left all the time. b. Now suppose both players are following the strategy of part (a) until player 1 decides to play up in the 6th stage. The the new NE after the 6th stage is (up,right). Find the total discounted utility for player 2 in this case. c. Using the grim trigger strategy, find the minimum value of δ2. Can you find any anomaly in your calculated value of δ2?
- parameter z = 5 List all pure strategies for either player that are weakly dominated.Consider the following game 1\2 Y Z A 10,3 3,9 B 8,5 6,1 Suppose Player 2 holds the following belief about Player 1: θ1 (A,B) = (9/10,1/10) What is the expected payoff from playing ‘Y’ ? What is the expected payoff from playing ‘Z’ ? Based on these beliefs, player 2 should respond by playing _____We have studied the 2-period alternating offer bargaining model with constant costs of delay. In other words, in that model, there were two agents 1 and 2 , each with a constant delay cost c1 and c2 respectively. That is the cost to each player i of each period of delay is ci . Now consider the 3 -period version of the model. In this model there are 3 periods t = 0, 1, 2 . As before the cost of each additional period of delay is ci to player i, i.e., if the period t = 2 proposal z = (z1, z2) is accepted then player i’s payoff is zi − 2ci and if the period t = 2 proposal is rejected, then it is (−2c1, −2c2). Assume c1 > c2. Find a SPE of the game?
- Given the stage game above/below. Suppose that the players play (C,C) in period t = 1; 3; 5; ::: and plays (D,D) in period t = 2; 4; 6; ::. Compute the discounted payoff of each player.What is the secure strategy for player B in the game presented in TableConsider the game of Chicken in which each player has the option to “get out of the way” and “hang tough” with payoffs: Get out of the way Hang tough Get out of the way 2,2 1,3 Hang tough 3,1 00 a. Find all pure strategy Nash equilibria, if they exist b. Let k be the probability that player 1 chooses “hang tough” and u be the probability that player two chooses “hang tough.” Find the mixed stragety Nash equilibria, if they exist
- A new government lottery has been announced. Each person who buys a ticket submits an integer between 0 and 100. The winner is the person whose submission is closest to four-fifths of the average of all submissions. If ties occur, the price will be shared. If Chloe expects other players to select numbers randomly, what number should she choose? Chloe should choose the number (a)_____ if you expect all other players to exhibit the same depth of reasoning as Chloe, what number would you choose? you should choose the number (b)______A strategy is a decision rule that describes the actions a player will take at each decision point. The normal-form game indicates the players in the game, the possible strategies of the players, and the payoffs to the players that will result from alternative strategies. In the game presented in Table Normal-Form Game, does player B have a dominant strategy? What is the secure strategy for player B in the game presented in Table Normal-Form Game?The mixed stratergy nash equalibrium consists of : the probability of firm A selecting October is 0.692 and probability of firm A selecting December is 0.309. The probability of firm B selecting October is 0.5 and probability of firm selecting December is 0.5. In the equilibrium you calculated above, what is the probability that both consoles are released in October? In December? What are the expected payoffs of firm A and of firm B in equilibrium?