a. Find the Nash equilibria, if any, in the game above. For each equilibrium, justify why it is a Nash equilibrium in a few sentences. b. Suppose this game was to be played infinitely many times between the two players. Find a condition for 8 such that the Pareto-optimal outcome can be attained. Find this for both T
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- Consider the following coordination game: Player 2P1 Comedy Show Concert Comedy Show 11,5 0,0 Concert 0,0 2,2 a. Find the Nash equilibrium(s) for this game.b. Now assume Player 1 and Player 2 have distributional preferences. Specifically, both people greatly care about the utility of the other person. In fact, they place equal weight on their outcome and the other person’soutcome, ρ = σ = ½. Find the Nash equilibrium(s) with these utilitarianpreferences.c. Now consider the case where Player1 and Player2 do not like each other. Specifically, any positive outcome for the other person is viewed as anegative outcome for the individual, ρ = σ = -1. Find the Nashequilibrium(s) with these envious preferences.on 8.1 Consider the following game: Player 1 A C D 7,6 5,8 0,0 Player 2 E 5,8 7,6 1, 1 F 0,0 1,1 4,4 a. Find the pure-strategy Nash equilibria (if any). b. Find the mixed-strategy Nash equilibrium in which each player randomizes over just the first two actions. c. Compute players' expected payoffs in the equilibria found in parts (a) and (b). d. Draw the extensive form for this game.Please no written by hand Two players bargain over how to split $10. Each player i ∈ {1, 2} choose a number si ∈ [0, 10] (which does not need to be an integer). Each player’s payoff is the money he receives. We consider two allocation rules. In each case, if s1 + s2 ≤ 10, each player gets his chosen amount si and the rest is destroyed. 1. In the first case, if s1 + s2 > 10, both players get zero. What are the (pure strategy) Nash equilibria? 2. In the second case, if s1 + s2 > 10 and s1 6= s2, the player who chose the smallest amount receives this amount and the other gets the rest. If s1 + s2 > 10 and s1 = s2, they both get $5. What are the (pure strategy) Nash equilibria? 3. Now suppose that s1 and s2 must be integers. Does this change the (pure strategy) Nash equilibria in either case?
- Paramter y = 0 If ⟨a, d⟩ is played in the first period and ⟨b, e⟩ is played in the second period, whatis the resulting (repeated game) payoff for the row player?Cameron and Luke are playing a game called ”Race to 10”. Cameron goes first, and the players take turns choosing either 1 or 2. In each turn, they add the new number to a running total. The player who brings the total to exactly 10 wins the game. a) If both Cameron and Luke play optimally, who will win the game? Does the game have a first-mover advantage or a second-mover advantage? b) Suppose the game is modified to ”Race to 11” (i.e, the player who reaches 11 first wins). Who will win the game if both players play their optimal strategies? What if the game is ”Race to 12”? Does the result change? c) Consider the general version of the game called ”Race to n,” where n is a positive integer greater than 0. What are the conditions on n such that the game has a first mover advantage? What are the conditions on n such that the game has a second mover advantage?There are three players who must each choose an “effort” level from 1 to 7, that is, Si = {1, 2, 3, ..., 7}. The payoff for each player i is ui(si, s−i) = 10 max{s1, s2, s3} − si. How many pure- strategy Nash equilibria are there? Select one: a.2 b.4 c.none of the other answers d.3 e.1
- Consider the extensive form game portrayed below. The top number at aterminal node is player 1’s payoff, the middle number is player 2’s payoff,and the bottom number is player 3’s payoff.a. Derive the strategy set for each player. (Note: If you do not want to listall of the strategies, you can provide a general description of a player’sstrategy, give an example, and state how many strategies are in thestrategy set.)b. Derive all subgame perfect Nash equilibria. c. Derive a Nash equilibrium that is not a SPNE, and explain why it isnot a SPNE.a) Find the Nash equilibria in the game (in pure and mixed strategies) and the associated payoffs for the players. b) Now assume that the game is extended in the following way: in the beginning Player 1 can decide whether to opt out (this choice is denoted by O) or whether to play the simultaneous-move game in a) (this choice is denoted by G). If Player 1 opts out (plays O) then both Player 1 and Player 2 get a payoff of 4 each and the game ends. If Player 1 decides to play G, then the simultaneous-move game is played. Find the pure-strategy Nash equilibria in this extended version of the game. (Hint: note that Player 1 now has 4 strategies and write the game up in a 4x2 matrix.) c) Write the game in (b) up in extensive form (a game tree). Identify the subgames of this game.E3 Bayesian Game]. Consider a Bayesian game described by a following payoff matrix. Please solve (show your solution). 1. Enumerate all pure strategies for each player. 2. Suppose that player 1 observes his type ?1 = 3. How does player 1 think of the probability of ?2? 3. Find a (pure strategy) Bayesian Nash equilibrium.
- (a) Find all the Nash Equilibria, if there is any. (no explanation needed for this part (b) Does player 1 (choosing rows) have any dominant action? If yes, which action(s)? Any dominated action(s)? If yes, which ones? Answer the same questions for player 2, too. (c) If player 1 moves first (and player 2 moves next), what would be the sequentially rational equilibrium (draw the game tree and use backward induction)?What if player 2 moves first (and then player 1 moves next)? (d) Looking at your findings in (c), would player 1 want to move first or second or is she indifferent (the order doesn’t matter)?Consider 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 existparameter z= 1 In the unique subgame perfect Nash equilibrium, what is the sum of the payoffs tothe two players?