Refer to Table Left-Right-ZigZag. How many pure strategy Nash equilibria does this game have? Table: Left-Right-ZigZag Player 1 Drive left Driver right Zigzag 1) 1 2) 2 3) 3 4) 4 5) None of the above. Drive left (1,1) (-1,-1) (0,0) Player 2 Driver right (-1,-1) (1,1) (0,0) Zigzag (0,0) (0,0) (0,0)
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- Two players bargain over $20. Player 1 first proposes a split of(n, 20 - n), where n is an integer in {0, 1, ..., 20}. Player 2 can either accept or reject this proposal. If player accepts it, player 1 obtains $n and player 2 obtains $(20 - n). If player 2 rejects it, the money is taken away from them and both players will get $0. Question: Find two subgame perfect Nash equilibria of this game and state clearly each player's equilibrium strategies (recall that in a dynamic game, a player's strategy is a complete-contingent plan). Explain why the strategy profiles form a subgame perfect equilibrium.PLAYER B LEFT RIGHT UP 5 FOR A, 30 FOR B 10 FOR A, 12 FOR B PLAYER A DOWN -2 FOR A, 10 FOR B 8 FOR A, 15 FOR B In the above game, the players are seeking to maximize the number they recieve. They choose at the same time. What is the Nash equillibrium? Player A will choose UP and player B will choose LEFT Player A will UP and player B will choose RIGHT Player A will choose DOWN and player B will choose LEFT Player A will choose DOWN and player B will choose RIGHT Player A will choose LEFT and player B will choose UP Player A will choose LEFT and player B will choose DOWN Player A will choose RIGHT and player B will choose UP Player A will choose RIGHT and player B will choose DOWNSuppose two players play a two-period repeated game, where the stage game is the normal-form game shown below. Is there a subgame perfect Nash equilibrium in which the players select (A, X) in the first period? If so, fully describe such equilibrium. If not, explain why not. Player 1 has choice A, B; Player 2 has choice X, Y, Z. Payoff: (A,X)-(5,7), (A,Y)-(2,4), (A,Z)-(3,8), (B,X)-(1,4), (B,Y)-(3,5), (B,Z)-(1,4)
- 10 1. Consider the game where initially She chooses between "Stay Home" and "Go Out". If She chooses "Stay Home" then She gets 2 and He gets 0. If She chooses "Go Out" then they each simultaneously choose "Movie" or "Concert" where the payoffs are 0,1 or 3 as in the Battle of the Sexes Game. What are the subgame perfect Nash Equilibria of this game ?Consider the following representation of a hockey shootout. The shooter can shoot on their forehand, or deke to their backhand, and the goalie can anticipate either move. The number in each cell in the table below represents the percentage chance that the shooter scores for each pair of pure strategies. Anticipate Forehand Anticipate Backhand Shoot Forehand 20 40 Deke Backhand 40 10 In the mixed strategy Nash equilibrium of this game, what is the percentage chance that the player scores? (ie. An 80% chance should be recorded as 80)Consider the game shown below. In this game, players 1 and 2 must move at the same time without knowledge of the other player’s move. Player 1’s choices are shown in the row headings (A, B, C, D), Player 2’s choices are shown in the column headings (E, F, G). The first payoff is for the row player (Player1) and the second payoff is for the column player (Player 2). Player 2 Player 1 E F G A 2, 7 7, 2 2, 6 B 5, 5 5, 4 8, 4 C 4, 6 8, 4 7, 5 D 1, 6 3, 5 6, 4 Highlight the correct answer: Player 1: Has a dominant strategy to choose A Has a dominant strategy to choose B Has a dominant strategy to choose C Has a dominant strategy to choose D Does not have a dominant strategy Player 2: Has a dominant strategy to choose E Has a dominant strategy to choose F Has a dominant strategy to choose G Does not have a dominant strategy The Nash equilibrium outcome to this game is: A/F B/E B/G C/F C/G There is no pure strategy Nash…
- Consider the game shown below. In this game, players 1 and 2 must move at the same time without knowledge of the other player’s move. Player 1’s choices are shown in the row headings (A, B, C, D), Player 2’s choices are shown in the column headings (E, F, G). The first payoff is for the row player (Player 1) and the second payoff is for the column player (Player 2). Player 2 Player 1 E F G A 2, 4 7, 7 2, 6 B 10, 6 1, 7 12, 4 C 4, 6 8, 8 7, 7 D 1, 6 3, 9 6, 7Consider the game shown below. In this game, players 1 and 2 must move at the same time without knowledge of the other player’s move. Player 1’s choices are shown in the row headings (A, B, C), Player 2’s choices are shown in the column headings (D, E, F). The first payoff is for the row player (Player 1) and the second payoff is for the column player (Player 2). Player 2 Player 1 D E F A 6, 8 4, 7 2, 9 B 2, 3 2, 6 4, 7 C 5, 4 7, 5 3, 6There 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
- 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.Consider the game shown below. In this game, players 1 and 2 must move at the same time without knowledge of the other player’s move. Player 1’s choices are shown in the row headings (A/B), Player 2’s choices are shown in the column headings (C/D). The first payoff is for the row player (Player1) and the second payoff is for the column player (Player 2). Player 2 Player 1 C D A 8, 3 2, 4 B 7, 4 3, 5 Pick the correct answer: Player 1: Has a dominant strategy to choose A Has a dominant strategy to choose B Has a dominant strategy to choose C Has a dominant strategy to choose D Does not have a dominant strategy Player 2: Has a dominant strategy to choose A Has a dominant strategy to choose B Has a dominant strategy to choose C Has a dominant strategy to choose D Does not have a dominant strategy The Nash equilibrium outcome to this game is: A/C A/D B/C B/D There is no pure strategy Nash equilibrium for this gameFor the game in FIGURE 7.6, derive players’ best-reply functions, plot them as shown in Figure 7.5, and find the Nash equilibrium.