Describe the game and find all Nash equilibria in the following situation: Each of two players chooses a non-negative number. In the choice (a1, a2), the payoff of the first player is equal to a1(a2 - a1), and the payoff of the second player is equal to a2(1 – a1 – a2).
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Describe the game and find all Nash equilibria in the following situation: Each of two players chooses a non-negative number. In the choice (a1, a2), the payoff of the first player is equal to a1(a2 - a1), and the payoff of the second player is equal to a2(1 – a1 – a2).
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- Two individuals are bargaining over the distribution of $100 in which payoffs must be in increments of $5. Each player must submit a one-time bid. If the sum of the bids is less than or equal to $100, each player gets the amount of the bid and the game ends. If the sum of the bids is greater than $100, the game ends and the players get nothing. Does this game have a Nash equilibrium? What is the most likely equilibrium strategy profile for this game?Find all of the Nash equilibrium of the following three player game. Player 1 chooses rows (a,b). Player 2 chooses columns (c,d). Player 3 chooses matrices (x.y). Player 3 receives the third listed payoff for cach outcome. 5,5,5 4,4,0 8,8,3 7,7,3 1,3,1 4,2,0 3,2,4 3,1,0 a by A) (b,d.x) and (a,d.y) B) (b.cy) OC) (b,d,x) D) (a,d,x) and (b,c.y) E) (b,c,x) OF) (a,d.x)Find all Nash equilibria in the next game
- 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), 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, 6Martin has a brother and can take a selfish action, which pays him $10 and his brother $0, or an altruistic action, which pays him $6 and his brother $6. The parents love their children equally and decide how to distribute $20 between them. The parents' payment is the minimum of their two children's payments. Assume that the game is simultaneous. Find the Nash equilibrium.Consider two players in the game-theoretic setting below: Player 2 Choice A Choice B Choice 20 |-30 A -20 30 Player 1 Choice |-30 40 B 30 -40 What is the (pure-strategy) Nash equilibrium? There are more than one (pure-strategy) Nash equilibria. O (P1, P2) = (A, B) is the only (pure-strategy) Nash equilibrium. O (P1, P2) = (A, A) is the only (pure-strategy) Nash equilibrium. O There is no (pure-strategy) Nash equilibrium. O (P1, P2) = (B, A) is the only (pure-strategy) Nash equilibrium. (P1, P2) = (B, B) is the only (pure-strategy) Nash equilibrium.
- Find the Nash Equilibria for each of the following 2x2 games (The left payoff value is associated with the strategies (1 or 2) while the right payoff value is associated with the strategies (A or B)): (a) (b) (c) (d) B 1 3,3 2 4,4 5,5 1 2 1 2 1 2 A 2,2 24 A B 2,5 0,3 4,0 -5,2 A 1,0 0,6 B 4,1 5,5 A B -1,-3 -3,2 -4,1 0,0Consider a game in which two players simultaneously pick an integer from 1 through 10. Let s1 denote choice of player 1, and let S2 denote the choice of player 2. If s1 7 82 then player 1's payoff is: Si + 82 2 and player 2's payoff is 10 – ,*. Si + 82 2 If s1 = 82 then each player gets 5. How many pure strategy Nash equilibria does the game have (you will get full points if you are within 2 of the correct answer)? Numerical answerConsider 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 following sequential game. Player 1 plays first, and then Player 2 plays after observing the choice of Player 1. At the bottom of the decision tree, the first number represents the payoff of Player 1, while the second number represents the payoff of Player 2. For player 2, A stands for Accommodate and F stands for Fight. Player 1 Enter Player 2 A F A Stay out Player 2 F (16,30) (-6,18) In the Nash equilibrium of this game, player 2 earns Player 2 player 1 would play (0,40) (0,20) ✓an incentive to threaten F because, if player 1 believed him then so that player 2 would earn ✓. However, this threat isFor a two-player game, the payoff function for player 1 is V1(x1, x2) = x1 + 10x1x2and for player 2 is V2(x1, x2) = x2 + 20x1x2.Player 1’s strategy set is the interval [0,100], and player 2’s strategy set is theinterval [0,50]. Find all Nash equilibria.A and B are competitors in the mobile phone industry. Both A and B have to decide whether to participate or not to participate in a Phone for the Future Trade Fair next month. The matrix payoff below shows the profits (USD million) corresponding to their actions. a) What is the Nash equilibrium of the above game? b) Is the Nash equilibrium Pareto Optima? Explain. c) Suppose B is pessimistic of A's rationality, what is B's strategy? Compare and comment on B's strategy in (a) and (c). A Participate Do not participate B Participate Do not participate 400,1000 200,200 500,500 1000,400