3. Consider the following game in normal form. Player 1 is the "row" player with strate- gies a, b, c, d and Player 2 is the "column" player with strategies w, x, y, z. The game is presented in the following matrix: a b W x y z 3,3 2,1 0,2 2,1 1,1 1,2 1,0 1,4 e 0,0 1,0 3,2 1,1 d 0,0 0,5 0,2 3,1 Find all the Nash equilibria in the game in pure strategies.
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- Consider the following extensive form game between player 1 and player 2. T B (2, 2) L R R (3, 1) (0, 0) (5, 0) (0, 1) (a). Find the normal form representation of this game. (show the bimatrix) (b). Find all pure strategy NE. (c). Which of these equilibria are subgame perfect?3. Consider the following game in normal form. Player 1 is the "row" player with strate- gies a, b, c, d and Player 2 is the "column" player with strategies w, x, y, z. The game is presented in the following matrix: W x y Z 2,1 0,2 2,1 a 3,3 b 1,1 1,2 1,0 1,4 c 0,0 1,0 3,2 1,1 d 0,0 0,5 0,2 3,1 Find all the Nash equilibria in the game in pure strategies.GAME 5 Player B B2 B1 Player A A1 7, 3 5, 10 A2 3, 8 9, 6 In Game 5 above, O there are no Nash equilibria in pure strategies. Player A choosing A1 and Player B choosing B2 is a Nash equilibrium. O Player A choosing A1 and Player B choosing B1 is a Nash equilibrium. O Player A choosing A2 and Player B choosing B1 is a Nash equilibrium.
- Let the following simultaneous game matrix be used L1 and L2 are the strategies of the player Line Cl and C2 are the strategies of the player Column We reason here in pure strategies and in mixed strategies С1 C2 L1(2, 1) (0, 0) L2 (0, 0) (1, 2) Please select one answer: a. In this game, there is no balance in pure strategies and the balance in mixed strategies is (2/3; 1/3) b. In this game, there is an equilibrium in pure strategies and the equilibrium in mixed strategies is (2/3; 1/3) c. In this game, there are two pure strategy equilibria and the mixed strategy equilibrium is (2/3; 2/3) d. In this game, there are two pure strategy equilibria and the mixed strategy equilibrium is (2/3; 1/3) e. In this game, there are two equilibria in pure strategies and no equilibrium in mixed strategies16. Consider a game with two players, each with three strategies. Given the payoff matrix shown below, determine which pair of strategies is a Nash equilibrium. Player One X Y 1 3. X' 3 1 4 Player Two Y 5 0. 3 Z' 4 (a) X, X' 21 1.2. Consider the following game: u 1 r 6,0 0,6 d 3,2 6,0 Are there any Nash equilibria in pure strategies for this game? Find the mixed strategy Nash equilibrium.
- Player 2 E F H A 6, 5 6, 7 9, 6 7,6 В Player 1 C 6, 7 6, 9 8, 5 9, 7 5, 8 5, 6 7,5 7,5 7,9 8, 7 11, 6 5, 6 (1) In the Unique Nash equilibrium of this game, which strategy does Player1 play? And why? (2) In the Unique Nash equilibrium of this game, which strategy does Player2 play? And why? (3) Is this game dominance solvable? And Why? (4) Does this game have at least one inadmissible Nash equilibrium? And Why?4. Consider the following game. There are two players and two possible simultaneous move games A and B. Player 1 first chooses which game is to be played; both player observe which game is played and choose their strategy. The payoff matrices for Game A and B are given by : where k> >0. Game A U D L (10,5) (0,0) R (0,0) (5,10) Game B: U D (a) Write down the extensive form of this game. (b) Assume k = 2. Solve for all subgame perfect Nash equilibria of this game. L R (kk) (0,0) (0,0) (0,0) (c) Can you find a value of k such that there exists a SPNE in which player 1 chooses game B? Can you find a value of k such that all SPNE involve player 1 choosing game B?GAME 5 Player B B1 B2 Player A A1 7,3 | 5, 10 A2 3, 8| 9, 6 In Game 5 above, O Neither player has a dominant strategy. O Player B has a dominant strategy. O Player A has a dominant strategy. O Both players have dominant strategies.
- The following table contains the possible actions and payoffs of players 1 and 2. Player 1 U M D L 2,10 2,5 8,7 Player 2 C 5,2 11,8 8,4 R 5,4 2,9 8,3 This is a simultaneous move game. In a pure strategy Nash equilibrium of this game, Player 1 receives a payoff of ✓and player 2 receives a payoff of If, instead of playing simultaneously, player 2 moves first, then in the Nash equilibrium player 1 receives a payoff of ✓and player 2 receives a payoff of2. Consider the following 2 → 2 normal form games: In both games Player 1 (the row player) chooses either strategy A or strategy B. Player 2 (the column player) chooses simultaneously strategy A or B. The outcomes are defined by the following matrices. The first number in each cell indicates the payoff of Player 1, the second number (after the comma) indicates the payoff of Player 2. A B Game I A B 50X, 30X 10X, 10X 10X, 10X 30X, 50X Game II A A 110X, 30X B 10X, 10X B 10X, 10X 30X, 50X where X > 0 is a payoff scale parameter. (a) For both games: find all Nash equilibria in pure strategies, if any, and find the Nash equilibrium in mixed strategies. How would you describe this strategic interaction in the two games?es Use the following payoff matrix for a simultaneous-move one-shot game to answer the accompanying questions. C 23, 5 27, 5 a. What is player 1's optimal strategy? Player 1 Strategy O Strategy B O Player 1 does not have an optimal strategy. Strategy A b. Determine player T's equilibrium payoff. 21 D 10, 16 4,9 Player 2 E 18, 18