a. What is player 1's optimal strategy? O Strategy A O Player 1 does not have an optimal strategy. Strategy B b. Determine player 1's equilibrium payoff.
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- 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, 185) Mixed strategy Nash equilibrium Consider a mixed strategy Nash equilibrium of the following coordination game: Player 2 Player 1 A B a 5.5 6.-2 b -2,6 1,1 a) In the above game, explain in words what condition player 1's probability p of playing strategy A must satisfy to induce player 2 to mix strategies between a and b in equilibrium. b) Solve for the mixed strategy Nash equilibrium where player 1 chooses A with probability P and player 2 chooses a with probability p, for 0 < p, P < 1.9. For the payoff matrix below, consider a sequential version of the game in which Player 2 moves first and then Player 1 moves second. Which of the following is not a correct statement? Player 1 Top Center Bottom Left 1,2 5,1 6,0 Player 2 Middle 7,1 11,2 -4,3 Right 10,4 1,0 8,5 The subgame perfect payoffs coincide with those of a Nash equilibrium b. Depending on Player 2's move, Player 1 will sometimes optimally choose Bottom c. Player 2 will choose Right in the subgame perfect equilibrium d. Player 2 would be willing to pay a positive $ amount to move first rather than second e. Player 1 would be willing to pay a larger $ amount than Player 2 to move first
- QUESTION 9 D 2 C D D, D OD, C C, D C II C (2,2) (4,0) (0,4) (3,3) In the dynamic game represented by the graph above, the numbers inside paranthesis show the payoffs of Player 1 and Player 2, respectively. The Nash equilibrium of this game is for Player 1 to choose O C, C and Player 2 to chooseExercise 6.8. Consider the following extensive-form game with cardinal payoffs: 1 R O player pay 000 2 1 M 3 b 010 O player 3's payoff 1 2 221 2 000 0 0 (a) Find all the pure-strategy Nash equilibria. Which ones are also subgame perfect? (b) [This is a more challenging question] Prove that there is no mixed-strategy Nash equilibrium where Player 1 plays Mwith probability strictly between 0 and 1.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?
- 2. Consider the following simultaneous move stage game. In each cell, player l's payoff is shown first. Player 2 L C R T 3, 1 0 ,0 5 , 0 Player 1 M 2, 1 I, 2 3 , 1 В 1, 0 , 1 4, 4 This game is played twice, without discounting of the second stage payoffs. Both players observe the outcome of the first stage prior to making their second stage choices. Determine whether or not (4,4) can be the first stage payoffs from a pure strategy subgame perfect Nash equilibrium. Explain your answer carefully.Use the following payoff matrix for a simultaneous-move one-shot game to answer the accompanying questions. Player 2 Strategy C D E F Player 1 A 9, 8 14, 14 18, 25 12, 19 B 23, 13 10, 18 14, 26 19, 21 a. What is player 1’s optimal strategy? multiple choice Strategy B Strategy A Player 1 does not have an optimal strategy. b. Determine player 1’s equilibrium payoff. PrevQuestion 5 ave Answer Player 1 has two options: L and R. If Player 1 chooses L, then Player 2 has two options; she can choose A or B. If Player 2 chooses A payoffs are (2; 1) and if she chooses B payoffs are (1; 2). If Player 1 chooses R, then Player 2 has two options; she can choose Y or N. If Player 2 shows Y payoffs are (0; 3) and if she chooses N payoffs are (1; 2). Which of the following statements is true? O A. (L; B,Y ) is a subgame perfect equilibrium of this game O B. (R; B,N ) is a subgame perfect equilibrium of this game O C.(L; A,Y) is a subgame perfect equilibrium of this game O D. (R; B,Y ) is a subgame perfect equilibrium of this game
- Subgames • • • At each node where a player makes a decision, we can think about the "subgame" starting at that node A subgame is the portion of the tree following a particular node How many subgames does our first sequential move example have? • How many subgames does our entry game have? • How many subgames does the game on the right have? • What are its Nash equilibria? Player 1 (1,4) T Player 2 U B (5,2) (3,3) р Player 1 D L 9 (2, 0) Player 2 R (6, 2)5. Consider a simultaneous game in which player A chooses one of two actions (Up or Down), and B chooses one of two actions (Left or Right). The game has the following payoff matrix, where the first payoff in each entry is for A and the second for B.(8 points) B Right Left 3,3 5,1 Down 2,2 4,4 a. Find the Nash equilibrium or equilibria. b. Which player, if any, has a dominant strategy? A UpQuestion 28 Suppose the following game: I Player 1 nice nice Player 2 50, 50 mean 90,20 Which of the following is true? mean 20,90 30, 30 O Both players play "mean" with a 70 % chance, and "nice" with a 30% chance. O There is no Nash Equilibrium in pure strategies, there is only one Nash Equilibrium in mixed strategies. Both players play "nice" with a 70% chance, and "mean" with a 30% chance. There is no Nash Equilibrium in mixed strategies, there is only one Nash Equilibrium in pure strategies.