Use the following payoff matrix for a simultaneous-move one-shot game to answer the accompanying question. Player 1 Strategy С 6, 14 12, 5 a. What is player 1's optimal strategy? Player 2 D 7, 11 15, 1 Strategy B Strategy A Player 1 does not have an optimal strategy. b. Determine player 1's equilibrium payoff. E 18, 20 7, 25 10, 19 16, 17
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- Consider the following centipede game consisting of two players, Pl and P2. The left/right number each terminal node represents Pl's/P2's payoff, respectively. Then, answer the following questions of [D5] and [M5]-|M8]: P1 G P2 P1 G -(0, 2) D (2, 0) (1, 1) (4, 0) Suppose that P2 chooses G or D randomly. Then, what is the Pl's best response of P1 for the P2's choice? And explain why. We assume that random choice is level-0 in the level-k theory. Then, answer the P2's choice in level-2. (a) D (b) G (c) random choice on (G, D) (d) G with probability 1/3 Answer all the properties of Nash equilibrium and subgame perfect equilibrium which is derives from the backward induction. (a) All subgame perfect equilibria are Nash equilibria in any game. (b) All Nash equilibria are subgame perfect equilibria in any game. (c) There is always a unique Nash equilibrium in any game. (d) There exist pure-strategy Nash equilibria in any game. (e) The Nash equilibrium in prisoners' dilemma game is socially…Use the following payoff matrix for a simultaneous-move one-shot game to answer the accompanying questions. Player 1 Strategy C 15, 7 8, 12 a. What is player 1's optimal strategy? Player 1 does not have an optimal strategy. Strategy A. Strategy B. b. Determine player 1's equilibrium payoff. Player 2 D 10, 11 19, 7 E 19, 15 12, 3 F 18, 20 15, 16Consider the following two-player game.First, player 1 selects a number x≥0. Player 2 observes x. Then, simultaneously andindependently, player 1 selects a number y1 and player 2 selects a number y2, at which pointthe game ends.Player 1’s payoff is: u1(x; y1) = −3y21 + 6y1y2 −13x2 + 8xPlayer 2’s payoff is: u2(y2) = 6y1y2 −6y22 + 12xy2Draw the game tree of this game and identify its Subgame Perfect Nash Equilibrium.
- Keith and Blake play a simultaneous one-shot game given by the following table: Blake Left Right Keith Top Bottom 4.00, -5.00 0.00, -6.00 5.00, 0.00 -2.00, 7.00 As there is no unique pure strategy Nash equilibrium, assume that each player plays each of his choices half of the time. What is the average payoff for Keith? | What is the average payoff for Blake? (Round to two decimals if necessary.) (Round to two decimals if necessary.)Game theory Consider a simultaneous move game with two players. Player 1 has three possible actions (A, B, or C) and Player 2 has two possible actions (D or E.) In the payoff matrix below, each cell contains the payoff for Player 1 followed by the payoff for Player 2. Player 2 7. Player 1 A B C D -3, -3 0, -11 -4, 3 -11, E 0 -7, -7 -12, 0 (a) Identify any dominated strategies in this game. If there are none, state this clearly. (b) Identify any pure strategy Nash Equilibria in this game. If there are none, state this clearlyi. ii. QUESTION ONE A. A Nash equilibrium is a strategy profile such that every player's strategy is the best response to all the other players. It requires that each player makes a best response and that expectations regarding the play of other players are correct. Below is the table showing strategies and payoff for Player 1 and Player 2. PLAYER 1 R1 R2 R3 R4 C1 0,7 5,2 7,0 6,6 C2 2,5 3,3 2,5 2,2 PLAYER 2 C3 7,0 5,2 0,7 4,4 CA 6,6 2,2 4,4 10,4 REQUIRED; Transform the normal form game above into an imperfect extensive game form Find the Nash equilibrium for the game above using iterative deletion of strictly dominated strategies. Find the Nash equilibrium using brute force or cell by cell inspection.
- Consider a sequential game where there are two players, Jake and Sydney. Jake really likes Sydney and is hoping to run in to her at a party this weekend. Sydney can't stand Jake. There are two parties going on this weekend and each player's payoffs are a function of whether they see one another at the party. The payoff matrix is as follows: Sydney Party 1 Party 2 Party 1 6, 18 18, 6 Jake Party 2 24,8 0,24 a) Does this game have a pure strategy Nash Equilibrium? b) What is the mixed strategy Nash Equilibrium? c) Now suppose Sydney decides what party she is going to first. Her roommate is friends with Jake and will call him to tell him which party they go to. Write the extensive form of this game (game tree). d) What is the subgame perfect Nash equilibrium from part c?Consider the following game. There are two payers, Player 1 and Player 2. Player 1 chooses a row (10, 20, or 30), and Player 2 chooses a column (10/20/30). Payoffs are in the cells of the table, with those on the left going to Player 1 and those on the right going to player 2. Suppose that Player 1 chooses his strategy (10, 20 or 30), first, and subsequently, and after observing Player 1’s choice, Player 2 chooses his own strategy (of 10, 20 or 30). Which of the following statements is true regarding this modified game? I. It is a simultaneous move game, because the timing of moves is irrelevant in classifying games.II. It is a sequential move game, because Player 2 observes Player 1’s choice before he chooses his own strategy.III. This modification gives Player 1 a ‘first mover advantage’. A) I and IIB) II and IIIC) I and IIID) I onlyE) II onlyConsider the payoff matrix for a game depicted below. Player 1 selects the row and Player 2 selects the column. Up Down Left 1, -1 -1, 1 Right -1, 1 1, -1 What is (are) the Nash equilibrium (equilibria)? Question 18Answer a. Player 1 plays right; Player 2 plays down b. Player 1 plays left; Player 2 plays down c. Player 1 plays down; Player 2 plays left d. Player 1 plays right; Player 2 plays up e. Player 1 plays up; Player 2 plays left f. There is no Nash equilibrium g. Player 1 plays down; Player 2 plays right h. Player 1 plays up; Player 2 plays right i. Player 1 plays left; Player 2 plays up
- Return to the game between Monica and Nancy in Exercise U10 in Chapter 5. Assume that Monica and Nancy choose their effort levels sequentially instead of simultaneously. Monica commits to her choice of effort level first. On observing this decision, Nancy commits to her own effort level. What is the subgame - perfect equilibrium of the game where the joint profits are 5m + 4n+ mn, the costs of their efforts to Monica and Nancy are m2 and n2, respectively, and Monica commits to an effort level first? Compare the payoffs to Monica and Nancy with those found in Exercise U10 in Chapter 5. Does this game have a first-mover or second - mover advantage? Using the same joint profit function as in part (a), find the subgame - perfect equilibrium for the game where Nancy must commit first to an effort level. U10. Return to the game between Monica and Nancy in Exercise U10 in Chapter 5. Assume that Monica and Nancy choose their effort levels sequentially instead of simultaneously. Monica commits…. Find the Nash equilibrium of the following modified Rock-Paper-Scissors game: • When rock (R) beats scissors (S), the winner’s payoff is 10 and the loser’s payoff is −10. • When paper (P) beats rock, the winner’s payoff is 5 and the loser’s payoff is −5. • When scissors beats paper, the winner’s payoff is 2 and the loser’s payoff is −2. • In case of ties, both players receive 0 payoff. You are suposed to create a system of equations and then solve for them and find 3 probabilities- please show how to do thatConsider a game in which there is $4 to be divided, and the first mover is only permitted to make one of three proposals: (a) $3 for the first mover and $1 for the second mover, (b) $2 for each, or (c) $1 for the first movier and $3 for the second mover. The second mover is shown the proposal and can either accept, in which case it is implemented, or reject and cause each to earn $0. Show this game in extensive (tree) form. Be sure to show the payoffs for each person, with the first mover listed on the left, for each of the six terminal nodes.