in game theory, when are there player 1 and palyer 2 and the payoff function is as follow (if player 1 get to choose first) (u1, u2). my question is if the order change like player 2 get to choose first, will the order of the payoff change as well like (u2, ul)?
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- a) Find the Nash equilibria in the game (in pure and mixed strategies) and the associated payoffs for the players. b) Now assume that the game is extended in the following way: in the beginning Player 1 can decide whether to opt out (this choice is denoted by O) or whether to play the simultaneous-move game in a) (this choice is denoted by G). If Player 1 opts out (plays O) then both Player 1 and Player 2 get a payoff of 4 each and the game ends. If Player 1 decides to play G, then the simultaneous-move game is played. Find the pure-strategy Nash equilibria in this extended version of the game. (Hint: note that Player 1 now has 4 strategies and write the game up in a 4x2 matrix.) c) Write the game in (b) up in extensive form (a game tree). Identify the subgames of this game.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…Consider the following sequential game. Player 1 plays first, and then Player 2 plays after observing the choice of Player 1 (if necessary). 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. In equilibrium, the payoff of Player 1 is ✓ and the payoff of player 2 is Player 2 L₂ (-1,8) L₁ Player 1 R₂ (100,1) R₁ (1,0)
- 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).In 'the dictator' game, one player (the dictator) chooses how to divide a pot of $10 between herself and another player (the recipient). The recipient does not have an opportunity to reject the proposed distribution. As such, if the dictator only cares about how much money she makes, she should keep all $10 for herself and give the recipient nothing. However, when economists conduct experiments with the dictator game, they find that dictators often offer strictly positive amounts to the recipients. Are dictators behaving irrationally in these experiments? Whether you think they are or not, your response should try to provide an explanation for the behavior.Three politicians are voting on whether to allow themselves a salary increase of$3,500per a year. If they vote in favor of a raise, then they lose some public support, costing them each$1,500 per year. The salary increase passes if two or more politicians vote in favor of it. What is this game’s Nash equilibrium (or equilibria)? Explain. There is no need to draw a payoff matrix.
- The centipede game, first introduced by Robert Rosenthal in 1981, is an extensive form game in which two players take turns choosing either to take a slightly larger share of an increasing pot, or to pass the pot to the other player. In other words, player 1 chooses between D (Down) and A (Across), where D is pocketing the pot and A is passing the pot to the player 2. Similarly, player 2 chooses between A and D. The payoffs are arranged so that if one passes the pot to one's opponent and the opponent takes the pot on the next round, one receives slightly less than if one had taken the pot on this round. A 2 A A 2 A 1 A (3,5) Ꭰ D D D D (1,0) (0,2) (3,1) (2,4) (4,3) 1. Find the subgame perfect Nash Equilibrium using backward induction.Question 1 Consider the following game. Player 1 has 3 actions (Top, middle,Bottom) and player 2 has three actions (Left, Middle, Right). Each player chooses their action simultaneously. The game is played only once. The first element of the payoff vector is player 1’s payoff. Note that one of the payoffs to player 2 has been omitted (denoted by x). 1. Determine the range of values for x such that Player 2 has a strictly dominant strategy.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 only
- Question 1 Consider the following game. Player 1 has 3 actions (Top, middle,Bottom) and player 2 has three actions (Left, Middle, Right). Each player chooses their action simultaneously. The game is played only once. The first element of the payoff vector is player 1’s payoff. Note that one of the payoffs to player 2 has been omitted (denoted by x). A) Suppose that the value of x is such that player 2 has a strictly dominant strategy. Find the solution to the game. What solution concept did you use to solve the game? B) Suppose that the value of x is such the player 2 does NOT have a strictly dominant strategy. Find the solution to the game. What solution concept did you use to solve the game?UNIT 9 CHAPTER 5 In a gambling game, Player A and Player B both have a $1 and a $5 bill. Each player selects one of the bills without the other player knowing the bill selected. Simultaneously they both reveal the bills selected. If the bills do not match, Player A wins Player B's bill. If the bills match, Player B wins Player A's bill. Develop the game theory table for this game. The values should be expressed as the gains (or losses) for Player A. Is there a pure strategy? Why or why not? Determine the optimal strategies and the value of this game. Does the game favor one player over the other? Suppose Player B decides to deviate from the optimal strategy and begins playing each bill 50% of the time. What should Player A do to improve Player A’s winnings? Comment on why it is important to follow an optimal game theory strategy.Suppose that Teresa and Caroline are both in the public eye. They get offers to sell secrets of the other to tabloids. If both keep the secrets, they are both better off than if they get exposed. If only one is exposed, the other person is better off than if no one was exposed. Their payoffs from each option are given in the payoff matrix. Suppose that Caroline and Teresa play the game over four television seasons, where each season is a new game. Consider the scenarios. Remember, a tit‑for‑tat strategy is one where the person starts by cooperating and then plays whatever strategy the other firm played last. Over four seasons, how much will Caroline make if she plays a tit‑for‑tat strategy and Teresa always exposes? $_______ Over four seasons, how much will Caroline make if she and Teresa both always expose? $_________ Does Caroline have a dominant strategy when she and Teresa play for four seasons? No, there is no dominant strategy…
