Consider the following variation to the Rock (R), Paper (P), Scissors (S) game: • Suppose that the Player 1 (row player) has a single type, Normal. • Player 2 (column player) has two types Normal and Simple. • A player of Normal type plays this zero-sum game as we studied in class whereas a player of type Simple always play P. • Player 2 knows whether he is Normal or Simple, but player 1does not. a) Suppose player 2 is of type Normal with probability 1/3 and of type Simple with probability (2/3). Find all pure strategy Bayesian Nash Equilibria. b) Suppose player 2 is of type Normal with probability 2/3 and of type Simple with probability (1/3). Find all pure strategy Bayesian Nash Equilibria.

Consider the following variation to the Rock (R), Paper (P), Scissors (S) game: • Suppose that the Player 1 (row player) has a single type, Normal. • Player 2 (column player) has two types Normal and Simple. • A player of Normal type plays this zero-sum game as we studied in class whereas a player of type Simple always play P. • Player 2 knows whether he is Normal or Simple, but player 1does not. a) Suppose player 2 is of type Normal with probability 1/3 and of type Simple with probability (2/3). Find all pure strategy Bayesian Nash Equilibria. b) Suppose player 2 is of type Normal with probability 2/3 and of type Simple with probability (1/3). Find all pure strategy Bayesian Nash Equilibria.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.7P

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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?
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. a. Develop the game theory table for this game. The values should be expressed as the gains (or losses) for Player A. b. 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? d. 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. с.
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, u1)?
i. 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.
2. 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?
Consider the following game - one card is dealt to player 1 ( the sender) from a standard deck of playing cards. The card may either be red (heart or diamond) or black (spades or clubs). Player 1 observes her card, but player 2 (the receiver) does not - Player 1 decides to Play (P) or Not Play (N). If player 1 chooses not to play, then the game ends and the player receives -1 and player 2 receives 1. - If player 1 chooses to play, then player 2 observes this decision (but not the card) and chooses to Continue (C) or Quit (Q). If player 2 chooses Q, player 1 earns a payoff of 1 and player 2 a payoff of -1 regardless of player 1's card - If player 2 chooses continue, player 1 reveals her card. If the card is red, player 1 receives a payoff of 3 and player 2 a payoff of -3. If the card is black, player 1 receives a payoff of 2 and player 2 a payoff of -1 a. Draw the extensive form game b. Draw the Bayesian form game
Consider 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.
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
Consider the following game : Stag Rabbit Stag 9, 9 0, 8 Rabbit 8, 0 7, 7 The first payoff is that of player 1 and the second that of player 2. a. ) Draw the extensive form of the simultaneous game. Find all the Nash equilibrium. p. Suppose player 1 moves first or we are in a sequential game now. Draw the extensive form in the sequential version. c. What is the subgame perfect Nash Equilibrium (SPNE) in the sequential version? d. ) Explain why it is an SPNE.
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Consider 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.
1) Suppose that Player A can take two actions, either Up or Down. Player A is thinking to choose Up 50 percent of the time, and Down 50 percent of the time. This type of strategy is called a ____ ? 2) Consider a payoff matrix of a game shown below. In each cell, the number on the left is a payoff for Player A and the number on the right is a payoff for Player B. In order for (Down, Right) to be a unique pure strategy Nash equilibrium, a must be (greater, or smaller) than 3 and b must be (greater, or smaller) than 3. refer to image