4: Beth and Dan are playing a game of chess. To simplify things, we'll treat this as astatic game. Suppose that at the critical point of the game, Beth will leave her queenhanging. The problem for Dan is that he doesn't know whether she will want him tocapture the queen (a true sacrifice) or if she will not want him to capture the queen(a Botez gambit). So, Dan has to choose whether to capture her queen or to furtherdevelop his other pieces. Beth is a strong player, so if Dan chooses the wrong move,she will definitely win. However, if Dan chooses the right move, he has a chance atwinning. The payoffs below represent the probability of each player winning giveneach of the possible outcomes:DanСapture1,0Botez Gambit 0.5, 0.5Develop0.75, 0.25True SacrificeBeth1,0(1) First, provide a definition for a dominant strategy. Does either player have adominant strategy in this game? Are there any pure-strategy Nash equilibriain this game? Explain.(2) Verify that Beth playing True Sacrifice with probability ? and Botez Gambitwith probability , and Dan playing Capture with probability and Developwith probability is a mixed-strategy Nash equilibrium of thisgame.Inyouranswer, be careful to show that this outcome satisfies the definition of Nashequilibrium.

A game is defined as a strategic situation model in which players make strategic decisions implying…

Dan Capture Develop 0.75, 0.25 1,0 True Sacrifice 1,0 Beth Botez Gambit 0.5, 0.5 (1) First, provide a definition for a dominant strategy. Does either player have a dominant strategy in this game? Are there any pure-strategy Nash equilibria in this game? Explain. (2) Verify that Beth playing True Sacrifice with probability and Botez Gambit with probability 3, and Dan playing Capture with probability and Develop with probability is a mixed-strategy Nash equilibrium of this game. In your

Dan Capture Develop 0.75, 0.25 1,0 True Sacrifice 1,0 Beth Botez Gambit 0.5, 0.5 (1) First, provide a definition for a dominant strategy. Does either player have a dominant strategy in this game? Are there any pure-strategy Nash equilibria in this game? Explain. (2) Verify that Beth playing True Sacrifice with probability and Botez Gambit with probability 3, and Dan playing Capture with probability and Develop with probability is a mixed-strategy Nash equilibrium of this game. In your

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.7P

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7. Solving for dominant strategies and the Nash equilibrium Suppose Rajiv and Simone are playing a game in which both must simultaneously choose the action Left or Right. The payoff matrix that follows shows the payoff each person will earn as a function of both of their choices. For example, the lower-right cell shows that if Rajiv chooses Right and Simone chooses Right, Rajiv will receive a payoff of 5 and Simone will receive a payoff of 5.
Exercise 7: The facility location game. Our example is a game in which two firms compete through their choice of locations. Suppose that two firms A and B are each planning to open a store in one of six towns located along six consecutive exits on a highway. We can represent the arrangement of these towns using a six-node graph as in Figure 1 Now, based on leasing agreements, Firm 1 has the option of opening its store in any of towns 1, 3, or 5, while Firm 2 has the option of opening its store in any of towns 2, 4, or 6. These decisions will be executed simultaneously. Once the two stores are opened, customers from the towns will go to the store that is closer to them. So for example, if Firm A open its store in town 3 and Firm B opens its store in town 2, then the store in town 2 will attract customers from 1 and 2, while the store in town 3 will attract customers from 3, 4, 5, and 6. If we assume that the towns contain an equal number of customers, and that payoffs are directly…
12.3 Armed Conflict: Consider the following strategic situation: Two rival armies plan to seize a disputed territory. Each army's general can choose either to attack (A) or to not attack (N). In addition, each army is either strong (S) or weak (W) with equal probability, and the realizations for each army are independent. Furthermore the type of each army is known only to that army's general. An army can capture the territory if either (i) it attacks and its rival does not or (ii) it and its rival attack, but it is strong and the rival is weak. If both attack and are of equal strength then neither captures the territory. As for payoffs, the territory is worth m if captured and each army has a cost of fighting equal to s if it is strong and w if it is weak, where s <w. If an army attacks but its rival does not, no costs are borne by either side. Identify all 12.7 Exercises • 267 the pure-strategy Bayesian Nash equilibria of this game for the following two cases, and briefly describe…
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.
Ma3. The payoff matrix below shows the payoffs for Stefan and Imani in a two strategy game. In the mixed strategy equilibrium, Stefan will play strategy Up with a probability of 1/5 and strategy Down with a probability of 4/5, and Imani will play strategy Left with a probability of 2/5 and strategy Right with a probability of 3/5. What is Stefan's expected payoff in the mixed strategy equilibrium? 10.6 5 4.56 10.4
5 Suppose two players play one of the two normal-form games shown in Figure 1. L U 0,-1 D 2,4 R 2,0 6,0 L U | 4,-1 D 2,-2 R 2,0Now suppose that Player 2 knows which game is being played, but Player 1 does not. Find the pure strategy Bayesian Nash equilibrium of this game.
3) Indicate whether the following statement is TRUE or FALSE and explain your answer: If one player is the leader in a game it means that s/he has the first move in the game. The key thing is that this move can be observed by the follower before making his/her own decision, otherwise it would be a simultaneous move game that cannot be solved by backwards induction.
Theo and Addy are deciding what toys to pick out at the toy store. Depending on what toys they pick, they can play different games together, but they can’t coordinate their choices. They can’t talk to one another at all until after that make their choice. Below is their payout matrix which shows their utility for each choice. All the bold figures are for Theo and all the non bold figures are for Addy. Addy Strategies Theo Strategies Toy Gas Pump Jump Rope Toy food 20 10 10 3 Ball 7 3 9 4 a) If Theo chooses Toy Food, what would be the possible outcomes for Addy? What would be best for Addy? b) If Addy chose a Toy Gas Pump, what are the possible outcomes for Theo? What would be best for Theo? c) Does Addy have a dominant strategy? If yes, what is her strategy? If not how can you tell? d) Does Theo have a dominant strategy? If yes, what is her strategy? If not how…
1. Consider the following simultaneous move game Player 2 C D Player 1 A 4,3 4,-2 B 2,2 3,-1 C 3,0 4,0 d) Suppose the game is now played sequentially where Player 1 chooses first, player 2 observes 1’s choice and then makes his own choice. What are the Nash equilibria of this sequential game?
John and Jane usually vote against each other’s party in the SSC elections resulting to negating or offsetting their votes. If they vote for their party of choice, each of them gains four units of utility (and lose four units of utility from a vote against their party of choice). However, it costs each of them two units of utility for the hassle of actually voting during the SSC elections. A. Diagram a game in which John and Jane choose whether to vote or not to vote.
Consider a two-player, sequential-move game where each player can choose to play right or left. Player 1 moves first. Player 2 observes player 1’s actual move and then decides to move right or left. If player 1 moves right, player 1 receives $0 and player 2 receives $25. If both players move left, player 1 receives –$5 and player 2 receives $10. If player 1 moves left and player 2 moves right, player 1 receives $20 and player 2 receives $20. a. Write this game in extensive form. b. Find the Nash equilibrium outcomes to this game. c. Which of the equilibrium outcomes is most reasonable? Explain.