**12) What is a Nash equilibrium? Why do we generally think that Nash equilibria will be likely outcomes of games?**

Nash equilibrium is an concept in game theory where the ideal result of a game is the place where…

Answered: 12) What is a Nash equilibrium? Why do…

Principles of Microeconomics

7th Edition

ISBN:9781305156050

Author:N. Gregory Mankiw

Publisher:N. Gregory Mankiw

Chapter17: Oligopoly

Section: Chapter Questions

Problem 5CQQ

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8) Consider the following game: There are 10 players. Each player is asked to write a number between 0 and 100. The player whose choice is closest to half of the average is the winner. Describe the game and discuss what would you expect to be the average number if the game were played in class with your colleagues. Do you expect them to play the Nash equilibrium strategy? Explain.
1. We can use game theory to model everyday interactions. Consider a game that you're well familiar with: The choice of which side of a hallway to select if someone is coming from the other direction and has not yet made their own selection clear. How many Nash equilibria are in this game? Remember, an outcome is a Nash equilibrium if nobody has a profitable unilateral deviation; if no one has reason to be the only one to change what they're doing. 2. Consider following scenario which we can think of as a game played between two oligopoly firms, Timmer's snow removal & Jack's snow removal services. It costs Timmer $200 per customer per year. It costs Jack's $225 per customer per year. If the firms compete by strategically setting prices, what Nash equilibrium price do we expect? Explain. Remember, an outcome is a Nash equilibrium if nobody has a profitable unilateral deviation; if no one has reason to be the only one to change what they're doing.
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3. For each of the following statements, say whether it is true or false, and explain your answer. (a) "A roll-back equilibrium is not a Nash equilibrium." (b) "If a simultaneous-move game has pure strategy equilibria, the game cannot have mixed strategy equilibria too. (c) "Players who have a strictly dominant pure strategy will never choose to play a mixed strategy."
4. Game theory: An example of the prisoner's dilemma in the real world is when two competitors are battling it out in the marketplace. Often, many sectors of the economy have two main rivals. There can be rivalry such as between du and Etisalat in Telecommunications services, Coca-Cola and Pepsi-Cola in soft drinks. Suppose du plans to cut its price. Etisalat will likely follow suit to retain its market share. This may end up with low profits for both companies. A price drop by either company may thus be construed as defecting since it breaks an implicit agreement to keep prices high and maximize profits. Thus, if du drops its price but Etisalat continues to keep prices high, du is defecting, while Etisalat is cooperating (by sticking to the spirit of the implicit agreement). In this scenario, du may win market share and earn incremental profits by selling more. Assume that the incremental profits that accrue to du and Etisalat are as follows: o If both keep prices high (Cooperate),…
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4: Beth and Dan are playing a game of chess. To simplify things, we'll treat this as a static game. Suppose that at the critical point of the game, Beth will leave her queen hanging. The problem for Dan is that he doesn't know whether she will want him to capture 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 further develop 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 at winning. The payoffs below represent the probability of each player winning given each of the possible outcomes: Dan Сapture 1,0 Botez Gambit 0.5, 0.5 Develop 0.75, 0.25 True Sacrifice Beth 1,0 (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…
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