AS.440.684 (80) Problem Set #1 Game Theory Summer 2023

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AS.440.684 (80)

Problem Set #1

Game Theory

Summer 2023

Assigned Wed, 5/24

Due Wed, 5/31 (before 4pm)

1. a) Describe in words a static game you observe or play in your life.

b) Explain the players, actions, information sets, outcomes, and payoffs.

c) Represent the game in normal form. Define the set of strategies available to each player using the notation from class.

d) Solve the game for any Nash equilibria, both pure and mixed.

e) True static games may be uncommon. Explain how your game might not be realistic.

2. Consider the normal form game below between two players:

a) Find all dominated strategies for both players. For each, list the dominated strategy, as well as the strategy that dominates it.

b) Find the set of strategies that survive iterated elimination of strictly dominated strategies. List the best response to each.

c) Find all Nash equilibria, both pure and mixed.

3. a) Represent the game rock-paper-scissors in normal form. (See Wikipedia for a description of the game: https://en.wikipedia.org/wiki/Rock_paper_scissors.) Players receive a payoff of 1 if they win, -1 if they lose, and zero if they tie.

b) Calculate player 1’s expected payoff of playing S (Scissors) when it believes its opponent will play each strategy with equal probability.

c) Calculate player 1’s expected payoff of playing Rock or Scissors with equal probability, but never playing paper, when it believes its opponent will play each strategy with equal probability.

d) What is player 1’s best response if it believes its opponent will play Rock and Scissors with equal probability?

d) Calculate the mixed strategy Nash equilibrium for the game. (Hint: define two probabilities for each player, say r₁ and r₂, with the third option being 1-r₁-r₂.)

4. Consider a model of crime where a person decides to commit a crime (C) or not (N). Simultaneously, a policeman decides to engage in a high level of enforcement (H) or low (L). If the person does not commit a crime, they receive a payoff of 0. If they commit a crime they receive a payoff of -1 if caught and 2 if not caught. The policeman will not catch a criminal if they have a low level of enforcement, but they also do not expend any effort, so their payoff is always zero. A policeman exerting a high level of enforcement receives a payoff of 5 if they catch the criminal and -5 if no crime is committed due to the wasted effort.

a) Represent this game in normal form.

b) Solve for all Nash equilibria.

c) Interpret the equlibria. Explain in words the economic significance.

d) Can the police force guarantee zero crime? Explain why or why not.

5.  The nations of Amazonia and Zootopia work together to clean up pollution and maintain Earth’s resources. Both nations want a clean and sustainable environment, but also want to avoid the hard work necessary to achieve this objective. In particular, we know the following information:

• Working hard to protect the environment costs a nation 60 units of happiness.

• If both nations work hard, the environment is very healthy, which gives each nation 120 units of happiness.

• If only one nation works hard, the environment is somewhat healthy, which gives each nation 70 units of happiness.

• If neither nation works hard, the environment suffers greatly, which leaves each nation only with 50 units of happiness.

a) Draw the 2x2 payoff matrix for this game.

b) Does either nation have a dominant strategy? Explain. What is the Nash equilibrium for this game?

c) Is there an outcome that would be better than the Nash equilibrium for both nations? If there is, what barriers prevent this outcome from being chosen? If there isn’t a better outcome, justify how you know this to be true.