Economics 482 Game Theory and Economics - Problem Set 3 : Nash equilibrium Pure Strategies
Economics 482 Spring 2021
Due Tuesday, February 16
Problem Set 3
1. There is a town with 99 residents. The residents must decide individually and simultaneously whether to spend time lobbying the local government for better trash collection. If at least 50 residents take the time to lobby, then the government will respond and improve collection services. If fewer than 50 residents lobby, the government will ignore them and do nothing about collection services. If services are improved, all residents earn a bonus payoff of 10. However, any resident who lobbies suffers a payoff loss of 1 because of the effort and time he expends. Thus, if services are improved, all residents who don’t lobby earn a payoff of 10, and all residents who do lobby earn a payoff of 9. If services do not improve, all residents who don’t lobby earn a payoff of 0, and all residents who do lobby earn a payoff of -1.
1. (a) For any profile of pure strategies, let x−i be the number of players other than player i who lobby. For what values of x−i is it a best response for player i to lobby? For what values of x−i is it a best response for player i not to lobby?
2. (b) Are there any pure strategy Nash equilibria of this game in which more than 50 residents lobby? Explain.
3. (c) Are there any pure strategy Nash equilibria of this game in which exactly 50 residents lobby? Explain.
4. (d) Are there any pure strategy Nash equilibria of this game in which fewer than 50 residents lobby? Explain.
5. (e) Suppose that the game is different: instead of 50 residents, it is now only required that at least one resident lobby in order for trash collection to improve. Find a mixed strategy Nash equilibrium of this game in which all residents use the same mixed strategy, i.e., all residents lobby with the same positive probability. This equilibrium can be characterized as the solution to a single equation, in which the probability that a resident lobbies is a variable. State this equation (it can be solved, but you need not do so).
2. Three candidates, Adams, Buchanan, and Cleveland, are running for political office against each other. Three voters will choose one of the candidates via a majority rule election: each voter votes for exactly one candidate, and if some candidate receives at least two votes, she wins the election. If there is a 3-way tie (i.e., all three candidates receive exactly one vote), then one candidate is chosen to be the winner at random, with the probability that any candidate wins equal to 1/3.
The voters’ preferences are summarized as follows. Voter 1 likes Adams better than Buchanan, and Buchanan better than Cleveland. Voter 2 likes Buchanan the most, then Cleveland, then Adams. Voter 3 likes Cleveland the most, then Adams, then Buchanan. Any given voter prefers a tie-breaking randomization to an outcome in which any but his favorite candidate wins outright (for example, voter 1 prefers a 3-way tie to either Buchanan or Cleveland receiving a majority number of votes, but prefers an outcome in which Adams receives a majority of votes to a 3-way tie).
1. (a) How many outcomes does this game have?
2. (b) Find all pure strategy Nash equilibria of the game and divide them into four categories:
Nash equilibria in which Adams wins, Nash equilibria in which Buchanan wins, Nash equilibria in which Cleveland wins, and Nash equilibria in which there is a 3-way tie.
3. (c) Does any player have a strategy that is weakly dominated by another strategy? Does any player have a weakly dominant strategy? Prove your answer.
4. (d) Suppose voter 3’s preferences are changed so that he still likes Cleveland, then Adams, then Buchanan, but now he likes a 3-way tie less than an outcome in which Adams wins a majority. Voters 1 and 2’s preferences are unchanged. Does this change the set of Nash equilibrium outcomes of the game? If so, say exactly how.
3. Consider the following two-player game in matrix form:
Player 2
L C R
Player U 2,7 1,1 4,2
1 M 1,2 4,6 2,0
D 5,1 2,3 0,4
1. (a) Find all Nash equilibria of the game in pure strategies.
2. (b) Find a Nash equilibrium in which players use mixed strategies that are not pure strategies.
