博弈论试题1

Introduction to Game Theory Problem Set 1

Shanghai University of Finance and Economics

Professor Derek Tai-wei Liu

1. Player A offers player B a gamble. For a price, he would throw a dice and pay player B $6 if it comes up 1. If it comes up 3 or 5, he will pay $3. But if it comes 2, 4 or 6, he pays nothing. At what price would this gamble be fair?

2. a) In the game Matching Pennies, suppose player 1 thinks player 2 plays Heads with probability .7 and Tails with probability .

3. What is the expected utility to player l from playing Heads, and what is his expected utility from playing Tails? What pure strategy should player 1 choose, given these beliefs? Find beliefs player 1 could have about player 2's strategy that would rationalize player 1’s other pure strategy.

b) Think about the steps you used in part (a) to compute player 1's expected utility, given her beliefs about player 2's strategy. Why is this procedure legitimate?

c) A casino offers the following gamble: a fair coin is flipped over and over until it comes up heads. If the coin comes up heads for the first time on flip n (which happens with probability 1/2n), you win $2n and the game ends. What is the expected value of this gamble? How much would you pay to take this gamble? Explain one reason why someone might pay less than the expected value.

d) A bookie offers the following lottery: If the high temperature in Los Angeles on January 28, 2012 is at least 45o F more than the high temperature in Chicago on January 28, you win $1. Otherwise, you win $0. How much would you pay for this gamble? Argue that this is a reasonable assessment of your subjective probability estimate of the event that the high temperature in LA exceeds the high in Chicago by at least 45o F on January 28.

3. Write down the strategic form of Rock, scissors, Paper and find all Nash equilibria. (In the game, each player has three choices: Rock, Scissors, and Paper. The game is zero-sum. Rock beats Scissors, Scissors beat Paper, and 'Paper beats Rock. If both players make the same choice they tie.)

4. For each of the following simultaneous games, identify any dominant

strategies and Nash Equilibrium. In cases where both players have dominant strategies, is the outcome of playing them Pareto Efficient?

Pareto Efficient: A strategy is Pareto Efficient if there is no other strategy in which a player is better off without making other players worse off.

1 Practice:

1. Player A offers player B a gamble. For a price, he would throw a dice and pay player B $6 if it comes up 1. If it comes up 3 or 5, he will pay $3. But if it comes 2, 4 or 6, he pays nothing. At what price would this gamble be fair?

2. For each of the following simultaneous games, identify any dominant strategies and Nash Equilibrium. In cases where both players have dominant strategies, is the outcome of playing them Pareto Efficient? Pareto Efficient: A strategy is Pareto Efficient if there is no other strategy in which a player is better off without making other players worse off.

5. ("Borrowed" from Myerson, 1991; and Moulin, 1983) Chair, Ranking

Member, and Scrub are voting in a committee to choose among three options, A, B, and C. Each player submits a secret vote for one option. If any option gets two or more votes, it is the outcome. Otherwise, if there is a (three-way) tie, Chair invokes her prerogatives and chooses her most preferred option (A) as the outcome. Are there any strictly or weakly dominated strategies? Solve the game using iterated deletion of weakly dominated strategies. Would the Chair be better off if she could commit to choose an option besides her favorite in the event of a tie?

(The table is read as follows: If option A is the outcome, Chair's payoff is 8,

Ranking Member's payoff is 0, and Scrub's payoff is 4. If option B is the outcome, Chair's payoff is 4, Ranking Member's payoff is 8, and Scrub's payoff is 0, etc.)