Consider the infinite repeated two-player game where at each stage the play- ers play the non-zero-sum game given by A В A (5,5) (0,7) B (7,0) | (2,2) (a) Consider the following strategies: SA: always play A. SB: always play B. • St: play A on the first stage, thereafter copy what the other player did in the preceding stage. Suppose that the total payoff involves a discount factor 8, and that both players are restricted to the pure strategies sA, 8B, and sr. Determine the set of d for which (sT, sT) is a Nash equilibrium. (b) Use the one stage deviation principle to determine whether the pair (ST, ST) is a subgame perfect Nash equilibrium for some range of val- ues of §.

Consider the game below. How many best responses does Player 1 have to strategy L of Player 2? 1) 0 2)1 3) 2 4)3 Player 1 T M B L 2,0 2, 2 2,3 Player 2 с 2,1 3,2 1, 2 R 3,2 0,1 0,3

Player 1 Player 2 2 X 4 2 y 1 3 X 2 2 b y O 6 3 1 4 2 X 1 5 y O Consider the game tree given above: a. Write out the set of pure strategies for all players. b. Find the Subgame Nash equilibria of this game.

Q5 A and B are sharing a room with a TV and a DVD player. DVD's can be rented for the weekend for a cost of $5. They would each get $4 worth of enjoyment from a weekend movie, so if both of them rent a movie a on a particular weekend they can each get $8 worth of enjoyment at a cost of $5 rental. (a) Model this as a strategic form game with both players having the strategies Rent (R) and No Rent (N), (b) What is the equilibrium if they share the room for one weekend only? (e) Draw the extensive form of the game if they stay together for two weeks, and identify the subgame perfect equilibria. (d) Explain whether or not the outcome of the game will change if they stay together for 10 weeks.

Consider the two-player game depicted below w А 1,5 2,2 0,4 3,11 В 13,0 4,1 2,5 4,2 с 1,3 2,6 5,2 2,0] a. What is Player 2's best response if Player 1 plays B? b. What is Player l's best response if Player 2 plays z? c. What are the pure strategy Nash Equilibria? సా

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This payoff matrix shows a two-player variant of the sushi game, but where B really loves sushi, so her payoffs from getting to eat sushi are higher than A's, and her payoffs from there not being enough sushi are lower than A's. Payoffs are ordered A, B. В A 10, 15 10, 25 20, 15 0, -5 A. Find and state all of the mixed strategy Nash equilibria of the game. If there are multiple equilibria, number them in preparation for answering part "c." B. (From the week 6 lectures) Draw a graph of the expected payoffs for each player. Use these graphs to formulate the decision rule for each player. C. (From the week 6 lectures) Draw a graph of the best replies of the two players. Carefully label each Nash equilibrium.

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2. For the following payoff matrix find all of the mixed strategy Nash equilibria (check for dominated strategies to eliminate). Player 1 X Y Z A 1,3 4,1 2,3 Player 2 B 2,4 3,1 0,0 C 2,3 5,0 4,2

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