PS135: Game Theory in the Social SciencesQuizUC Berkeley · Department of Political ScienceSummer 20

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PS135: Game Theory in the Social SciencesQuizUC Berkeley · Department of Political ScienceSummer 2015 · Professor Sean GailmardScores are out of 100 points possible. Each numbered questionhas equal weight toward that total. Each lettered subparthas equal weight in the point value of its question.1. Consider the following game:R1L2A2B(4, 1)(2, 2)MA(0, 0)B(3, 0)(0, 1)(a) Find all the pure strategy Nash equilibria.(b) Find all the pure strategy subgame perfect Nash equilibria.(c) Find all the pure strategy perfect Bayesian equilibria.2. Consider the following game:R1L2A(3, 0)(2, 2)M2BA(0, 1)(0, 1)1B(3, 0)(a) Show that there is no pure strategy perfect Bayesian equilibriumfor this game.(b) Find the mixed strategy perfect Bayesian equilibrium.3. Find all the separating and pooling equilibria of the following signalinggame.(1, 1)(2, 2)t1(2, 0)N2(0, 0)(0, 0)0.50.52(1, 0)t2(0, 1)(1, 1)4. A student, player 1, has to hand in a problem set at the other end ofBerkeley’s campus but needs to rush into a midterm exam. She hastwo options. She can deliver the problem set after the exam (call thisL) and incur a late penalty. Alternatively, she can give the problemset to player 2, a random student who happens to be next to player 1(call this S). Player 2 can either deliver the problem set on time (callthis D) or throw it away in the nearest compost bin (call this T ). Forplayer 1, the payo↵ is 1 if the problem set is delivered on time, 1 ifit is thrown away and 0 if it is delivered late. The payo↵s for player 2are x if he delivers and y if he throws it away.(a) Draw the game tree for this game.(b) What conditions do you have to place on x and y in order forplayer 1 to trust that player 2 will deliver the problem set in anequilibrium?2(c) Now, assume that some proportion of students are “nice guysâ€(N ) for which x = 1 and y = 0, while a proportion 1 p are“jerks†(J) for which x = 0 and y = 1. Modify the game to allowNature to choose what type player 2 is before the game begins.Only player 2 knows his type. Draw the new game tree.(d) Assume p = 3 . What are the pure strategy perfect Bayesian4equilibria of this game?3