n sgame, player 1 is powerless; her proposal at he start of he game is irrelevant. Every subgame following player 2's rejecion of a proposal of player 1
is a variant of he ulimatum game n which player 2 moves irst. hus evey suh subgame has a uique subgame perfect equilibrium, n wich player 2 offers noting to player 1, and player 1 accepts all proposals. Using backward induction, player 2' s opimal acion ater any offer ( xv X2) of player 1 with X2 < 1 is rejecion (N). Hence in every subgame perfect equilibrium player 2 obtans all he pie.
This is an introductory course in game theory, which will provide you with mathematical tools for analyzing strategically interdependent situations, i.e., the situations in which your optimal decision depends on what other people will do. In particular, we will study central solution concepts in game theory such as Nash equilibrium, subgame perfect equilibrium, and Bayesian equilibrium. To illustrate the analytical value of these tools, we will cover a variety of applications, e.g., international relations, business competition, auctions, marriage market, and so forth. There is no prerequisite for this course, although some background on microeconomics and familiarity of probabilistic thinking would be helpful.
where J (/ M ) is the number of hours per day spent studying for Japanese (/ Math- ematics). You only care about your GPA. Then, answer the following questions.
(a) What is your optimal allocation of study time?
(b) Suppose T = 10. If you follow this optimal strategy, what will be your GPA?
Strategy and Outcome
Strategy in dynamic game = Complete plan of actions What each player will do in every possible chance of move.
Even if some actions will not be taken in the actual play, players specify all contingent action plan.
Q = K 1 =4
L 1 =8 Then, answer the following questions.
(a) In the short run, the …rm is committed to hire a …xed amount of capital K(+1), and can vary its output Q only by employing an appropriate amount of labor L . Derive the …rm’s short-run total, average, and marginal cost functions. (b) In the long run, the …rm can vary both capital and labor. Derive the …rm’s
(5) Suppose that this game is played finitely many times, say T (≥ 2) times. De- rive the subgame perfect Nash equilibrium of such a finitely repeated game. Assume that payoff of each player is sum of each period payoff.
(6) Now suppose that the game is played infinitely many times: payoff of each player is discounted sum of each period payoff with some discount factor δ ∈ (0, 1). Assume specifically that A = 16, c = 8. Then, derive the condition under which the trigger strategy sustains the joint-profit maximizing prices you derived in (3) (as a subgame perfect Nash equilibrium).
Two neighboring homeowners, 1 and 2, simultaneously choose how many hours to spend maintaining a beautiful lawn (denoted by l 1 and l 2 ). Since the appearance of one’s property depends in part on the beauty of the surrounding neighborhood, homeowner’s benefit is increasing in the hours that neighbor spends on his own lawn. Suppose that 1’s payoff is expressed by
Players can reach Nash equilibrium only by rational reasoning in some games, e.g., Prisoners’ dilemma.
However, rationality alone is often insufficient to lead to NE. (see Battle of the sexes, Chicken game, etc.) A correct belief about players’ future strategies
(a) Show that there is no pure-strategy equilibrium in this game.
(b) Is there any strictly dominated strategy? If yes, describe which strategy is dominated by which strategy. If no, briefly explain the reason.
(c) Derive the mixed-strategy Nash equilibrium.
3. Partial Equilibrium (10 points)
Consider the following partial equilibrium analysis. Let CS(p) and P S(p) be the consumer surplus and producer surplus (for a given market price p), respectively. Show that the competitive price minimizes the total surplus, i.e., CS(p) + P S(p). Why does the equilibrium price minimize rather than maximize the welfare? 4. Exchange Economy (20 points)
C) Now suppose that the rule of the game is modified as follows. If exchange occurs, each individual receives 3 times as much amount as the bill she will have. For example, if individual 1 receives $5 and 2 receives $10 initially and both wish to exchange, then 1 will receive $30 (= $10 x 3) and 2 will receive $15 (= $5 x 3). Nothing happens if they do not exchange. Then, does trade occur in a Bayesian Nash equilibrium? Explain.