$3000 to Jerry and $2000 to Freddie. If the car is a lemon, then it is worth
$1000 to Jerry and $0 to Freddie. Note that, in both cases, Jerry values the car more than does Freddie, so eficiency requires that the car be traded and the surplus (in each case $1000) be divided between them. But there is incomplete information; Freddie observes nature's choice, whereas Jerry knows only that the car is a peach with probability q. Then the players simultaneously and independently decide whether to trade (T) or not (N) at the market price p. If both elect to trade, then the trade takes place. Otherwise, Freddie keeps the car.
1. Course Description
This is an introductory course in game theory, which will provide you with mathematical tools for analyzing strategic situations ‐ 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. Game theory has been widely recognized as an important analytical tool in such fields as economics, political science, phycology and biology. To illustrate its analytical value, we will cover a variety of applications that include international relations, development, 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.
Consider a consumer problem. Suppose that a choice function x(p; !) satis…es Walras’s law and WA. Then, show that x(p; !) is homogeneous of degree zero. 6. Lagrange’s Method
You have two …nal exams upcoming, Mathematics (M) and Japanese (J), and have to decide how to allocate your time to study each subject. After eating, sleeping, exercising, and maintaining some human contact, you will have T hours each day in which to study for your exams. You have …gured out that your grade point average (G) from your two courses takes the form
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
5. Bayesian Nash Equilibrium (12 points)
There are three different bills, $5, $10, and $20. Two individuals randomly receive one bill each. The (ex ante) probability of an individual receiving each bill is therefore 1/3. Each individual knows only her own bill, and is simultaneously given the option of exchanging her bill for the other individual’s bill. The bills will be exchanged if and only if both individuals wish to do so; otherwise no exchange occurs. That is, each individuals can choose either exchange (E) or not (N), and exchange occurs only when both choose E. We assume that individuals’ objective is to maximize their expected monetary payoff ($).
(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.
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.