How do people actually play this game? In one experiment involving $10 prizes and 32 pairs of subjects, only two trustors invested nothing; ive of them invested every thing. Overall, trustors invested about half their initial unds. Trustees varied widely in their choices. Twenty percent retuned nothing and another 27 percent returned only $1. But others paid back large amounts. Among the 30 trustees who invested something, II received more than their investments, 16 received less, and 3 received the same amount. Overall, trustors received about $0.95 in return for every dollar invested.38 These results are fairly typical of experimental trust games. Trustors show trust by investing sizable 37Michael Farrand (cd.). Records of the Federal Convention, Vois. I-III, New Haven, Conn.: Yale University Press, 1966, pp. 578-579. lIJoyce Berg. John Dickhaut, and Kevin McCabe, "Trust, Reciprocity. and Social History," Games and Economic Behavior 10, July 1995, pp. 122-142.
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, management, 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
3. Auction (14 points)
Suppose that a seller auctions one object to two buyers, = 1, 2. The buyers submit bids simultaneously, and the buyer with higher bid receives the object. The loser pays nothing while the winner pays the average of the two bids b + b
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.
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) If an agent is risk averse, her risk premium is ALWAYS positive.
(b) When every player has a (strictly) dominant strategy, the strategy profile that consists of each player’s dominant strategy MUST be a Nash equilibrium. (c) If there are two Nash equilibria in pure-strategy, they can ALWAYS be Pareto