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.
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.
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
5. Bayesian Nash Equilibrium (12 points)
There are three different bills, $5, $10, and $20. Two individuals randomly receive one bill each. The (ex ante) probability of an individual receiving each bill is therefore 1/3. Each individual knows only her own bill, and is simultaneously given the option of exchanging her bill for the other individual’s bill. The bills will be exchanged if and only if both individuals wish to do so; otherwise no exchange occurs. That is, each individuals can choose either exchange (E) or not (N), and exchange occurs only when both choose E. We assume that individuals’ objective is to maximize their expected monetary payoff ($).
e z . The prices of the three goods are given by (p, q, 1) and the consumer’s wealth is given by ω.
(a) Formulate the utility maximization problem of this consumer.
(b) Note that this consumer’s preference can be expressed in the form of U (x, y, z) = V (x, y) + z. Derive V (x, y).