Game Theory (Fall 2005)
By: Yuval Peres
Game theory is a fascinating subject, which sheds light on diverse topics including economics, statistical decision theory, voting and even evolutionary biology. We will first study combinatorial games and zero sum games for which there is a satisfying general theory, and continue with more general games where the key notion is Nash equilibrium, and the problem of deciding between Nash equilibria is a topic of active research. To gain a feeling for the strategies involved, we will try some of the games in class. Additional topics: 1. The notion of Shapley value helps identify the power of an agent when multiple coalitions are possible. 2. We will compare voting systems such as simple majority vote versus the electoral college. 3. We'll consider hide and seek games related to the assigment problem. 4. Well known games like hex or chess change dramatically when instead of alternating moves, a coin is tossed at each turn to decide who moves. 5. We will discuss Arrow's impossibility theorem for a "rational" scheme of jointly deciding between more than two options. We will also explore some of the beautiful mathematical tools that give game theory its power: von Neumann's minimax theorem will be established using supporting hyperplanes for convex sets; we'll prove the existence of Nash equilibria via the Brouwer fixed point theorem, following Nash's original argument.
Link to material: http://www.stat.berkeley.edu/users/peres/155.html