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ECON 401

# ECON 401 - CS 573 Algorithmic Game Theory Instructor...

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CS 573: Algorithmic Game Theory Lecture date: March 14, 2008 Instructor: Chandra Chekuri Scribe: James Lai Contents 1 Bayesian Nash Equilibrium 1 2 Mechanisms Without Money 2 2.1 Single-Peaked Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Motivation for Single-peaked preferences . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.3 House exchange/ Kidney allocation problem . . . . . . . . . . . . . . . . . . . . . . . 3 2.4 Stable Matchings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1 Bayesian Nash Equilibrium Our previous definition of mechanisms is suitable if there exist dominant strategy equilibriums. However, this may be too strong a requirement. For example, there is no dominant strategy equilibrium for a fixed price auction. Also, economists favor the Bayesian approach to knowledge in which instead of the strict private value model, they assume that there is prior distributional knowledge about other people’s valuations. Definition 1.1 (Bayesian Game) A Bayesian game on a set of N players has the following: every i N has a typespace T i every i N has a set of possible actions X i every i N has a probability distribution D i on T i every i N has a utility function u i : T 1 × X 1 × X 2 × ... × X n R We assume that All players know D 1 , ..., D n The type t i of i is the outcome drawn from D i independently of other players Players are risk neutral expectation maximizers Definition 1.2 (Bayesian Nash Equilibrium) A strategy for i is a function s i : T i X i . A profile of strategies ( s 1 , ..., s n ) is a Bayesian Nash Equilibrium if and only if for all i , for all t 1 , ..., t n , and for all x i X i E D - i [ u i ( t i , s i ( t i ) , s i ( t i )] E D - i [ u i ( t i , x i , s i ( t i )] Note that the expectation is over the random choices of t i from D i

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We can define mechanisms as before except now the D i is public information. A mechanism implements a social choice function f if for all t 1 , ..., t n , there is a Bayesian Nash equilibrium ( s 1 , ..., s n ) such that a ( s 1 ( t 1 ) , ..., s n ( t n )) = f ( t 1 , ..., t n ) Remark 1.3 It can be shown that a direct revleation mechanism exists whenever a mechanism exists (both with respect to Bayesian Nash equilibrium) 2 Mechanisms Without Money Recall that the Gibbard-Satterthwaite theorem precludes incentive compatible mechanisms for social choice functions f if | A | ≥ 3 and f is onto, where A is the outcome space. However, the theorem applies because players have arbitrary preferences profiles over A . One way to circumvent the theorem is to use payments and we have described mechanisms such as VCG and explored various settings under which mechanisms can be designed. We give a few examples of mechanisms
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