Notes19 - CS 573 Algorithmic Game Theory Lecture date March...

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Unformatted text preview: CS 573: Algorithmic Game Theory Lecture date: March 26th, 2008 Instructor: Chandra Chekuri Scribe: Qi Li Contents Overview: Auctions in the Bayesian setting 1 1 Single item auction 1 1.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Second Price Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 First Price Auction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Auctions in the Bayesian setting We have seen mechanism design in the “worst cass” setting. That is the dominant strategy im- plementations we sought. Although dominant strategy implementation is ideal, it is a very strong requirement. Economists also favor Bayesian-Nash implementation since it allows for more mechanism as well as capturing more realistically “prior” information that players have about the valuation of other players. We will study some types of auctions in the Bayesian setting which allows us to show: • The revenue equivalence theorem • Optimal mechanism design both due to Roger Myerson (also done independently by Riley and Samuelson) who won the Nobel prize in 2007. 1 Single item auction 1.1 Setting • N players • Player i ’s valuation is in [0, ω ] and there is a probility distribution on [0, ω ] that gives the value of i • The valuation of players are independent • All players know the distribution of all other players, but not the actual realization of the value. We will mostly focus our attention on the “symmetric” case where the distribution of all the players are the same. This assumption is reasonable in a setting with a large number of players. 1 Let X 1 ,...,X n denote the independent identically distributed random variables that define the value of the players. We use F and f to denote the cumulative distribution function ( cdf ) and the density function repectively. Assumption 1.1 F is continuous and differentiable Note that d F d x = f ( x ) In this simple setting we can ask several interesting questions: 1. Does the first price auction have an equilibrium (Bayesian Nash)?...
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Notes19 - CS 573 Algorithmic Game Theory Lecture date March...

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