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# Lecture20-new - 6.254 : Game Theory with Engineering...

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Unformatted text preview: 6.254 : Game Theory with Engineering Applications Lecture 20: Mechanism Design II Asu Ozdaglar MIT May 4, 2010 1 Game Theory: Lecture 20 Introduction Outline Mechanism design from social choice point of view Implementation in dominant strategies Revelation principle VCG Mechanisms and examples Budget-balancedness dAGV Mechanisms Reading: Microeconomic Theory, MasColell, Whinston and Green, Chapter 23. Algorithmic Game Theory, edited by Nisan, Roughgarden, Tardos, and Vazirani, Chapter 9, by Noam Nisan. 2 Game Theory: Lecture 20 Social Choice Viewpoint Introduction Our goal is to analyze how individual preferences can be aggregated into desirable social or collective decisions. An important feature of such settings in which collective decisions must be made is that individuals’ actual preferences are not publicly observable. As a result, in one way or another, individuals must be relied upon to reveal this information ⇒ mechanism design problem . 3 Game Theory: Lecture 20 Social Choice Viewpoint Model Consider a setting with I agents. These agents must make a collective choice from some set Y (of possible alternatives). Each agent privately observes his preferences over the alternatives in Y . We model this by assuming that agent i privately observes a signal θ i that determines his preferences, i.e., θ i is agent i ’s type . The set of possible types for agent i is denoted by Θ i and we use the notation Θ = ∏ I i = 1 Θ i . Each agent i is assumed to be an expected utility maximizer, whose Bernoulli utility function is given by u i ( y , θ i ) with y ∈ Y . As in all incomplete information settings, we assume that agents’ types are drawn from a commonly known prior distribution over the type θ = ( θ 1 , . . . , θ I ) , and the type distribution and the utility functions u i ( y , θ i ) are common knowledge among the agents. 4 Game Theory: Lecture 20 Social Choice Viewpoint Social Choice Function Since agents’ preferences depend on the realization of their types, the agents may want the collective decision to depend on θ . Definition A social choice function is a function f : Θ 1 × ··· × Θ I → Y that for each possible profile of agents’ types ( θ 1 , . . . , θ I ) , assigns a collective choice f ( θ 1 , . . . , θ I ) ∈ Y . Definition The social choice function f is ex-post efficient if for no profile θ = ( θ 1 , . . . , θ I ) is there a y ∈ Y such that u i ( y , θ i ) ≥ u i ( f ( θ ) , θ i ) for all i, and u i ( y , θ i ) > u i ( f ( θ ) , θ i ) for some i. The problem is that the θ i ’s are not publicly observable, so for the social choice f ( θ 1 , . . . , θ I ) to be chosen, each agent must be relied on to disclose their type correctly. But for a given f ( · ) an agent may not find it in his best interest to reveal this information truthfully....
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## This note was uploaded on 05/08/2010 for the course CS 6.254 taught by Professor Asuozdaglar during the Spring '10 term at MIT.

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Lecture20-new - 6.254 : Game Theory with Engineering...

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