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Unformatted text preview: 6.254: Game Theory with Engineering Applications Guest Lecture: Social Choice and Voting Theory Daron Acemoglu MIT May 6, 2010 1 Game Theory: Lecture 21 Introduction Outline Social choice and group decisionmaking Arrow&s Impossibility Theorem GibbardSatterthwaite Impossibility Theorem Single peaked preferences and aggregation Group decisions under incomplete information Reading: Microeconomic Theory, MasColell, Whinston and Green, Chapters 21 and 23. 2 Game Theory: Lecture 21 Group and Collective Choices Social Choice Functions Recall that mechanism design, in an environment with H players each with possible type & i 2 & 1 and a set of feasible alllocations Y , started with a social choice function f : & 1 & ::: & & H ! Y : But where does this social choice function come from? Presumably, it re&ects some ¡social objective¢such as fairness or e£ ciency. But how do we arrive to such an objective? More general question: How do groups make collective/group/political decisions? Two sets of issues: 1 Aggregating up to collective preferences from individual preferences. 2 Using dispersed information of the group e£ ciently. 3 Allocation of ¡power¢in a group (not discussed in this lecture). 3 Game Theory: Lecture 21 Group and Collective Choices Setup Let us investigate these questions in a slightly more general setup, where we take the types, the & &s, to be rankings over all possible allocations. For this purpose, let us restrict attention to a society with a ¡nite set of individuals H , with the number of individuals denoted by H , and a ¡nite set of allocations denoted by P . Individual i 2 H has an indirect utility function de¡ned over choices available to the group or ¢policies£ p 2 P U ( p ; ¡ i ) ; where ¡ i indexes the utility function (i.e., U ( p ; ¡ i ) = U i ( p ) ). The bliss point of individual i is de¡ned as: p ( ¡ i ) = arg max p 2P U ( p ; ¡ i ) : 4 Game Theory: Lecture 21 Group and Collective Choices Preferences More Generally Individual i weakly prefers p to p , p & i p and if he has a strict preference, p ¡ i p : Assume: completeness , re&exivity and transitivity (so that z & i z and z & i z 00 implies z & i z 00 ). 5 Game Theory: Lecture 21 Group and Collective Choices Collective Preferences? Key question: Does there exist welfare function U S ( p ) that ranks policies for this group (or society)? Let us &rst start with a simple way of ¡aggregating¢the preferences of individuals in the group: majoritarian voting . This will lead to the Condorcet paradox . 6 Game Theory: Lecture 21 Voting and the Condorcet Paradox The Condorcet Paradox Imagine a group consisting of three individuals, 1, 2, and 3, three choices and preferences 1 a & c & b 2 b & a & c 3 c & b & a Assume &open agenda direct democracy¡system for making decisions within this group....
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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.
 Spring '10
 AsuOzdaglar

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