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Lecture19-new

Lecture19-new - 6.254 Game Theory with Engineering...

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6.254 : Game Theory with Engineering Applications Lecture 19: Mechanism Design I Asu Ozdaglar MIT April 29, 2010 1
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Game Theory: Lecture 19 Introduction Outline Mechanism design Revelation principle Incentive compatibility Individual rationality “Optimal” mechanisms Reading: Krishna, Chapter 5 Myerson, “Optimal Auction Design,” Mathematics of Operations Research , vol. 6, no. 1, pp. 58-73, 1981. 2
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Game Theory: Lecture 19 Auction Theory Viewpoint Introduction In the next 3 lectures, we will study Mechanism Design, which is an area in economics and game theory that has an engineering perspective. The goal is to design economic mechanisms or incentives to implement desired objectives (social or individual) in a strategic setting–assuming that the different members of the society each act rationally in a game theoretic sense. Mechanism design has important applications in economics (e.g., design of voting procedures, markets, auctions), and more recently finds applications in networked-systems (e.g., Internet interdomain routing, design of sponsored search auctions). 3
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Game Theory: Lecture 19 Auction Theory Viewpoint Auction Theory Viewpoint We first study the mechanism design problem in an auction theory context, i.e., we are interested in allocating a single indivisible object among agents. An auction is one of many ways that a seller can use to sell an object to potential buyers with unknown values. In an auction, the object is sold at a price determined by competition among buyers according to rules set by the seller (auction format), but the seller can use other methods. The question then is: what is the “best” way to allocate the object? Here, we consider the underlying allocation problem by abstracting away from the details of the selling format. 4
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Game Theory: Lecture 19 Auction Theory Viewpoint Model We assume a seller has a single indivisible object for sale and there are N potential buyers (or bidders) from the set N = { 1, . . . , N } . Buyers have private values X i drawn independently from the distribution F i with associated density function f i and support X i = [ 0, w i ] . Notice that we allow for asymmetries among the buyers, i.e., the distributions of the values need not be the same for all buyers. We assume that the value of the object to the seller is 0. Let X = N j = 1 X j denote the product set of buyers’ values and let X - i = j 6 = i X j . We define f ( x ) to be the joint density of x = ( x 1 , . . . , x N ) . Since values are independently distributed, we have f ( x ) = f 1 ( x 1 ) × · · · × f N ( x N ) . Similarly, we define f - i ( x - i ) to be the joint density of x - i = ( x 1 , . . . , x i - 1 , x i + 1 , . . . , x N ) . 5
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Game Theory: Lecture 19 Auction Theory Viewpoint Mechanism A selling mechanism ( B , π , μ ) has the following components: A set of messages (or bids/strategies) B i for each buyer i , An allocation rule π : B → Δ , where Δ is the set of probability distributions over the set of buyers N , A payment rule μ : B → R N .
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