Review! November 28, 2011 () November 28, 2011 1 / 23
Mechanism Design Design mechanisms/auctions such that when participants play selfishly, the designer’s goals are achieved. Some typical settings: markets: Given individual preferences for goods and money, determine the right way to reallocate the goods and money, e.g to maximize social welfare auctions: given buyers with preferences over items being sold, determine winners of auction and payments so as to maximize auctioneer’s profit. resource allocation in distribution systems: given resource owners that incur costs when their resources are used, and users with different needs and willingness to pay for resources, determine the allocation of resources to user to optimize global objective function. () November 28, 2011 2 / 23
Single Parameter Allocation Problems Auctioneer offering/allocating good/service. n agents, agent i has value v i for receiving good. There are constraints on which subsets of agents can simultaneously be served. For example: single item auctions, k -unit auctions digital goods auctions ad auctions multiple markets – can only sell in one of them The auction takes as input a bid bector b 1 , . . . , b n and chooses as output a feasible subset S of winning bidders (specified by allocation vector x ) and a price p i ( ≤ b i ) for each i ∈ S . Agents bid to maximize their own utility u i = v i x i − p i . Goal: Design auction to achieve goals, which are usually: profit maximization, i.e. maximize � i p i social welfare maximization, i.e. maximize � i v i x i . () November 28, 2011 3 / 23
Strategies and Equilibria Definition A strategy for an agent in an auction is a mapping from values to bids (or actions), i.e. s i ( v i ) describes how player i plays when his value is v i . Definition Dominant Strategies : An auction has dominant strategies ( s 1 , . . . s n ) if for all i and all b − i , agent i ’s utility is maximized by playing s i ( v i ) . When the dominant strategy s i ( v i ) = v i , we say the auction is truthful . Theorem The Revelation Principle: For any auction with dominant strategies, there is an equivalent direct-revelation auction which results in same outcome and payments and is truthful. () November 28, 2011 4 / 23
Characterization of Truthful Auctions Definition An auction is truthful if it is a dominant strategy for each player to bid their true value. Also, Let x i ( v ) be the probability that agent i wins when players values are v . let p i ( v ) be expected payment of agent i when players values are v . ( Monotonicity :) An auction is truthful iff for any fixed v − i , x i ( v i , v − i ) is monotone non-decreasing in v i . For deterministic mechanisms, this means that there is a threshold t ( v − i ) such that x i ( v i , v − i ) is 1 when v i > t ( v − i ) and x i ( v i , v − i ) is 0 when v i < t ( v − i ) . ( Allocation rule determines payments: ) � ∞ p i ( v ) = v i x i ( v ) − x i ( w , v − i ) dw + p i ( 0 , v − i ) . 0 For deterministic mechanisms, the payment when the player wins is the threshold bid t ( v − i ) . () November 28, 2011 5 / 23
Truthful auction examples Vickrey (Second Price) Auction for a single item: Sell to bidder with the highest bid (= by assumption his value); Payment: 2nd highest bid. k-item auction: sell to bidders with top k bids; Payment: k+1st bid. Social welfare maximization: Choose feasible subset S of bidders such that � i ∈ S v i maximized. S are the winners. i ∈ S \ j v i where S ′ is the feasible Payment of bidder j ∈ S is: � i ∈ S ′ v i − � subset of bidders excluding j such that � i ∈ S ′ v i maximized. () November 28, 2011 6 / 23
Bayes-Nash Equilibrium Assume players values are drawn from known prior distributions, say v i drawn independently from F i . All players and auctioneer know the priors F i , but only player i knows his draw v i . Definition Bayes-Nash Equilibrium : A set of strategies ( s 1 , . . . s n ) is a Bayes-Nash equilibrium for an auction if, for all i , agent i ’s expected utility (taken over the random draws of v − i ) is maximized by playing s i ( v i ) . When the BN equilibrium is s i ( v i ) = v i , we say the auction is Bayes-Nash incentive compatible . Theorem The Revelation Principle: For any auction with a Bayes-Nash equlibrium there is an equivalent direct-revelation auction which results in same expected outcome and payments that is BN incentive compatible. () November 28, 2011 7 / 23
BN equilibrium examples First price auction, 3 players values uniform [0,1], s ( v ) = 2 v / 3. What do you need to check to verify this? All pay auction, 3 players values uniform [0,1], s ( v ) = 2 v 3 / 3. () November 28, 2011 8 / 23
Characterization of Bayes-Nash Equilibrium Fix an auction and assume each agent i uses strategy s i . Let x i ( v i ) be the probability that agent i wins, taken over the random draws of v − i , when each agent j plays according to strategy s j . let p i ( v i ) be the expected payment of agent i , where the expectation is taken over the random draws of v − i . The set of strategies s = ( s 1 , . . . , s n ) is a Bayes-Nash equilibrium iff ( Monotonicity :) x i ( v i ) is monotone non-decreasing in v i . ( Payment Rule: ) � ∞ p i ( v i ) = v i x i ( v i ) − x i ( w ) dw + p i ( 0 ) . 0 Corollary Revenue Equivalence: All auctions with the same allocation rule (in equilibrium) result in same auctioneer revenue. () November 28, 2011 9 / 23
Revenue Equivalence expected revenue of 1st price auction = expected revenue of 2nd price auction = expected revenue from all-pay auction. Useful for computing equilibrium strategies. Example: All pay auction, 3 players uniform [0,1]. Allocation rule is same as that of 2nd price auction. () November 28, 2011 10 / 23
Profit Maximization Assume agents’ values are drawn from known priors F i . Suppose truthtelling is a BN equilibrium for some mechanism such that x i ( v i ) is the probability that agent i wins in equilibrium. By the payment identity we have � ∞ p i ( v i ) = v i x i ( v i ) − x i ( w ) dw . 0 Therefore � ∞ � ∞ � � E v i ( p i ( v i )) = v i x i ( v i ) − x i ( w ) dw f i ( v i ) dv i . 0 0 Rearranging gives: � ∞ � v i − 1 − F ( v i ) � E v i ( p i ( v i )) = x i f i ( v i ) dv i = E v i ( x i ψ i ( v i )) , f ( v i ) 0 where ψ ( v ) = v − 1 − F ( v ) . f ( v ) () November 28, 2011 11 / 23
Myerson Mechanism � ∞ � v i − 1 − F ( v i ) � E v i ( p i ( v i )) = x i f i ( v i ) dv i = E v i ( x i ψ i ( v i )) , f ( v i ) 0 where ψ ( v ) = v − 1 − F ( v ) . f ( v ) Therefore, if we want to maximize profit, we should choose the feasible allocation that maximizes � ψ ( v i ) x i . i ψ ( v ) called v ’s virtual value Potential Issue: This may not be truthful. This is a monotone allocation rule as long as ψ i ( v i ) is monotone increasing in v i . The condition (on a probability distribution) that ψ ( v ) is monotone non-decreasing is called “regularity”. () November 28, 2011 12 / 23
Profit Maximization with priors For regular distributions, choosing the allocation that maximizes the sum of virtual values results in an allocation rule such that for each v − i , x i ( v i , v − i ) is monotone non-decreasing in v i . Therefore, truthtelling is a dominant strategy! This mechanism maximizes expected auctioneer profit! () November 28, 2011 13 / 23
Myerson Mechanism Examples: two players, values drawn uniform [0,1] ψ ( v ) = v − 1 − F ( v ) = v − 1 − v = 2 v − 1 , 1 f ( v ) so ψ ( v ) ≥ 0 iff v ≥ 1 / 2. Resulting allocation rule: Allocation to highest bidder if value is at least 1/2 This is equivalent to Vickrey with reserve price of 1/2. Digital goods, n players values drawn iid from regular prior F . Let r be the value such that ψ ( r ) = 0. Profit maximizing auction: Vickrey with reserve price of r . Resulting payment rule: pay r if you win, 0 if lose. Note that payment rule slightly more complicated when agents values are drawn from different prior distributions. () November 28, 2011 14 / 23
Profit Maximization without Prior Example: digital goods auctions. () November 28, 2011 15 / 23
Profit Maximization: No priors How do we design truthful auctions that are (approximately) optimal in the worst-case in absence of prior? Characterize Bayesian optimal mechanism Opt F for every i.i.d. 1 distribution F . Opt F is to charge each agent the price p that maximizes p ( 1 − F ( p )) . Define distribution independent benchmark 2 G ( v ) = sup Opt F ( v ) . F Best fixed price auction! Design a prior-free truthful mechanism that approximates G ( v ) on every 3 v . Random sampling auctions are constant competitive. (Deterministic auctions don’t work.) Randomly partition bidders into two sets. Compute best fixed price profit in each part. Attempt to extract profit of each part from other part using truthful auction from homework. () November 28, 2011 16 / 23
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