Algorithmic Game Theory

Transcription

1 Algorithmic Game Theory Lecture 10 06/15/10 1 A combinatorial auction is defined by a set of goods G, G = m, n bidders with valuation functions v i :2 G R + 0. $5 Got $6! More? Example: A single item for sale, each bidder has a value vi for $3 receiving the item. $10 $6 $7! In multiple rounds of interactions, the auctioneer makes announcements a t, bidders submit bids b i,t (vi,a 1,...,a t ). bi,t is bidder i s strategy. 2 t 1

2 Classical forms of auctions for a single item: English Auction (open ascending price auction) Dutch Auction (open descending price auction) Sealed First-Price Auction... An auction mechanism is a direct revelation mechanism, if the auctioneer asks all bidders to declare their valuations in the first round and then immediately announces the result. For any auction mechanism which possesses a Nash equilibrium, there exists a direct revelation mechanism in which it is Nash equilibrium for all bidders to declare their true valuation and which produces the same outcome as the original mechanism when everyone plays their Nash strategy. This is called the revelation principle. 3 A combinatorial auction is defined by a set of goods G, G = m, n bidders with valuation functions v i :2 G R + 0. An auction mechanism takes as an input a vector of valuations v =(v 1,...,v n ) declared by the bidders and consists of an allocation rule A(v) = A 1 (v),...,a n (v) where A i (v) denotes the set of goods allocated to bidder i given declared valuations v, satisfying A i (v) for all, and a payment scheme p(v) = A j (v) = p 1 (v),...,p n (v) i = j, where p i (v) denotes the payment of bidder i for receiving A i (v). 4

3 Given allocation A(v) and payments p(v), we call bidder i s utility. u i (v) =v i Ai (v) p i (v) We assume that bidders act strategically and will bid in way that maximizes their utility. For a given allocation by its social welfare. A(v) = A 1 (v),...,a n (v), we denote SW(A(v)) = j v j Aj (v) 5 We say that a mechanism M =(A, p) is truthful, if for every bidder i with true valuation vi and any valuations v i declared by the remaining bidders, it holds that for all v i. u i (v i,v i ) u i (v i,v i ) Truthfulness requires that it is a bidder s best strategy to declare her true valuation no matter what the other bidders do. We say that truth-telling is a dominant strategy. This is much stronger than requiring truth-telling to be a Nash equilibrium. Bidders don t have to assume that others are acting rational. 6

4 The VCG Mechanism The VCG-Mechanism selects a social welfare maximizing allocation A(v) argmax A SW(A ) charges each bidder i price p i (v) =v i Ai (v) v j Aj (v) v j Aj (v i ). j j=i Theorem 1 The VCG-Mechanism is truthful. 7 The VCG Mechanism The VCG-Mechanism gives each bidder a rebate equal to the amount by which the world is better place because of her. p i (v) =v i Ai (v) v j Aj (v) v j Aj (v i ). j j=i maximum social welfare obtainable maximum social welfare obtainable if bidder i hadn t submitted her bid A bidder is charged her bid minus the value by which her bid increases the social welfare. 8

5 Computing Allocations The VCG-Mechanism requires finding the welfare maximizing allocation. This is bad from an algorithmic point of view... In the weighted set packing problem, given a ground set U, U = m, of elements and sets S 1,...,S n over U with weights w 1,...,w n, we want to find a collection P {1,...,n} of sets with S i S j = for all i, j P, i = j, maximizing w(p )= i P w i. Fact 1 Set packing is hard to approximate within NP=ZPP. 9 m 1/2 ε, unless Computing Allocations So what if we can solve the allocation problem only approximately? Considering algorithmic issues also brings up the question of how bidders specify their valuations: A full specification will in general take exponential space. We will assume that bids are given as oracles which we can query for information. As an example, consider a rank-query oracle. Given a function φ :2 G R + 0 R+ 0, the oracle for bidder i returns a set S maximizing φ S, v i (S). 10

6 Computing Allocations Let φ(s, v) = v. S The Greedy Allocation Algorithm Set G = G,B = {1,...,n}. While there exists a bidder in B with non-zero value for a subset of the items in G, query the oracle of each bidder i B maximizing v i (S i )/ S i, for their set S i G let k argmax i B v i (S i )/ S i and allocate set S k to bidder k, let G = G\S k and B = B\{k}. 11 Computing Allocations -approxi- Theorem 2 The Greedy Allocation Algorithm computes a 2m mation with respect to the optimal social welfare. So we can efficiently approximate the welfare maximizing allocation. But how do we get a truthful auction mechanism based on this? Unfortunately, combining approximate allocation algorithms with VCG payments does not yield truthful mechanisms in general. Next we ll see how to fix this problem for auctions among a special class of bidders. 12

7 Auctions for Single-Minded Bidders We say that a bidder i with valuation function v i is singleminded, if there exists a set of goods and a value, such that S i ṽ i ṽi, v i (S) = if S i S 0, else. We will denote a single-minded bid as (S i,v i ), for short. In the single-minded setting, we will assume that any allocation rule will allocate to a bidder declaring bid (S i,v i ) either set or nothing at all. S i Thus, an allocation rule A is completely defined by the set A(v) {1,...,n} of winning bidders it selects. 13 Auctions for Single-Minded Bidders An allocation rule A is called monotone, if for all v i vi arbitrary v i, if i A(v i,v i ) then i A(vi,v i ). and Under a monotone allocation rule, increasing a winning bid can never turn it into a losing bit. A monotone allocation rule defines critical values θ i = θ i (v i ), such that i A(v i,v i ) v i θ i. The critical value θ i (v i ) denotes the minimum value bidder i has to bid in order to win when the remaining bids are. v i 14

8 Auctions for Single-Minded Bidders We say that a single-minded bidder is known, if the set of goods she desires is public knowledge. S i An auction mechanism satisfies the voluntary participation property (VP), if all bidders are guaranteed to have non-negative utility. We call an auction mechanism normalized, if all nonwinning bidders pay 0. Theorem 3 A normalized auction mechanism M =(A, p) satisfying the VP property for known single-minded bidders is truthful, if and only if allocation rule A is monotone and payment scheme p is based on critical values, i.e., p i (v) =θ i (v i ). 15 Auctions for Single-Minded Bidders Theorem 4 The Greedy Allocation Algorithm for single-minded bidders is monotone. Prove: Exercise. 16