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##### Bayesian Combinatorial Auctions

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**Bayesian Combinatorial Auctions**• Giorgos Christodoulou, Annamaria Kovacs, Michael Schapira האוניברסיטה העברית בירושלים The Hebrew University of Jerusalem**Combinatorial Auctions**opt=9**Combinatorial Auctions**bidders items valuations (normalized) (monotone) Objective: Find a partition of the items that maximizes the social welfare**Valuations**• Submodular (SM) The marginal value of the item decreases as the number of items increases. • Fractionally-subadditive (FS) additive**FS Valuations**items add. valuations**Combinatorial Auctions - Challenges**• Strategic • We want bidders to be truthful. • VCG implements the opt. (exp. time) • Computational • approximation algorithms (not truthful)**Huge Gaps**• Submodular (SM) 1-1/e-e[Feige-Vondrak] • Fractionally-subadditive (FS) 1-1/e[Dobzinski-Schapira] O(log(m) log log(m))[Dobzinski]**Solution?**• We do not know whether reasonable truthful and polynomial-time approximation algorithms exist. • How can we overcome this problem? • An old/new approach.**Partial Information**is drawn from D**Player i will bid**• Strategy Profile • Algorithm = allocation + payments Utility of player i Auction Setting**Bayesian Combinatorial Auctions**Question:Can we design an auction for which anyBayesian Nash Equilibrium provides good approximation to the social welfare?**(**) (Pure) Bayesian Nash [Harsanyi] • Bidding function • Informal: In a Bayesian Nash (B1,…,Bn), given a probability distribution D, Bi(vi) maximizes the expected utility of player i (for all vi).**Expected Social of a B.N.E.**Bayesian PoA Optimal Social Welfare for fixed v Bayesian PoA = biggest ratio between SW(OPT) and SW(B) (over all D, B)**Price of Anarchy**Bayesian PoA [Gairing, Monien, Tiemann, Vetta]**Utility of player i**Second Price • Player i will bid • Strategy Profile • Algorithm: • Give item j to the player i with the highest bid. • Charge I the second highest bid.**Second Price**Social Welfare = 1**Second Price**Social Welfare =**PoA=1/**Second Price Social Welfare =**Supporting Bids**• Bidders have only partial info (beliefs) • They want to avoid risks. (ex-post IR) Supporting Bids:(for all S)**Lower Bound**opt=2**Lower Bound**Nash=1 PoA=2**Our Results**• Bayesian setting: • The Bayesian PoA for FS valuations (supporting bids, mixed) is 2. • Complete-information setting: • FS Valuations:Existence of pure N.E. • Myopic procedure for finding one. • PoS=1. • SM Valuations: Algorithm for computing N.E. in poly time.**Valuations**• Submodular (SM) The marginal value of the item decreases as the number of items increases. • Fractionally-subadditive (FS) additive**Upper Bound(full-info case)**Lemma. For any set of items S, where is the maximizing additive valuation for the set S.**Upper Bound**Let be a fixed valuation profile**Upper Bound**Let be a fixed valuation profile optimum partition: Nash partition:**maximum additive valuation wrt**Upper Bound Let be a fixed valuation profile optimum partition: Nash partition: Since b is a N.E**maximum additive valuation wrt**Upper Bound Let be a fixed valuation profile optimum partition: Nash partition: Since b is a N.E**Since b is a N.E**Upper Bound and so**Since b is a N.E**Upper Bound and so using lemma we get**Since b is a N.E**Upper Bound and so using lemma we get and so**Upper Bound**summing up**But…**• Open Question: Does a (pure) BN with supporting bids always exist? • Open Question:Can we find a (mixed) BN in polynomial time? • We consider the full-information setting.**The Potential Procedure**• Start with item prices 0,…,0. • Go over the bidders in some order 1,…,n. • In each step, let one bidder i choose his most demanded bundle S of items. Update the prices of items in S according to i’s maximizing additive valuation for S. • Once no one (strictly) wishes to switch bundle, output the allocation+bids.**The Potential Procedure**• Theorem: If all bidders have fractionally-subadditive valuation functions then the Potential Procedure always converges to a pure Nash (with supporting bids). • Proof: The total social welfare is a potential function.**The Potential Procedure**• Theorem: After n steps the solution is a 2-approximation to the optimal social welfare (but not necessarily a pure Nash). [Dobzinski-Nisan-Schapira] • Theorem: The Potential Procedure might require exponentially many steps to converge to a Pure Nash.**The Potential Procedure**• Open Question: Can we find a pure Nash in polynomial time? • Open Question: Does the Potential Procedure converge in polynomial time for submodular valuations?**The Marginal-Value Procedure**• Start with bid-vectors bi=(0,…,0). • Go over the items in some order 1,…,m. • In each step, allocate item j to the bidder i with the highest marginal value for j. Set bij to be the second highest marginal value.**The Marginal-Value Procedure**• Theorem: The Marginal-Value Procedure always outputs an allocation that is a 2-approximation to the optimal social-welfare. [Lehmann-Lehmann-Nisan] • Proposition: The bids the Marginal-Value Procedure outputs are supporting bids and are a pure Nash equilibrium.**Open Questions**Design an auction that minimizes the PoA for B.N.E. • Can a (mixed) Bayesian Nash Equilibrium be computed in poly-time? • Algorithm that computes N.E. in poly time for FS valuations. Second Price