Bayesian Combinatorial Auctions Giorgos Christodoulou, Annamaria Kovacs, Michael Schapira האוניברסיטה העברית בירושלים The Hebrew University of Jerusalem

Combinatorial Auctions

opt=9

Combinatorial Auctions Objective: Find a partition of the items bidders items valuations that maximizes the social welfare (normalized) (monotone)

Valuations Submodular (SM) The marginal value of the item decreases as the number of items increases. Fractionally-subadditive (FS) additive

FS Valuations abc items add. valuations

Combinatorial Auctions - Challenges Strategic We want bidders to be truthful. VCG implements the opt. (exp. time) Computational approximation algorithms (not truthful)

Unknown Valuations

Huge Gaps Submodular (SM) Fractionally-subadditive (FS) 1-1/e-   [Feige-Vondrak] 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

Complete Information

Auction Setting Player i will bid Strategy Profile Algorithm = allocation + payments Utility of player i

Bayesian Combinatorial Auctions Question: Can we design an auction for which any Bayesian Nash Equilibrium provides good approximation to the social welfare?

(Pure) Bayesian Nash [Harsanyi] Bidding function Informal: In a Bayesian Nash (B 1,…,B n ), given a probability distribution D, B i (v i ) maximizes the expected utility of player i (for all v i ). ()

Bayesian PoA Optimal Social Welfare Expected Social of a B.N.E. for fixed v Bayesian PoA = biggest ratio between SW(OPT) and SW(B) (over all D, B)

Bayesian PoA Price of Anarchy [Gairing, Monien, Tiemann, Vetta]

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. Utility of player i

Second Price Social Welfare = 1

Second Price Social Welfare =

Second Price Social Welfare = PoA=1/

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 Letbe a fixed valuation profile

Upper Bound Letbe a fixed valuation profile optimum partition: Nash partition:

Upper Bound Since b is a N.E Letbe a fixed valuation profile optimum partition: maximum additive valuation wrt Nash partition:

Upper Bound Since b is a N.E Letbe a fixed valuation profile optimum partition: maximum additive valuation wrt Nash partition:

Upper Bound Since b is a N.E and so

Upper Bound Since b is a N.E and so using lemma we get

Upper Bound Since b is a N.E 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.

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 Potential Procedure

The Marginal-Value Procedure Start with bid-vectors b i =(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 b ij to be the second highest marginal value.

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. The Marginal-Value Procedure

Open Questions Can a (mixed) Bayesian Nash Equilibrium be computed in poly-time? Algorithm that computes N.E. in poly time for FS valuations. Second Price Design an auction that minimizes the PoA for B.N.E.

Thank you!