Bayesian Combinatorial Auctions: Expanding Single Buyer Mechanisms to Many Buyers

1. Bayesian Combinatorial Auctions: Expanding Single BuyerMechanisms to Many Buyers Saeed Alaei University of Maryland e-mail: saeed@cs.umd.edu

2. Introduction • We want to allocate items (heterogeneous) to buyers. • Each buyer has some relevant private information which we refer to as her type • E.g., valuation for various subsets of items, budget, etc. • Buyers’ types are distributed independently (not identically) according to known distributions. • In addition to supply constraints, there might be single buyer constraints • E.g., budget constraints, downward set constraints on bundles of items. • Goal: design a truthful mechanism to maximize the expected value of some objective function • E.g., welfare, revenue, etc.

3. Highlights of Our Approach • A general reduction from multi-buyer problem to single-buyer sub-problems. • The reduction requires no assumptionsabout: • nature of types • distributions (except independence) • valuations (additive, submodular, etc.). • other single buyer constraints (budget, etc.). • Works for any objective function that is linearly separable over buyers (e.g. welfare, revenue, etc.). • Solving the single buyer problem is often much easier • Compare Myerson’s auction vs. the single price for maximizing revenue

4. Related Work • The problem is well-understood in single dimensional-settings (i.e., each is a number). • Myerson (Math of OR, 1981) Optimal Auction Design. • The problem is much harder in multi-dimensional settings • Armstrong (Econometrica , 1996) – Multi-product non-linear pricing. • Hartline, Kleinberg, Malekian, (SODA, 2011) – Bayesian Incentive Compatibility via Matchings. • Chawla, Hartline, Malec, Sivan (STOC 2010) – Multi-parameter mechanism design and sequential posted pricing. • Bhattacharya, Goel, Gollapudi, Munagala (STOC 2010)- Budget constrained auctions with heterogeneous items.

5. Summary of improved mechanisms AIP = Asymmetric Item Pricing

6. Main Result • Single Buyer Problem:For buyer given a vector of probabilities compute the following: • : optimal single buyer mechanism subject to the expected probability of allocating each item being no more than . • : the expected objective value of . Theorem: If we can compute and , we can construct a multi-buyer mechanism which is a -approximation of the optimal mechanism, assuming that there are at least copies of each item. If we can only compute an -approximation of and , then the resulting mechanism is -approximation of the optimal.

7. Notation • : the number of copies of item . • : allocation of item to buyer • Assume each buyer needs at most one copy of each item, i.e., . • : payment of buyer .

8. Overview of Approach • Relax the problem by requiring the supply constraints to hold only in expectation. • i.e., replace with . • We show how to construct an optimal mechanism for the relaxed problem by running independent single buyer mechanisms where are computed using a convex program. • We use a rounding scheme to convert this to a mechanism that satisfies the supply constraints at every instance. i.e., to ensure .

9. Relaxation & Decomposition Relax the problem by requiring that the supply constraint to hold only in expectation. i.e. Claim: There is an optimal mechanism of the following form for the relaxed problem:

10. Relaxation & Decomposition (proof) 2. Create copies of. Each copy is run on one of the buyers and dummy buyers with random types drawn from the corresponding distributions. 1. Suppose is the optimal mechanism for the relaxed problem. 4. Suppose is the expected probability that allocates a copy of item to buyer . Replace each with .

11. How to compute the optimal ? Convexity: Consider any two feasible mechanisms and any , a mechanism , that runs with prob and with prob , must also be feasible. Theorem: the optimal can be computed by solving the following program: Theorem: If the space of feasible mechanisms is convex, then are always concave.

12. The Convex Program Remark: Often is the optimal objective value of a convex program of the following form. All such programs can be merged with the main convex program.

13. Rounding • Enforcing The Supply Constraints • We present two approaches, Pre-Rounding and Post-Rounding. • (Pre-Rounding): Run sequentially. If item is sold out, set before running . • Problem: buyers have a lower chance of getting the items when they are served later. • Solution: randomly preclude items from earlier buyers to equalize the probability of preclusion for all buyers.

14. Magician’s Problem • Problem Definition: • A sequence of boxes arrive online. There is a prize hidden in some box • A magician has wands that can open boxes. • A wand may break with prob. if used on box . However, . • Adversarial setting, but fixed in advance. To ensure that the prize is obtained with a prob, say , each box must be opened with prob at least . Theorem: For any , there exists a magician parameterized by that guarantees that each box is opened with ex-ante probability at least . We call it the -conservative magician. Also define to be the optimal for a magician with at least wands.

15. -Pre-Rounding • -Pre-Rounding (high level description): • Compute the optimal for the convex program. • For each item create a -conservative magician with wands. • Serve the buyers sequentially . For each buyer : • For each : write on a box and present to the magician. • If the magician does not open the box, set . • Run to compute the final outcome for buyer . • For each : break the magician’s wand If allocated a copy of item . Break wand if allocated item 1 If not open: If there are at least copies of each item, then choosing ensures that each item is available to each with probability at least which is at least . Break wand if allocated item 2 Remark: The -pre-rounding mechanism is an online mechanism, the order in which the buyers are served can be arbitrary. +1

16. Conclusion In a Bayesian Combinatorial setting, to solve a multi-buyer (independent) mechanism design problem, it is enough to solve the single-buyer problem. By going from single-buyer to multi-buyer, we lose at most in the approximation (i.e., the approximation factor is multiplied by assuming there are at least copies of each item). Questions?