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Bayesian Combinatorial Auctions: Expanding Single Buyer Mechanisms to Many Buyers

Saeed Alaei

arXiv:1106.0961v4cs.GT

TL;DR

The paper addresses the difficulty of designing multi-buyer Bayesian combinatorial auctions under supply constraints. It decomposes the problem into single-buyer mechanisms and rounds their outcomes, obtaining a γ_kα approximation when α-approximate single-buyer mechanisms are available. The framework also yields generalized prophet inequalities and improved mechanisms for several settings.

  • Problem

    Multi-buyer mechanism design requires coordinated decisions over an exponentially large joint type space, despite buyers having independently distributed types.

  • Method

    The framework relaxes supply constraints to ex-ante expectations, independently runs single-buyer mechanisms, and uses generic rounding mechanisms to enforce supply constraints pointwise.

  • Results

    The generic mechanisms achieve a γ_kα-approximation when each buyer has an α-approximate single-buyer mechanism and concave benchmark, with γ_k at least 1 − 1/sqrt(k+3).

  • Takeaways & Limitations

    The main coordination difficulty can be reduced to constructing single-buyer mechanisms, while the framework also improves approximation factors in several literature settings.

  • Takeaways & Limitations

    The stated mechanisms may run in time polynomial in |T| rather than in the input size when the type space is exponentially large and the distribution has a compact representation.

Abstract

from arXiv · show

For Bayesian combinatorial auctions, we present a general framework for approximately reducing the mechanism design problem for multiple buyers to single buyer sub-problems. Our framework can be applied to any setting which roughly satisfies the following assumptions: (i) buyers' types must be distributed independently (not necessarily identically), (ii) objective function must be linearly separable over the buyers, and (iii) except for the supply constraints, there should be no other inter-buyer constraints. Our framework is general in the sense that it makes no explicit assumption about buyers' valuations, type distributions, and single buyer constraints (e.g., budget, incentive compatibility, etc). We present two generic multi buyer mechanisms which use single buyer mechanisms as black boxes; if an $α$-approximate single buyer mechanism can be constructed for each buyer, and if no buyer requires more than $\frac{1}{k}$ of all units of each item, then our generic multi buyer mechanisms are $γ_kα$-approximation of the optimal multi buyer mechanism, where $γ_k$ is a constant which is at least $1-\frac{1}{\sqrt{k+3}}$. Observe that $γ_k$ is at least 1/2 (for $k=1$) and approaches 1 as $k \to \infty$. As a byproduct of our construction, we present a generalization of prophet inequalities. Furthermore, as applications of our framework, we present multi buyer mechanisms with improved approximation factor for several settings from the literature.

1 Introduction

The paper develops a general decomposition framework that reduces multi-buyer Bayesian combinatorial-auction design to single-buyer problems under independent types, linearly separable objectives, and limited inter-buyer constraints.

  • Motivation: Multi-buyer mechanism design must coordinate buyers despite an exponentially large joint type space, while individual mechanisms may also face difficult incentive-compatibility constraints.The framework primarily addresses the coordination challenge by decomposing the problem into buyer-specific sub-problems.
  • Framework: The framework first relaxes supply constraints to ex-ante expectations, enabling independent optimization through single-buyer mechanisms.It then converts the ex-ante solution into one satisfying supply constraints at every instance.
  • Framework: Two generic multi-buyer mechanisms use single-buyer mechanisms as black boxes and resolve allocation conflicts while preserving approximation quality.The construction includes the magician’s problem as a key ingredient and supports buyer-specific single-buyer mechanisms.
  • Applications: The framework yields improved approximation factors for several previously studied settings, including sequential posted pricing and budget-constrained mechanisms.Applications cover multiple settings involving matroid, demand, budget, additive, and correlated valuation constraints.
  • Prophet inequalities: The paper improves the prior bound for prophet inequalities involving the sum of k choices to 1 − 1/sqrt(k+3).The new bound is stated to be tight for k = 1 and useful for small k.

2 Preliminaries

The preliminaries formalize Bayesian combinatorial auctions, their feasibility and decomposability assumptions, and the multi-buyer and single-buyer optimization problems connected by ex-ante allocation rules.

  • Model: The model sells m heterogeneous indivisible items with limited supplies to n buyers whose publicly known types are independently distributed.A buyer’s type may itself be multidimensional and need not have product-distributed coordinates.
  • Model: The objective is the expected value of a function linearly separable across buyers, including objectives such as welfare and revenue.Each buyer contributes through her type, allocations, and payment.
  • Assumptions: Feasible mechanism spaces impose incentive compatibility, convexity, and decomposability, with no implicit inter-buyer constraints beyond supply constraints.The paper illustrates that buyer-specific prices satisfy decomposability, whereas identical prices across buyers do not.
  • Optimization problems: The multi-buyer problem maximizes expected objective value over feasible mechanisms, while the single-buyer problem optimizes each buyer’s mechanism subject to upper bounds on ex-ante item allocations.The single-buyer benchmark is a concave function of the allocation bound when the required conditions hold.
  • Decomposition: Without the supply constraints, the multi-buyer optimization decomposes into independent buyer optimizations.This observation motivates the reduction from the multi-buyer problem to single-buyer problems.
  • Allocation rules: The ex-ante allocation rule records each buyer-item allocation probability averaged over all type profiles and obeys aggregate supply bounds by linearity of expectation.For every item j, the expected allocations satisfy sum_i x_ij ≤ k_j.

3 Decomposition via Ex ante Allocation Rule

The paper uses ex-ante allocation rules to upper-bound the multi-buyer optimum, then rounds independently optimized single-buyer mechanisms into feasible mechanisms with controlled loss.

  • Benchmark: Each buyer’s optimal benchmark R_i is concave, supporting the convex-program formulation used by the decomposition.Concavity follows by combining single-buyer mechanisms while respecting ex-ante allocation bounds.
  • Benchmark: The optimal value of the convex benchmark program upper-bounds the expected objective value of the optimal multi-buyer mechanism.The proof induces a feasible single-buyer mechanism for each buyer by simulating the other buyers.
  • Rounding: An independently assembled ex-ante solution may over-allocate items with nonzero probability, so it must be rounded to satisfy supply constraints pointwise.The paper presents two generic mechanisms for resolving these conflicts.
  • Pre-Rounding: Pre-rounding serves buyers sequentially and may withhold available items from earlier buyers to preserve them for later buyers.With at least k units of each item, the mechanism bounds each item’s preclusion probability using the magician’s construction.
  • Post-Rounding: Post-rounding runs all single-buyer mechanisms independently, then randomly deallocates over-allocated items while equalizing deallocation probabilities across buyers.Payments are adjusted accordingly, and each allocation is preserved with probability at least 1 − 1/sqrt(k+3).
  • Approximation guarantee: If each single-buyer mechanism and concave benchmark is an α-approximation, the final multi-buyer approximation factor is multiplied by α.The resulting generic mechanisms achieve γ_kα under the paper’s additional assumptions.

4 The Magician’s Problem

The Magician’s Problem models online selection with limited wands and adversarially arranged boxes. Its near-optimal solution supports the paper’s multi-buyer mechanisms and generalized prophet inequalities.

  • The Magician’s Problem: The magician sees boxes online, each hiding a possible prize and carrying a probability bound on wand breakage, while the sequence is adversarially arranged.The villain fixes the sequence in advance and the magician lacks prior information about it.
  • γ-Conservative Magician: For γ ≤ 1 − 1/√(k+3), the algorithm uses at most k wands while opening every box with ex-ante probability at least γ.With exact breakage probabilities, each box is opened with probability exactly γ.
  • γ-Conservative Magician: A γ-conservative magician adaptively compares previously broken wands with thresholds and randomizes when the count equals the threshold.Thresholds are computed before seeing the corresponding box.
  • Guarantees and hardness: The parameter γ_k is non-decreasing, is at least 1/2 for k = 1, and approaches 1 as k grows.The paper also gives a hardness result showing that the guarantee cannot be improved arbitrarily.
  • Prophet Inequalities: The construction yields a k-choice prophet inequality guaranteeing at least γ_k of the prophet’s expected payoff using one threshold and randomized skipping.The gambler sets a threshold with total exceedance probability k and uses the magician to control selection.

5 Generic Multi Buyer Mechanisms

The paper gives two generic mechanisms that combine approximate single-buyer mechanisms while enforcing item supplies through magician-based rounding. Their guarantees depend on the magician parameter and additional structural assumptions for incentive compatibility and objective preservation.

  • Framework: If each buyer has an α-approximate single-buyer mechanism and k is the minimum item supply, the generic mechanisms achieve a γ_kα-approximation.The bound uses γ_k ≥ 1 − 1/√(k+3).
  • Framework: The framework first solves an ex-ante supply relaxation, which decomposes into independent single-buyer problems, then converts the result to pointwise feasible allocations.The mechanisms use the single-buyer mechanisms as black boxes.
  • 5.1 Pre-Rounding: Pre-rounding serves buyers sequentially, uses item-specific magicians to filter tentative allocations, and adjusts payments while equalizing expected preclusion across buyers.The buyer order may be arbitrary or unknown, including in online settings.
  • 5.1 Pre-Rounding: With budget-balanced cross-monotonic cost sharing, γ-pre-rounding is DSIC and a γα-approximation for any γ ≤ γ_k.The cost-sharing scheme need only be shown to exist; it is not used computationally by the mechanism.
  • 5.1 Pre-Rounding: The pre-rounding construction implies that, when the feasible class contains all BIC mechanisms, the optimal DSIC-to-BIC gap is at most 1/γ_k, at most 2 for k = 1, and vanishes as k grows.This follows because the constructed mechanism is always DSIC while approximating the optimal mechanism in the class.
  • 5.2 Post-Rounding: Post-rounding independently computes tentative buyer outcomes, deallocates items to enforce supply, and preserves each tentative allocation with controlled probability.The resulting mechanism is BIC under assumptions A′1–A′4.

6 Single Buyer Mechanisms

The section develops single-buyer mechanisms for several constrained valuation settings and plugs them into the paper’s generic multi-buyer framework. These mechanisms provide approximation guarantees, including improved multi-buyer factors for settings from prior literature.

  • General framework: IPBR offers buyer-specific menus of item prices, with budget randomization allowing fractional payment and probabilistic receipt under budget constraints.Item pricing is presented as simple and practical, while budget randomization addresses the budgeted buyer’s knapsack difficulty.
  • General framework: The framework applies the single-buyer mechanisms to several settings, with each resulting multi-buyer approximation factor equal to the single-buyer factor multiplied by γ_k.Table 1 summarizes the mechanisms obtained using this framework.
  • 6.1 Single Item, Unit Demand, Budget Constraint: Theorem 10 establishes an optimal revenue-maximizing single-buyer IPBR mechanism that also satisfies γ-pre-rounding requirements.For the unit-demand, single-item setting, the mechanism is characterized through the modified distribution and price randomization framework.
  • 6.3 Multi Item (Independent), Additive, Budget Constraint: For independent additive valuations with budget constraints, the mechanism obtains at least 1−1/e of optimal single-buyer IPBR revenue and supports γ-pre-rounding.The mechanism is constructed item by item through the corresponding convex programs.
  • 6.4 Multi Item (Correlated), Additive, Budget and Matroid Constraints: For correlated additive valuations with budget and matroid constraints, an optimal truthful-in-expectation single-buyer mechanism supports γ-post-rounding and yields a γ_k-approximate multi-buyer BIC mechanism.For welfare, the same approach yields a γ_k-approximate welfare-maximizing BIC mechanism.

7 Analysis of γ-Conservative Magician

The γ-conservative magician is analyzed through a dynamic program and an equivalent sand displacement process. The analysis shows that, under a bound on total wand-breaking probabilities, it guarantees box-opening probabilities while controlling wand usage.

  • Opening probability: The analysis also gives an upper bound of 1−O(1/k) for non-adaptive strategies and shows that stronger opening guarantees cannot generally be obtained.The cited theorem states both the achieved opening probability scale and a corresponding impossibility bound.
  • Dynamic program: The dynamic program computes thresholds, state distributions, and opening probabilities for the γ-conservative magician.Its first proof part establishes ex-ante opening probability at least γ when enough wands are available.
  • Sand displacement process: The sand displacement process selects the leftmost γ-fraction of sand and moves an x_i fraction one position right each round.The process exactly reproduces the thresholds and cumulative state distributions computed by the magician’s dynamic program.
  • Sand displacement process: Theorem 16 bounds the sand distribution and its average distance from the moving barrier throughout the process.These bounds are used to control how far the barrier can advance.
  • Wand usage bound: If γ ≤ 1−1/√(k+3), the thresholds never exceed k−1 whenever the cumulative breaking probabilities satisfy the stated k-bound.Consequently, the magician requires no more than k wands.

8 Multi Unit Demands

Multi-unit demand markets are transformed into markets with at least k units per item and unit demand per buyer. The transformation preserves feasible mechanisms’ allocations and payments, enabling the simpler model to represent the original one.

  • Market transformation: When no buyer requires more than 1/k of an item’s total supply, the market can be reduced to one where buyers demand at most one unit of each item.The transformed market has at least k units of every item.
  • Market transformation: The transformation divides each item’s units into nearly equal bins and treats each bin as a distinct item type.Each bin contains either c_j or c_j+1 units, with c_j defined from the original supply and k.
  • Mechanism equivalence: Theorem 18 states that every feasible mechanism in the original market corresponds to one in the transformed market, and vice versa, with identical allocations and payments.Thus, finding the optimal mechanism in the transformed market suffices for the original market.
  • Mechanism equivalence: The correspondence allocates repeated units by cycling through bin lists, ensuring that no buyer receives two units from the same bin.This establishes feasibility in the transformed representation.
  • Scope boundary: The generic multi-buyer mechanisms require single-buyer mechanisms capable of correlated valuations in the transformed market because duplicated units are perfect substitutes.Among the paper’s single-buyer mechanisms, only the mechanism in §6.4 handles correlated valuations.

9 Conclusion

The paper concludes that its approximate reduction makes multi-buyer mechanism design tractable through single-buyer mechanisms, with approximation quality governed by market demand relative to supply rather than buyer count.

  • The framework reduces Bayesian combinatorial auction design from multiple buyers to single-buyer problems while preserving approximation guarantees.
  • Market size: As maximum demand-to-supply ratio 1/k decreases, buyers require less coordination and the optimal mechanism approaches independent treatment.
  • Market size: Approximation factors depend on γ_k and not on the number of buyers n.
  • Market size: The relevant asymptotic market parameter may be maximum demand relative to supply, since buyer count is irrelevant to the approximation factors.
  • Computational hardness: A small constant loss, 1/(k+3), can avoid coordinated optimization, shifting the main difficulty toward designing single-buyer mechanisms and handling their incentive constraints.

A Missing Proofs

This section supplies proofs for the paper’s approximation and incentive-compatibility claims, using randomized box-and-magician arguments, concavity, linear programs, and cost-sharing properties.

  • The magician construction models independent prizes in boxes, with expected total prize k but realized prizes sometimes exceeding the k-dollar winning limit.
  • The resulting bound is based on the expected truncated prize E[min(Σ_i X_i,k)], rather than the unrestricted expected total prize.
  • For sufficiently large n, no strategy can guarantee more than the stated asymptotic threshold, while γ = 1 − 1/√k is achievable for opening-box probabilities.
  • The pre-rounding and post-rounding constructions establish expected objective guarantees of at least γα times the relevant optimal value.
  • Pre-rounding remains DSIC and preserves ex-post properties, whereas post-rounding is only BIC because each buyer’s selection probability depends on other buyers’ reports.
  • Concavity of benchmark functions and feasible primal-dual assignments provide the structural arguments needed for the approximation bounds.
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