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An n-to-1 Bidder Reduction for Multi-item Auctions and its Applications

Andrew Chi-Chih Yao

arXiv:1406.3278v3cs.GT

TL;DR

The paper addresses how to maximize revenue in multi-item auctions when general mechanisms and closed-form guarantees are difficult to obtain. It introduces Best-Guess reduction from multi-bidder to single-bidder auctions and derives constant-factor, DSIC/BIC, and closed-form revenue results, including a simple 2nd-Price Bundling mechanism.

  • Problem

    Multi-item revenue-maximization mechanisms remain less understood because prior results often restrict distributions or yield computational procedures rather than closed-form formulas.

  • Method

    Best-Guess reduces k-item n-bidder additive auctions to k-item 1-bidder auctions and uses β-exclusive revenue benchmarks and deterministic mechanisms.

  • Results

    Deterministic Best-Guess achieves a constant fraction of the best randomized mechanism, DSIC achieves a constant fraction of BIC under full independence, and 2nd-Price Bundling achieves the stated closed-form revenue in the identically distributed case.

  • Takeaways & Limitations

    The results provide simple mechanisms and revenue characterizations for arbitrary valuation distributions, including regular and irregular distributions.

  • Takeaways & Limitations

    The Best-Guess reduction may yield zero revenue for some valuation profiles, such as when all valuations are zero.

Abstract

from arXiv · show

In this paper, we introduce a novel approach for reducing the $k$-item $n$-bidder auction with additive valuation to $k$-item $1$-bidder auctions. This approach, called the \emph{Best-Guess} reduction, can be applied to address several central questions in optimal revenue auction theory such as the power of randomization, and Bayesian versus dominant-strategy implementations. First, when the items have independent valuation distributions, we present a deterministic mechanism called {\it Deterministic Best-Guess} that yields at least a constant fraction of the optimal revenue by any randomized mechanism. Second, if all the $nk$ valuation random variables are independent, the optimal revenue achievable in {\it dominant strategy incentive compatibility} (DSIC) is shown to be at least a constant fraction of that achievable in {\it Bayesian incentive compatibility} (BIC). Third, when all the $nk$ values are identically distributed according to a common one-dimensional distribution $F$, the optimal revenue is shown to be expressible in the closed form $Θ(k(r+\int_0^{mr} (1-F(x)^n) \ud x))$ where $r= sup_{x\geq 0} \, x(1 - F(x)^n)$ and $m=\lceil k/n\rceil$; this revenue is achievable by a simple mechanism called \emph{2nd-Price Bundling}. All our results apply to arbitrary distributions, regular or irregular.

1 Introduction

The paper studies revenue-maximizing incentive-compatible mechanisms for multi-item auctions, where the multiple-item setting remains harder and less understood than the single-item case. It introduces Best-Guess reduction to address open questions about mechanism simplicity, randomization, and DSIC versus BIC.

  • 1 Introduction: The single-item case is solved by Myerson under independently distributed bidder values, whereas the general multiple-item case is provably harder.Existing work includes computational methods for discrete inputs and simple mechanisms for approximating optimal revenue.
  • 1 Introduction: Prior results often restrict valuation distributions, while computational approaches typically provide optimization procedures rather than closed-form revenue formulas.These limitations motivate studying elegant formulas and simple mechanisms for broader settings.
  • 1 Introduction: Best-Guess reduces k-item n-bidder auctions with additive valuations to k-item 1-bidder auctions.The reduction is used to study randomization and Bayesian versus dominant-strategy implementations.
  • 1 Introduction: The paper presents Deterministic Best-Guess, which achieves at least a constant fraction of the best randomized mechanism when items have independent valuation distributions.This addresses the power of randomization in that setting.
  • 1 Introduction: It also gives constant-factor DSIC-to-BIC revenue comparison under mutual independence and a closed-form identically distributed case achieved by 2nd-Price Bundling.The stated results apply to arbitrary distributions, including regular and irregular ones.

2 Preliminaries

The preliminaries define multi-item auction mechanisms, incentive-compatibility and rationality conditions, revenue benchmarks, and β-exclusive mechanisms. They then introduce β-Bundling as a deterministic IR-IC mechanism and establish that it approximates optimal β-exclusive revenue for product distributions.

  • 2 Preliminaries: A mechanism specifies allocation probabilities and payments, with DSIR requiring nonnegative truthful utility and DSIC preventing profitable misreporting against fixed other reports.The Bayesian counterparts evaluate these conditions in expectation over other buyers’ values.
  • 2 Preliminaries: Seller revenue is expected total payment, and REV(F) and REVBayesian(F) optimize it over DSIR-DSIC and BIR-BIC mechanisms, respectively.DREV(F) denotes the supremum over deterministic DSIR-DSIC mechanisms.
  • 2 Preliminaries: The itemwise Vickrey mechanism awards each item to its highest bidder at the second-highest bid and earns the expected sum of second-highest values.This provides the lower bound REV(F) ≥ E[X[2nd]].
  • 2 Preliminaries: A β-exclusive mechanism never allocates item j when its bid is at most threshold β_j, and REV^X(L,β) is the best IR-IC revenue under this restriction.Myerson’s per-item reserve price is a special case of β-exclusion.
  • 2 Preliminaries: β-Bundling is deterministic and IR-IC, and for product distributions it achieves a constant fraction of optimal β-exclusive revenue.The mechanism uses thresholds and a possible bundle surcharge.

3 Main Results

The paper develops Best-Guess reductions from multi-bidder, multi-item auctions to single-bidder auctions, then derives constant-factor mechanisms and revenue characterizations under progressively stronger independence assumptions.

  • β-Exclusion Theorem: The β-Exclusion Theorem compares optimal β-exclusive and β-adjusted revenue, providing the key bridge for the Best-Guess reduction.For β-exclusive mechanisms, adjusted and ordinary revenue coincide, making the comparison directly useful for the reduction.
  • Best-Guess Reduction: Best-Guess runs separate single-bidder auctions using each bidder’s values and rivals’ itemwise maxima as the exclusion thresholds.Only the top bidder for each item can receive it, enforced through a β-exclusive mechanism optimized for the bidder’s conditional distribution.
  • Best-Guess Reduction: For independent item distributions, combining Best-Guess with a second-price mechanism guarantees a constant fraction of optimal revenue, while preserving DSIR and DSIC.The mechanism chooses the better of Best-Guess and the second-price Vickrey mechanism; Best-Guess alone can fail when all valuations equal the same constant.
  • Deterministic Best-Guess Reduction: Deterministic Best-Guess is DSIR-DSIC and gives an 8.5-approximate Best-Guess mechanism, yielding a deterministic constant-factor guarantee against optimal revenue.It uses deterministic single-bidder mechanisms based on each bidder’s conditional distribution and can likewise be combined with deterministic second-price Vickrey.
  • BIC versus DSIC: With independent value distributions, optimal BIC revenue is at most nine times optimal DSIC revenue.Thus, in this setting, Bayesian and dominant-strategy optimal revenues are equivalent up to a constant factor.
  • Second-Price Bundling: For iid values, optimal revenue is Θ(k(r + ∫_0^{mr}(1 − F(x)^n) dx)), and Second-Price Bundling achieves Θ(REV(F)) with universal constants.Here r = sup_x≥0 x(1 − F(x)) and m = ⌈k/n⌉; SPB assigns each item to a maximum bidder and offers that bidder the resulting bundle at a surcharge plus the second price.

4 Theorem 1: Effect of β-Exclusion

Theorem 1 bounds adjusted revenue using β-exclusive mechanisms by comparing shifted valuation distributions and constructing threshold-respecting mechanisms.

  • β-Exclusion Lemmas: REV A(Y_u, β) ≤ REV X(Y_u, u) for every u in the coordinatewise interval [0, β].Any IR-IC mechanism can be converted into a u-exclusive one without reducing adjusted revenue.
  • Proof Strategy: The proof first selects u so that REV A(L, β) ≤ REV A(Y_u, β), then establishes REV A(Y_u, β) ≤ REV A(Y_γ, β).The two-step comparison reduces the original distribution to a threshold-adjusted distribution.
  • Mechanism Construction: The constructed mechanism for the shifted distribution is IR, IC, and β-exclusive while preserving the relevant expected revenue.Its bids are transformed coordinatewise using max{z′_j, β_j}.
  • Menu Transformation: The proof modifies the original menu by deleting entries and lowering payments, with parameters a, b, and c optimized for the resulting bound.Profitable values determine the retained menu entries, and the lowered payments support the effective-payment comparison.
  • Revenue Bound: For each profitable value, the transformed mechanism’s effective payment is bounded below by a fraction of the corresponding original payment.Lemma 4.4 supplies the key payment inequality used to complete the β-exclusion argument.

5 Proof of Theorem 2

Theorem 2 analyzes Best-Guess mechanisms through an adjusted-revenue relaxation and shows that, with independent item distributions, Best-Guess is within a universal constant factor of optimal revenue.

  • Upper Bound: For arbitrary distributions, Best-Guess mechanisms are DSIR-DSIC, and optimal revenue is bounded above by adjusted Best-Guess revenue plus second-highest-value revenue.The upper-bound argument applies to every distribution F.
  • Independent Items: When item distributions are independent, Best-Guess revenue is within a universal constant factor of optimal revenue and separate-sale revenue.Theorem 5.3 states the resulting comparison with SREV(F).
  • Independent Items: BGA(F) ≤ 8 BGR(F) when the items’ valuation distributions are independent across items.Theorem 5.2 compares relaxed adjusted revenue with the exclusive Best-Guess revenue.
  • Proof Ingredients: The proof invokes a logarithmic approximation for one-bidder separate sales to derive the independent-item guarantee.The cited approximation result is used after applying the Best-Guess reduction.

6 Deterministic Best-Guess Reduction

The deterministic Best-Guess construction uses β-exclusive one-buyer mechanisms and a constant-factor bundling bound to obtain a deterministic approximation.

  • Preliminaries: A shifted distribution L − c is defined by translating the origin so that Pr{z > y} under L − c equals Pr{z > y + c} under L.This notation supports the one-buyer analysis used in the deterministic construction.
  • Preliminaries: The proof relates β-exclusive revenue to general mechanisms through the threshold exceedance probabilities ξ_j(L).Lemma 6.1 supplies the comparison between exclusive and unrestricted revenue terms.
  • Bundling Bound: REV X(L, β) ≤ 7.5 max{SREV(L), BREV(L)} for any product distribution L.The bound uses optimal separate selling and grand bundling as benchmarks.
  • Deterministic Best-Guess: β-Bundling is an 8.5-approximation to the ideal optimal β-exclusive mechanism.The proof identifies DBGR as deterministic DSIR-DSIC and establishes its BGRα guarantee with α = 8.5.
  • Deterministic Best-Guess: The deterministic mechanism’s guarantee follows by combining the 8.5 bundling approximation with the reduction theorem.Theorem 3 is obtained from the corollary to Theorem 2 and Theorem 6.1.

7 A General Implementation of Best-Guess Reduction

The general implementation transforms any approximately optimal one-buyer mechanism into a β-exclusive mechanism, extending Best-Guess beyond the specific deterministic bundling construction.

  • Approximation Guarantee: The resulting n-buyer mechanism has approximation ratio 1/(8α+9) when the input one-buyer mechanism has ratio α.The deterministic Best-Guess mechanism is the special case using the mechanism studied earlier.
  • Transformation Definition: Φ_β(M) is defined by comparing two transformed mechanisms and selecting the one with higher expected revenue.The first shifts bids and payments through M, while the second allocates items whose bids exceed β.
  • Definitions: An α-approximate mechanism is defined by achieving at least a 1/α fraction of the optimal revenue for the relevant distribution.This approximation notion is used throughout the general implementation theorem.
  • General Transformation: An α-approximate one-buyer mechanism for L^+_β − β transforms into a β-exclusive mechanism M′ for L.The transformation preserves IR and IC and provides the revenue guarantee stated in Theorem 7.1.

8 Bayesian vs. Dominant Strategy Revenue

The section bounds optimal BIC revenue by Best-Guess and separate-item revenue, then relates these quantities to DSIC revenue through the reduction framework.

  • REV_Bayesian(F) ≤ BGA(F) + SREV(F), separating Bayesian revenue into Best-Guess and separate-sale components.The bound is stated directly as the principal comparison for the Bayesian setting.
  • Each bidder-specific mechanism M_i is individually rational and incentive compatible after averaging allocation and payment over the other bidders.This constructs one-bidder mechanisms from an optimal BIR-BIC multi-bidder mechanism.
  • The analysis bounds each item’s maximum-value Myerson revenue by separate-item revenue, yielding REV(X_j[max]) ≤ SREV(F).
  • The Best-Guess and randomized Best-Guess revenues satisfy BGA(F) ≤ 8BGR(F) ≤ 8REV(F), connecting the Bayesian upper bound to feasible revenue benchmarks.

9 Optimal Revenue in I.D.D. Case

For identically distributed values, the paper characterizes optimal revenue using an auxiliary distribution and shows that 2nd-Price Bundling achieves the matching order under arbitrary distributions.

  • The auxiliary quantity satisfies r_F + C_l(F) ≤ A_l(F) ≤ 2r_F + C_l(F), relating monopoly revenue and the concentration term to A_l(F).
  • REV(F^n⊗k) = Θ(k A_m(F_hat)), where F_hat(x) = F(x)^n and m = ⌈k/n⌉.This restates the section’s main revenue characterization in terms of the auxiliary distribution F_hat.
  • REV(F) ≤ F_X^β(F) + ||β|| for any fixed β, providing the general adjusted-revenue upper bound used in the proof.
  • SPB is individually rational and incentive compatible because it makes take-or-leave offers and truthful reporting weakly maximizes utility.
  • 2nd-Price Bundling achieves expected revenue at least Ω(k A_m(F_hat)) in Case 2, completing the lower-bound analysis.The mechanism uses a parameter w selected according to the relative sizes of C_m(F_hat) and r_F_hat.

Note 1: Proof of Lemma 6.1

The proof of Lemma 6.1 transforms a β-exclusive mechanism into a nonnegative-support mechanism while preserving incentive compatibility and controlling adjusted revenue.

  • The constructed mechanism M′ maps valuations through ψ(z)_j = max{z_j, β_j} and selects a representative preimage maximizing adjusted payment.
  • β-exclusivity ensures M′ is individually rational because allocations occur only on coordinates exceeding the corresponding β threshold.
  • The construction preserves incentive compatibility by making utilities equal within each ψ-preimage, so misreporting provides no advantage.
  • A shifted mechanism M′′ on nonnegative valuations is defined by q′′(z)=q′(z+β) and s′′(z)=s′(z+β)−βq′(z+β), and remains IR and IC.

Note 5: Proof of Lemma 9.3

The proof establishes that 2nd-Price Bundling earns a constant-order fraction of the target benchmark by analyzing acceptance probabilities in two cases.

  • 2nd-Price Bundling’s expected revenue is bounded below through the probability that a bidder receives at least m qualifying items and accepts the offer.Fact 8 gives the key lower-bound form Pr{|T_i| ≥ m} P_w(m) · w.
  • The proof uses canonical pairs formed by each group’s maximum and second maximum values to express acceptance probabilities.
  • Case A: In Case A, where C_m(F_hat) is large relative to r_F_hat, the chosen offer parameter yields a constant lower bound on the relevant acceptance event.
  • Case B: In Case B, where C_m(F_hat) is small relative to r_F_hat, the analysis obtains wP_w(m) ≥ (1/20m)D_m(F_hat).
  • Combining Facts 8–10 yields the desired lower bound for both k ≤ n and k > n, with m = ⌈k/n⌉.
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