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Welfare-Preserving \(\varepsilon \)-BIC to BIC Transformation with Negligible Revenue Loss

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Web and Internet Economics (WINE 2021)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 13112))

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Abstract

In this paper, we provide a transform from an \(\varepsilon \)-BIC mechanism into an exactly BIC mechanism without any loss of social welfare and with additive and negligible revenue loss. This is the first \(\varepsilon \)-BIC to BIC transformation that preserves welfare and provides negligible revenue loss. The revenue loss bound is tight given the requirement to maintain social welfare. Previous \(\varepsilon \)-BIC to BIC transformations preserve social welfare but have no revenue guarantee [4], or suffer welfare loss while incurring a revenue loss with both a multiplicative and an additive term, e.g., [9, 14, 28]. The revenue loss achieved by our transformation is incomparable to these earlier approaches and can be significantly less. Our approach is different from the previous replica-surrogate matching methods and we directly make use of a directed and weighted type graph (induced by the types’ regret), one for each agent. The transformation runs a fractional rotation step and a payment reducing step iteratively to make the mechanism Bayesian incentive compatible. We also analyze \(\varepsilon \)-expected ex-post IC (\(\varepsilon \)-EEIC) mechanisms [18]. We provide a welfare-preserving transformation in this setting with the same revenue loss guarantee for uniform type distributions and give an impossibility result for non-uniform distributions. We apply the transform to linear-programming based and machine-learning based methods of automated mechanism design.

Part of the work was done when Zhe Feng was a PhD student at Harvard University, where he was supported by a Google PhD fellowship. See the full version of this paper for the complete proofs and Appendix at https://arxiv.org/pdf/2007.09579.pdf.

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Notes

  1. 1.

    This allocation-invariance is ex ante, i.e., it is with respect to the prior distribution over types.

  2. 2.

    For discrete type settings, 0-EEIC is exactly DSIC. For the continuous type case, a 0-EEIC mechanism is DSIC up to zero measure events.

  3. 3.

    In this paper, we consider the revenue and welfare performance of the untruthful mechanisms with truthful reports, which is commonly used in the literature. It is an interesting future direction to consider the performance of untruthful mechanisms under equilibrium reporting.

  4. 4.

    The previous \(\varepsilon \)-BIC to BIC transformations [9, 14, 28] don’t state the welfare loss guarantee clearly. Consider Example 1 shown in Sect. 1.3, the original \(\varepsilon \)-BIC mechanism already maximizes welfare and the optimal allocation is unique, any unmatched type in replica-surrogate matching creates a welfare loss. Particularly, the welfare loss is unbounded when (inappropriately) choosing \(\eta < \frac{\varepsilon }{\sqrt{m} - 1}\) in replica-surrogate matching.

  5. 5.

    Dughmi et al. [17] propose a general transformation from any black-box algorithm \(\mathcal {A}\) to a BIC mechanism that only incurs negligible loss of welfare, with only polynomial number queries to \(\mathcal {A}\), by using Bernoulli factory techniques. This approach has no guarantee on the revenue loss. Cai et al. [9] generalize Bernoulli factory techniques in the replica-surrogate matching to transform any \(\varepsilon \)-BIC mechanism to a BIC mechanism that only incurs negligible loss of revenue, with polynomial number queries to the original \(\varepsilon \)-BIC mechanism and polynomial number samples from the type distribution.

  6. 6.

    This need to transform an infeasible, IC mechanism into a feasible and IC mechanism also arises in [27], who use a method from [23] to correct for feasibility violations that result from statistical machine learning while preserving strategy-proofness.

  7. 7.

    By contrast, the previous transformations [9, 14, 28] cannot preserve the distribution of the allocation, even for the single agent and uniform type distribution case.

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Conitzer, V., Feng, Z., Parkes, D.C., Sodomka, E. (2022). Welfare-Preserving \(\varepsilon \)-BIC to BIC Transformation with Negligible Revenue Loss. In: Feldman, M., Fu, H., Talgam-Cohen, I. (eds) Web and Internet Economics. WINE 2021. Lecture Notes in Computer Science(), vol 13112. Springer, Cham. https://doi.org/10.1007/978-3-030-94676-0_5

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