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Combinatorial Reallocation Mechanisms

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Abstract

We consider reallocation problems in settings where the initial endowment of each agent consists of a subset of the resources. The private information of the players is their value for every possible subset of the resources. The goal is to redistribute resources among agents to maximize efficiency. Monetary transfers are allowed, but participation is voluntary. We develop incentive-compatible, individually-rational and budget-balanced mechanisms for two settings in which agents have complex multi-parameter valuations, both settings include double auctions as a special case. The first setting is combinatorial exchanges, where we provide a mechanism that achieves a logarithmic approximation to the optimal efficiency when valuations are subadditive. The second setting is Arrow–Debreu markets for a single divisible good, where we present a constant approximation mechanism. The first result is given for a Bayesian setting, where the latter result is for prior-free environments.

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Notes

  1. The budget balance requirement is common in the cost-sharing literature (e.g., [34, 37]) but there the idea is to charge the participants an amount that suffices to cover the cost of providing the service.

  2. In fact, using noisy estimations of the medians decreases the performance of our mechanism in a rate proportional to the noise. Hence, even if we only have a black box access to the distributions, we can use the black box to estimate the medians within an arbitrary precision, and preserve very similar performance guarantees.

  3. A valuation v is subadditive if for ever two bundles S and T we have that \(v(S)+v(T)\ge v(S\cup T)\).

  4. If the optimal welfare is smaller than \(H_n\cdot r\), then the mechanism is not required to allocate the items, but if it does so, the revenue is guaranteed to be at least r.

  5. Our general model allows limitations on the endowments, for example, we may impose a size limit. Our main result for combinatorial exchanges does not impose limits on the allowed endowments, while our result for Arrow–Debreu markets assumes that the endowments are not too large.

  6. Note that this definition holds for every realization of v, i.e., ex post (and not only in expectation, which is usually a key for achieving budget balance in Bayesian domains, e.g., in [15]).

  7. The items are produced only if a sale is made. Here we assume for simplicity that the cost of producing the first item is r and the cost of producing any additional item is 0. This corresponds to the case that the production cost of items is governed by the start-up cost. A more realistic setup assumes a production cost for each item, or more generally for bundles of items. Indeed, the mechanism of Sect. 3.2 essentially provides a solution for this case.

  8. XOS is a subclass of subadditive valuations that includes all submodular valuations, see [30].

  9. When \(v_i(\cdot )\) is twice differentiable, we simply assume that \(v_i^{\prime \prime }(x) \le 0\) for every x.

  10. To accomplish that, greedily add agents to \(N_1\) while the sum of the \(r_i\)’s of agents in \(N_1\) is at least \(\frac{1}{4}\). Stop adding agents to \(N_1\) when we add an agent that makes an “overflow”: \(\Sigma _{i\in N_1}r_i> \frac{1}{4}\). Since each \(r_i\le \frac{1}{8}\), we also have that \(\Sigma _{i\in N_1}r_i\le \frac{3}{8}\). Continue similarly, only with agents that were not added to \(N_1\), to construct \(N_2\) and \(N_3\).

  11. Formally, order the buyers arbitrarily, and let \(x_i=\max \{0,\frac{1}{8} - \Sigma _{i'>i}x'_{i'}\}\) if \(x'_i+\Sigma _{i'>i}x'_{i'}\ge \frac{1}{8}\) (otherwise, \(x_i=x'_i\)).

  12. To see this, note that the derivative of \(x\cdot p\) is p for every x. Since \(x_i^{\prime }\) maximizes profit, and since \(v_i\) satisfies the decreasing marginals property, for every value \(x<x_i^{\prime }\) the derivative of \(v_i(r_i-x)\) is negative with absolute value of at most p. Therefore, the marginal profit is non-negative for \(x<x_i^{\prime }\). A similar argument holds also when \(v_i\) is not differentiable.

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Acknowledgements

The first author was supported by Israel Science Foundation grants number 2570/19 and by the Asper center at the Hebrew University Business School. The second author supported by the U.S.-Israel Binational Science Foundation grant number 2016192 and Israel Science Foundation grant number 2185/19.

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  1. The Hebrew University Business School, Jerusalem, Israel

    Liad Blumrosen

  2. Computer Science and Applied Math, Weizmann Institute of Science, Rehovot, Israel

    Shahar Dobzinski

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L.B. and S.D. wrote the whole paper together.

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Correspondence to Liad Blumrosen.

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A preliminary version of the results in this paper was presented in ACM EC 2014 under the title “Reallocation Mechanisms” and appeared in a 1-page abstract [5].

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Blumrosen, L., Dobzinski, S. Combinatorial Reallocation Mechanisms. Algorithmica 86, 1246–1262 (2024). https://doi.org/10.1007/s00453-023-01191-3

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