Reallocation mechanisms

Abstract

We consider exchange economies, where the initial endowment of each agent consists of some set of resources. The private information of the agents is their value for every possible subset of all the resources. The goal is to redistribute resources among agents to maximize efficiency, where monetary transfers are allowed. As agents may simultaneously play the role of buyers and sellers, the standard method for implementing efficient outcomes -VCG mechanisms -may not be budget balanced. In fact, it is known (Myerson and Satterthwaite, 1983) that even in the simplest bilateral-trade setting no mechanism can simultaneously achieve full efficiency, individual rationality and budget balance in (Bayes Nash) equilibrium.In this paper, we develop incentive-compatible, individually-rational and budget balanced mechanisms for several classic settings. All our mechanisms (except one) provide a constant approximation to the optimal efficiency in these settings, even in ones where the preferences of the agents are complex multi-parameter functions. Some of our mechanisms achieve approximation guarantee in expectation, given prior distributions on the preferences of the agents, and other approximation results hold for every realization of the preferences.We start with analyzing the simple one-buyer one-seller bilateral trade setting, where some design challenges start revealing. We present a mechanism that achieves in expectation at least half of the expected value of the efficient allocation. We then show how this mechanism can be used to achieve the same approximation ratio for the more general Partnership Dissolving setting, where several agents initially hold fractions of a divisible item.Both bilateral trade and partnership dissolving are single-parameter domains. Our main technical results are two mechanisms for more complex multi-parameter domains. The first domain is combinatorial exchange, where a set of m indivisible heterogeneous items is initially distributed among the agents. Agents have combinatorial sub-additive preferences. We show that there exists a truthful, individually rational, weakly budget balanced mechanism that provides an 8Hm-approximation to the optimal welfare (where Hm is the m'th harmonic number). Our second multi-parameter domain is the classic exchange model of Arrow and Debreu, where agents have concave valuation functions for a single divisible good. We devise a truthful, prior-free, individually rational, weakly budget balanced mechanism that provides a constant approximation to the optimal social welfare (as long as no single agent holds a huge chunk of the good).

Approximately Efficient Two-Sided Combinatorial Auctions

“…Technically, one could e.g. apply it to the WBB mechanism of Blumrosen and Dobzinski [1] for combinatorial exchange markets; however, the trick is unsatisfactory in practice as it essentially consists of giving the le over money to a random agent. is demonstrates a weakness in the current de nition of SBB, which motivates the introduction of a strengthened version, that we call direct-trade strong budget balance (DSBB).…”

“…Anyway all the mechanisms proposed in this paper are oblivious to the order in which sellers and buyers are presented. 2 e mechanism proposed in [1] achieves a constant approximation to the optimal social welfare if the size of the initial endowment of each agent is bounded by a constant; otherwise the approximation factor is of logarithmic order. 3 is question also applies to standard demand queries [10], which may be computationally hard to answer or may involve To additionally achieve a mechanism that results in a high social welfare, we exclude some items from trade and introduce randomness into the mechanism.…”

“…e mechanism is IR, SBB and DSIC and its gain from trade approaches the optimum when the market is su ciently large. Mechanisms that are IC, IR, and SBB have been given for bilateral trade in [1]. In addition to this, the authors proposed a WBB mechanism for a general class of combinatorial exchange markets.…”

“…However, it is known from the work of Myerson and Sa erthwaite [17] that it is generally impossible for an IR, BIC, and WBB mechanism to maximise social welfare in such a market, even in the bilateral trade se ing, i.e., when there is just one seller and one buyer. 1 e practical contexts noted above need the application of two-sided market mechanisms that can work in a combinatorial se ing, i.e., where there are multiple distinct items in the market and agents having possibly complex valuations over the subsets of items that they may receive. However, we are not aware of any such mechanism that approximates the social welfare while meeting the IR, DSIC and SBB requirements.…”

Modularity and greed in double auctions

“…This mechanism has been generalized to more complex settings in [2,8,18,3,10]. More recent work that falls into this category is [24,6] (where [24] actually applies ex post incentive compatibility, as appropriate for interdependent values). A second line of work that seeks to escape the impossibility results was recently initiated by [9], by analyzing the trade-off between incentives and efficiency while insisting on budget balance.…”