Ordinal allocation
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Ordinal allocation Christopher P. Chambers1 · Michael Richter2 Received: 4 September 2019 / Accepted: 19 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract A generalization of the well-known Vickrey auctions are lottery qualification auctions–where the m highest bidders win the good with uniform probability, and pay the m + 1 st highest bid upon winning. A random lottery qualification mechanism decides the integer m randomly. We characterize the class of mechanisms which are payoff equivalent to the random lottery qualification auctions. The key property characterizing this class of mechanisms is one which states that only the ordinal comparison of willingness-to-pay across individuals is relevant in determining the allocation. The mechanisms can be seen as compromising between ex-post utility efficiency and monetary efficiency.
1 Introduction A mechanism designer seeks to allocate an object. She wants to base this allocation on the participants’ valuations, but she is not necessarily concerned with revenue maximization. For example, a government may worry that allocating the object to the participant with the highest valuation may tend to induce a monopoly. Or, there may be legal constraints involved with using money in certain ways; as in kidney exchange. To this end, we discuss, in a private-value setting, a class of mechanisms which naturally generalize the Vickrey mechanism. These mechanisms do not guarantee ex-post efficiency (Green and Laffont 1977), the property that the good goes to a Dedicated to John Weymark with sincere thanks. We are grateful to an anonymous referee for very useful comments and suggestions. * Christopher P. Chambers [email protected] Michael Richter [email protected] 1
Department of Economics, Georgetown University, Washington, DC, USA
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Department of Economics, Royal Holloway, Egham, UK
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participant with the highest valuation. The building blocks of the mechanisms we study here are not new: they generalize the Vickrey (Vickrey 1961; Clarke 1971; Groves 1973) auction in a natural way. We imagine a single-unit auction with transfers and quasilinear preferences. The class of lottery qualification auctions, introduced by Bordley and Harstad (1996), is parametrized by an integer. The participants bid, and the m highest bidders each have a (uniform) chance of winning the object. The winner gets the object, and pays the m + 1 st highest price. This primitive class generalizes both the second-price Vickrey auction (when m = 1 ), and a straight uniform lottery with no transfers (as in models of kidney exchange Roth et al. 2004). Ex-post efficiency (Green and Laffont 1977) usually has the interpretation that, after all transfers have been paid (the money presumably burned or given to the mechanism designer), no reallocation of the goods (or money) results in Pareto dominance. However, in a well-known phenomenon in mechanism design (Green and Laffont 1977; Holmström 1979; Laffont and Mask
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