Abstract
We propose a measure of approximate ex-ante Pareto efficiency in matching markets. According to this measure, a lottery over matchings is \(\gamma \)-approximately efficient if there is no alternate lottery in which each agent’s ex-ante expected utility increases by an \(\gamma \) factor. A mechanism is \(\gamma \)-approximately efficient if every lottery produced in equilibrium is \(\gamma \)-approximately efficient. We argue this is the natural extension of approximate efficiency in transferable-utility settings to our nontransferable-utility setting. Using this notion, we are able to quantify the intuited efficiency improvement of the so-called Boston mechanism and the recently-proposed choice-augmented deferred acceptance mechanism over the random serial dictatorship mechanism. Furthermore, we provide the first formal statement and analysis of the Raffle mechanism, which is conceptually simpler than the Boston mechanism and has a comparable efficiency guarantee.
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Notes
- 1.
See also the considerable research in the optimization literature on the so-called “Santa Claus problem,” or maximin welfare optimization [7]. We are not aware of work that studies the maximin welfare of existing matching mechanisms.
- 2.
It is not hard to make this example more extreme in a market with n agents, producing an approximately efficient lottery with ex ante value 1 for all agents whereas the efficient lottery has value 1 for all but one agent and value n for a select agent.
- 3.
For example, the lottery that with probability 1/2 allocates the skateboard to Jieming and the bicycle to Jolene, and otherwise allocates nothing to either, would only be 4-approximate under this stricter definition, since an alternative would be to always allocate the bicycle to Jolene and the skateboard to Jieming.
- 4.
By the Birkoff von Neumann theorem, a lottery is feasible if and only if it is a distribution over deterministic assignments in which each consumer is assigned at most one good and each good is not over-capacitated.
- 5.
Another mechanism, the “Probabilistic Serial (PS)” mechanism, is fairly elaborate to describe so we do not formally do so here [8]. In the continuum model on which we focus most of our analysis, this mechanism is equivalent to RSD [17]. Thus our negative results on RSD apply to the PS mechanism in this continuum model.
- 6.
As they focus on the school choice setting, they assume goods have priorities and actually build off of c-DA, but as noted above, c-DA with single tie-breaking and RSD are equivalent in our setting.
- 7.
To handle discontinuities due to tie-breaking at 0 ticket allocation for under-demanded goods, we impose a technical restriction that if \(x^i_j > 0\) then \(x^i_j \ge \epsilon \) for some arbitrarily small \(\epsilon > 0\). This restriction impacts all value calculations by a quantity proportional to \(\epsilon \) which tends to 0, so we will omit these terms for brevity.
- 8.
This limit can be infinite if good j is underdemanded.
- 9.
In the full version of the paper, we discuss how to approximate these marginal gains—and hence the order \(\pi \)—in polynomial time, and argue that a small approximation error leads only to a small loss in approximation factor.
- 10.
This also implies a linear lower bound for the consumer-proposing deferred acceptance mechanism (c-DA). See the full version of the paper for more details.
- 11.
See, for example, the displayed information for the last round of the Singapore system at http://www.hdb.gov.sg/cs/infoweb/residential/buying-a-flat/new/application-status.
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Acknowledgements
We are grateful to Eric Budish, Peng Shi and especially Christina Lee for useful comments. All errors are our own.
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Immorlica, N., Lucier, B., Weyl, G., Mollner, J. (2017). Approximate Efficiency in Matching Markets. In: R. Devanur, N., Lu, P. (eds) Web and Internet Economics. WINE 2017. Lecture Notes in Computer Science(), vol 10660. Springer, Cham. https://doi.org/10.1007/978-3-319-71924-5_18
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