Abstract
We consider the problem of allocating several homogeneous indivisible goods when monetary transfers among agents are possible. We study the possibility of constructing strategy-proof and Pareto efficient mechanisms on restricted domains of agents’ valuation profiles. We show that there is no strategy-proof and Pareto efficient mechanism under the weak domain condition that all agents’ sets of possible valuations share at least four common valuations satisfying a certain inequality. Moreover, we prove that this impossibility result is robust to any affine transformation of domains, and we examine this impossibility result when the set of agent’s possible valuations consists of finite integers.
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Notes
We mention two important contributions in quasilinear environments. For queueing problems, Mitra (2001) provides a characterization of restricted domains (e.g., linear waiting costs) for the existence of a strategy-proof and Pareto efficient mechanism. For the public good provision, Liu and Tian (1999) establish a characterization of restricted domains (e.g., quadratic preferences) for the existence of a strategy-proof and Pareto efficient mechanism.
Schummer (2000) provides a comprehensive set of results about strategy-proof mechanisms in the problem of allocating heterogeneous indivisible goods when monetary transfers are possible. The second result is essentially the same as his impossibility result (Theorem 5) in the case of two agents and two heterogeneous indivisible goods.
The definitions depend on n and s. Because there is no fear of confusion, we omit to write n and s in the definitions.
When \(k=0\), this sentence is not necessary for the discussion.
When \(k=s\), we can understand the discussion more easily by replacing \(t_{i}(d_{i},v_{-i}^{S})\) with \(t_{i}(d_{i},v_{-i}^{S\backslash \{i\}})\).
When \(k=n\), this sentence is not necessary for the discussion.
It is important to study general models where each agent receives any number of the (homogeneous or heterogeneous) indivisible goods. We predict the impossibility results of strategy-proof and Pareto efficient mechanisms on small domains. However, Theorem 1 can not be easily extended to those models since the structure of decision-efficient allocations of indivisible goods is more complicated (in particular, when there are ties). Therefore, we leave the question of whether the impossibility results hold on small domains in those models.
The existence of four common valuations requires \(l\ge 4\), and we will show that the premise of this theorem requires \(l\ge 4\).
Ohseto (2000) shows that when \(s=1\) and \(l>\frac{2^{n}+2n^{2}-n-3}{n-1}\), there is no strategy-proof and Pareto efficient mechanism. The minimum l in the new result is less than half of the minimum l in his result since \(\frac{2^{n}+2n^{2}-n-3}{n-1}\ge 2(\frac{2^{n-1} +2n-3}{n-1})+1\). Therefore, the new result is a stronger impossibility result.
Let \(n=2,\) \(s=1,\) \(v_{1}=3,\) \(v_{2}=2\), and \(Z^{\prime }=\{((s_{1},t_{1}),(s_{2},t_{2}))\in Z\mid \) \(t_{1}\in \{4m|\) m is an integer\(\}\) and \(t_{2}\in \{5m|\) m is an integer\(\}\}\). Then, a feasible allocation \(((s_{1},t_{1}),(s_{2},t_{2}))=((0,-8),(1,5))\) is neither decision-efficient nor budget balanced, but is Pareto efficient.
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Acknowledgements
We would like to thank an associate editor, two anonymous referees, Miki Kato, and Zhen Zhao for helpful suggestions and detailed comments. This research was partially supported by JSPS KAKENHI Grant number 17K03614.
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Ohseto, S. Strategy-proof and Pareto efficient allocation of indivisible goods: general impossibility domains. Int J Game Theory 50, 419–432 (2021). https://doi.org/10.1007/s00182-021-00754-4
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DOI: https://doi.org/10.1007/s00182-021-00754-4