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Efficient Combinatorial Allocations: Individual Rationality versus Stability

Hitoshi Matsushima

Abstract

We investigate combinatorial allocations with opt-out types and clarify the possibility of achieving efficiency under incomplete information. We introduce two distinct collective decision procedures. The first procedure assumes that the central planner designs a mechanism and players have the option to exit. The mechanism requires interim individual rationality. The second procedure assumes that players design a mechanism by committing themselves to participate. The mechanism requires marginal stability against blocking behavior by the largest proper coalitions. We show that the central planner can earn non-negative revenue in the first procedure, if and only if he cannot do so in the second.

JEL Classification: D44; D61; D82

Notes

This study was supported by a grant-in-aid for scientific research (KAKENHI 25285059) from the Japan Society for the Promotion of Science (JSPS) and the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of the Japanese government.


Appendix

Proof of Lemma 1

From payoff equivalence, without loss of generality, we can assume x~X^. According to Arrow (1979) and D’Aspremont and Gérard-Varet (1979), it is evident that there exists xX such that (g,x) satisfies BIC and the balanced budgets in the sense that

iNxi(ω)=0 for all ωΩ.

From payoff equivalence, it is evident that there exists (bi)iNRn such that

E[x~i(ω)|ωi]=E[xi(ω)|ωi]+bi for all iN and all ωiΩi,

where note that

iNbi=E[iNx~i(ω)].

We specify xX by

xi(ω)=xi(ω)+bi for all iN and all ωΩ.

From this specification, it is clear that (g,x) satisfies BIC,

E[xi(ω)|ωi]=E[xi(ω)|ωi]+bi

=E[x~i(ω)|ωi] for all iN and ωiΩi,

and

iNxi(ω)=iNxi(ω)+iNbi=E[iNx~i(ω)] for all ωΩ.

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Published Online: 2018-01-04

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