Implementation of the Ordinal Shapley Value for a three-agent economy



1 Introduction

The Ordinal Shapley Value (OSV ) is a way to allocate gains realized by cooperation
in general economic environments. It is invariant with respect to the utility representa-
tion of the agents’ preferences and enjoys several desirable properties such as efficiency,
monotonicity, anonymity, and individual rationality (see Pérez-Castrillo and Wettstein
[2005]). It provides a reasonable outcome for a large class of environments even where
competitive equilibria or core allocations may fail to exist.

The OSV is a normative solution concept. An alternative approach to the problem of
sharing gains from cooperation consists of proposing mechanisms whose equilibria yield
“good” outcomes.1 In this paper, we propose the use of a bidding mechanism, which
combines and adapts to exchange economies proposals suggested in Pérez-Castrillo and
Wettstein [2001] and [2002]. Informally, the mechanism proceeds as follows: In stage 1
the agents bid to choose the proposer. Each agent bids by submitting an n-tuple of real
numbers, one number for each agent (including himself). The number submitted by agent
i for an agent j , is a commitment to forego a commodity bundle in case j is chosen as the
proposer. The bids submitted by an agent must sum up to zero. The agent for whom the
aggregate bid (sum of bids submitted for him by all agents including himself) is the highest
is chosen as the proposer. Before moving to stage 2, all the agents (including the proposer)
pay the “bid” (i.e., the promised commodity bundles) they submitted for the proposer. In
stage 2 the proposer offers a feasible allocation of the total initial resources. The offer is
accepted if all the other agents agree. In case of acceptance each agent receives the bundle
suggested for him in this allocation. In the case of rejection all the agents other than the
proposer play the same game again where the new initial endowments incorporate the
allocations paid and received by the end of stage 1.

We prove that the proposed bidding mechanism implements in Subgame Perfect equi-
librium the OSV correspondence for economies with at most three agents.

1 See Moore and Repullo [1988] and Maniquet [2003] for papers dealing with implementation in general
environments. Winter [1994], Dasgupta and Chiu [1996] and Vidal-Puga and Bergantinos [2003] deal with
the implementation of the Shapley value in Transferable Utility games.



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