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

2 The Ordinal Shapley Value and the Bidding Mech-
anism

We consider a pure exchange economy E consisting of a set N = {1, 2, ..., n} of agents and
k ≥ 2 commodities. Agent i ∈ N is described by {°ⁱ, wⁱ}, where wⁱ ∈ R^k is the vector of
initial endowments and °^l is a continuous and monotonic preference relation defined over
R^k. We denote by Âⁱ and ~^l the strict preference and indifference relationships associated
with °i, and e ≡ (1,..., 1) ∈ R^k.

For this economy, Pérez-Castrillo and Wettstein [2005] propose and prove the existence
of a solution concept, called the Ordinal Shapley Value (OSV ), the construction of which
relies on the notion of concessions. The use of concessions allows to “measure” the benefits
from cooperation. Concessions are made in terms of the reference bundle e.

Definition 1 The Ordinal Shapley Value is defined recursively.

(n = 1) In the case of an economy with one agent with preferences °¹ and initial
endowments α¹ ∈ R^k, the OSV is given by the initial endowment: OSV(°¹,α¹) = {a¹} .

For n ≥ 2, suppose that the solution has been defined for any economy with (n - 1) or
less agents.

(n) In the case of an economy (°i,ai)i∈N with a set N of n agents, the OSV ((°^l,a^l)_l∈N)
is the set of efficient allocations (xⁱ)i∈N for which there exists an n-tuple of concession
vectors (cⁱ)i∈N,cⁱ ∈ R^n-1 for all i ∈ N that satisfy:

n.1) for all j ∈ N, there exists y(j) ∈ OSV ((°i,ai + cje)i∈N∖_j∙) such that x^l ~^г y(j)i
for all i ∈ N\j , and

n.2) ^P c_i^j = ^P cⁱ_j for all j ∈ N.

i∈N \j      i∈N \j

By definition, the OSV is efficient.Itisalsoconsistent in the sense that any set of
(n - 1) agents is indifferent between keeping their allocation or taking the concessions
made by the remaining agent and reapplying the solution concept to the (n - 1) economy
(property n.1). Moreover, to ensure fairness, the concessions balance out (property n.2).
In fact, Pérez-Castrillo and Wettstein [2005] show that the concessions associated with
OSV allocations satisfy the stronger condition that they are symmetric, that is, the

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