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

Lemma 1 (a) For a two-agent economy, both agents are indifferent among the OSV
allocations.

(b) The concession c is the same for every OSV allocation and is a continuous function
of the initial endowments.

(c) Let x ∈ OSV ((°i, wi)i=₁,₂) and x⁰ ∈ OSV ((°i,wi + λe)i=₁,₂), λ > 0. Then,
xⁱ⁰ Âⁱ xⁱ for i =1, 2.

Proof. (a) As shown in Pérez-Castrillo and Wettstein [2004], the OSV for a two-
agent economy consists of the efficient allocations (x¹,x²) such that x¹ ~ w¹ + ce and
x² ~ w² + ce for some c ≥ 0. Consider now x,y ∈ OSV ((°i, ai)i=₁,₂) and denote by
c = c¹₂ = c₁² and d = d₂¹ = d²₁ the concessions associated respectively with x and y. Then,
x¹ ^¹ a¹ + ce, x² ^² a² + ce, y¹ ^¹ a¹ + de, and y² ^² a² + de. It is immediate that x¹ √¹ y¹
if and only if x² √² y². The efficiency of both allocations x and y implies x¹ ^¹ y¹ and
X² ^² y².

(b) The previous argument also shows that c = d, while the continuity of preferences
implies that the concession varies continuously with the initial endowments.

(c) There exist c and c⁰ such that: x¹ ~ w¹ + ce, x² ~ w² + ce, x⁰¹ ~ w¹ + (λ + c⁰)e
x⁰² ~ w² + (λ + c⁰)e. The allocation x⁰ is Pareto efficient in (°i, wⁱ + λe)_i=₁,₂ whereas x is
feasible, yet not Pareto efficient for the economy (°i, wⁱ + λe)_i=₁,₂. Hence, it must be the
case that λ + c⁰ > c and x⁰ⁱ Âi xⁱ for i = 1, 2. ■

The next theorem shows that the set of Subgame Perfect equilibrium outcomes (SP E)
of the bidding mechanism coincides with the OSV for economies with at most three agents.

Theorem 1 The bidding mechanism implements the Ordinal Shapley Value in Subgame
Perfect Equilibrium in economies with n ≤ 3.

Proof. The proof proceeds by induction.

(a) We first prove that every SPE outcome of the bidding mechanism is in the
OSV ((°i,wi)i∈N).

For n = 1 the proof is trivial. Note also that for economies with one agent, there is
only one OSV allocation.

Claim 1. In any SPE, any agent i different than the proposer α accepts the proposal x
at t = 3 if xi  yi for every i ∈ N∖{α}, where y ∈ OSV ((°^j, w^j — (j — B_α∕n)e)_j∙∈N∖{_α^ .

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