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, wj — (j — Bα∕n)e)j∙∈N∖{α^ .