all agents in N \{1} are worse off in the OSV of the second economy than in the OSV
of the former. Hence, at stage 2, agent 1 can offer an allocation that is worse off for all
j ∈ N \{1} and, by Pareto efficiency, better off for himself. Therefore, agent 1 is better
off after bidding according to b10 than after bidding according to b1.
Claim 4. In any SPE, the offer x made by the proposer at t = 2 always belongs to
OSV ((°j, wj)j∈N). Moreover, the agents’ bids at t = 1 are bj = cj for all i,j ∈ N, i = j,
where c is the matrix of concessions of x, and bii = - j ∈N \{i} cij .
To prove Claim 4, denote by x = x(i) ∈ OSV ((°j', wj — bje)j∙∈N∖{i}) the proposal that
agent i ∈ N makes if he is chosen as the proposer (we notice that Claim 3 states that
Bi = 0). We are going to prove that x ∈ OSV ((°j, wj)j∙∈N). First, according to Claim 2,
x is a Pareto efficient allocation. Moreover, the n—tuple of vectors of bids (bi)i∈N satisfies:
i) By Claim 3, xk ~k x(j)k for j ∈ N, k = j, where x(j) ∈ OSV ((°k, ak — bkke)k∈N\{j•}¢,
ii) P bij = P bij for all j ∈ N (by Claim 3, Bj =0, i.e., P bij = —bjj ;
i∈N∖{j} i∈N∖{j} i∈N∖{j}
moreover, the rules of the mechanism impose that bij =0, i.e., —bjj = bij).
i∈N i∈N∖{j}
Therefore, the allocation x is in the set OSV ((°j , wj)j∙∈N) taking the matrix of con-
cessions cij = bij for all i, j ∈ N, i 6= j.
(b) We now prove every allocation x in the set OSV ((°i, wi)i∈N) is an SPE outcome
of the bidding mechanism. We denote c the matrix of concessions of x. We propose the
following strategies for the case of n agents:
At t =1, each agent i, i ∈ N, announces bij = cij for every j ∈ N \{i} and bii =
— j ∈N∖{i} cij .
At t =2, agent i, if he is the proposer, proposes an allocation z that is Pareto efficient
and satisfies that zj ~ yj for every j ∈ N∕{i}, where y ∈ OSV ((°j', wj — (bj — Bi∕n)e)j∈N∖{i}) .
(We recall that, according to Lemma 1 (a), in economies with one or two agents either
there is only one OSV allocation or agents are indifferent among the several OSV allo-
cations.)
At t =3, agent i, if agent j ∈ N/{i} is the proposer, accepts any offer z such that
zi % yi, where y ∈ OSV (°°k, wk — (bjk — Bj/n)e)k∈N∖{j∙}) and rejects it otherwise.
First of all, we notice that if the agents make the previous bids, then the aggregate
bid to each one is zero. This is a direct consequence of the symmetry of the concessions.