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

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, w^j)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 bⁱ_i = - _{j ∈N \{i}} cⁱ_j .

To prove Claim 4, denote by x = x(i) ∈ OSV ((°^j', w^j — 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, w^j)_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, x^k ~^k x(j)^k for j ∈ N, k = j, where x(j) ∈ OSV ((°^k, a^k — b^kke)_k∈N\{_j•}¢,

ii) ^P b_i^j = ^P bⁱ_j for all j ∈ N (by Claim 3, Bj =0, i.e., ^P bⁱ_j = —b^j_j ;

i∈N∖{j}       i∈N∖{j}                                                         i∈N∖{j}

moreover, the rules of the mechanism impose that    b_i^j =0, i.e., —b^j_j =       b_i^j).

i∈N                      i∈N∖{j}

Therefore, the allocation x is in the set OSV ((°^j , w^j)_j∙∈N) taking the matrix of con-
cessions cⁱ_j = bⁱ_j 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 bⁱ_j = cⁱ_j for every j ∈ N \{i} and b_iⁱ =

^—   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 z^j ~ y^j for every j ∈ N∕{i}, where y ∈ OSV ((°^j', w^j — (bj — B_i∕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, w^k — (bj^k — 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.

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