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

concession cⁱ_j of agent i to agent j is equal to the concession c_i^j of agent j to i. Also, the
matrix of concessions associated with any OSV allocation is unique.

The bidding mechanism to implement the OSV is recursively defined as follows:

If there is only one agent {i}, he receives his initial endowments, so he obtains utility
uⁱ (wⁱ). (If only one player plays, there is no bidding stage.)

Given the rules of the mechanism for at most n - 1 agents, the mechanism for N =
{1,. . . ,n} proceeds as follows:

t =1: Each agent i ∈ N makes bids bⁱ_j ∈ <, one for every j ∈ N, with _j∈N bⁱ_j =0.
Hence, at this stage, a strategy for player i is a vector (bⁱ_j)j_∈N ∈ Hⁿ, where Hⁿ =
ⁿz ∈ <nlPj∈N Zj = o}.

For each i ∈ N , define the aggregate bid to player i by Bi = _j∈N b_i^j.Letα =
argmax_i (B_i ) where an arbitrary tie-breaking rule is used in the case of a non-unique
maximizer. Once the proposer α has been chosen, every player i ∈ N pays bⁱ_αe and
receives (B_α∕n) ∙ e.

t =2: The proposer α offers a feasible allocation (x¹,..., xⁿ) ∈ R^kn given the initial
resources (wⁱ)_i∈N.

t =3: The agents other than α, sequentially, either accept or reject the offer. If an
agent rejects it, then the offer is rejected. Otherwise, the offer is accepted.

If the offer is accepted, each agent i receives xⁱ . Therefore, the final payoff to an agent
i is uⁱ(xⁱ). On the other hand, if the offer is rejected, all players other than α proceed
to play the same game where the set of agents is N∖{α} and the initial resources for
these players are (wⁱ — (b^l_α — B_α∕n) e)_i∈N∖{_α}; while player α is on his own with resources
w^α - (b^α_α - B_α∕n)e. The final payoff to α is u^α(w^α - b_α^αe +(B_α∕n)e).Thefinal payoff to
any agent i = α is the payoff he obtains in the game played by N∖{α}.

3 The implementation for economies with at most
three agents

We start by proving several properties of the OSV allocations for economies with two
agents.

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