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



concession cij of agent i to agent j is equal to the concession cij 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
u
i (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 bij <, one for every j N, with jN bij =0.
Hence, at this stage, a strategy for player i is a vector
(bij)jN Hn, where Hn =
nz <nlPjN Zj = o}.

For each i N , define the aggregate bid to player i by Bi = jN bij.Letα =
argmaxi (Bi ) 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 biαe and
receives
(Bα∕n) ∙ e.

t =2: The proposer α offers a feasible allocation (x1,..., xn) Rkn given the initial
resources
(wi)iN.

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 u
i(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 (blα — Bα∕n) e)iN{α}; while player α is on his own with resources
wα - (bαα - Bα∕n)e. The final payoff to α is uα(wα - bααe +(Bα∕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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