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

Second, if they make these bids, the proposal at t =2will certainly be x, given that
x is efficient and guaranties the rest of the agents their OSV of the game without the
proposer and with the proposer’s concessions added to their initial endowment. Hence, if
the agents follow the previous strategies, the final outcome is always x.

We prove that the strategies are indeed SPE strategies. By the induction argument,
what the agents other than the proposer, say agent j, expect after the bids is some
allocation in OSV ((°^k, wk — (bj — Bj/n)e)_k∈N∖{_j∙}) . Therefore, it is easy to check that
the previous strategies are SPE strategies from t =2 on. Consider now the strategies at
t =1. Remember that we have shown that B_i =0for all i ∈ N. If agent i changes his
bid, the proposer will be the agent (or one of the set of agents) to whom i increases his
bid. Denote by α the proposer, and B_α⁰ > 0 the new aggregate bid. If α = i, then the
other agents will face a situation where all their initial endowments increase by the same
amount B_i⁰/n. By Lemma 1 (c), all these agents are better off in the new situation, hence
agent i is worse off. If the new proposer is α 6= i, then the outside option for agent i will
be a situation where all the agents other than i and α will see their initial endowment
increased by B_α⁰ /n while agent i’s initial endowment will decrease by (n — 1)B_α⁰ /n. An
argument similar to that of Lemma 1 (c) shows that agent i’s situation is worse off after
the change. Therefore, deviating is not profitable. ■

The major difficulty with extending this result for any number of agents is intimately
related to the transfer paradox (Safra [1984]). We briefly explain here this difficulty.

It is crucial for our result that (as stated in Claim 3) the equilibrium aggregate bids
are zero for every agent. For this result, it must be the case that a proposer can not
gain by increasing the bid for himself and facing at the proposal stage agents with larger
endowments. However, similar to the transfer paradox, an agent can be worse off in an
OSV allocation when the initial endowments of all agents (including himself) increase. If
this happens, the proposer may find it “easier” (less costly in terms of his own welfare)
to make an acceptable proposal to the set of agents with larger endowments.

The mechanism constructed provides a non-cooperative foundation for the OSV for
environments with a small number of agents. It also shows that the concessions underlying
the OSV concept can be interpreted in terms of bids.

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