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 b
10 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)jN). 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)jN{i}) the proposal that
agent i
N makes if he is chosen as the proposer (we notice that Claim 3 states that
B
i = 0). We are going to prove that x OSV ((°j, wj)jN). First, according to Claim 2,
x is a Pareto efficient allocation. Moreover, the n—tuple of vectors of bids
(bi)iN satisfies:

i) By Claim 3, xk ~k x(j)k for j N, k = j, where x(j) OSV ((°k, ak bkke)kN\{j•}¢,

ii) P bij = P bij for all j N (by Claim 3, Bj =0, i.e., P bij = —bjj ;

iN{j}       iN{j}                                                         iN{j}

moreover, the rules of the mechanism impose that    bij =0, i.e., —bjj =       bij).

iN                      iN{j}

Therefore, the allocation x is in the set OSV ((°j , wj)jN) taking the matrix of con-
cessions c
ij = bij for all i, j N, i 6= j.

(b) We now prove every allocation x in the set OSV ((°i, wi)iN) 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 z
j ~ yj for every j N∕{i}, where y OSV ((°j', wj (bj Bi∕n)e)jN{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
z
i % yi, where y OSV (°°k, wk — (bjk Bj/n)e)kN{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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