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

Proof: First, by induction, in case of rejection the agents expect to obtain an allocation
in the OSV in the economy without the proposer (and where the concessions have been
added to or substracted from their initial endowment); second, by Lemma 1 (a), agents in
a two-agent economy are indifferent among OSV allocations (as is the case for a one-agent
economy).

Claim 2. In any SPE of the game that starts at t =2, the proposer α proposes
an allocation x that is Pareto efficient and satisfies xⁱ ~ yi for every i ∈ N∕{α}, where
y = OSV (°^{j ,w}j - (bα ^- BαM^e)j∈N\{_α}} . Moreover, every agent i ∈ N∖{α} accepts
any offer x such that xⁱ % yi for every i ∈ N∕{α}.

Proof: These are clearly equilibrium strategies for the agents other than the proposer.
As regards the proposer, he cannot gain by switching to another offer that is accepted.
If he makes an unacceptable offer he obtains the bundle w^α - (bα - B_α∕n)e) which if
preferred to x^α would violate the Pareto efficiency of the proposal x.

Claim 3. In any SPE, Bi =0for i ∈ N. Moreover, each agent is indifferent about
the identity of the agent who is chosen as the proposer.

Proof: Denote by M the set of agents for whom the aggregate bid is the largest, that
is, M ≡ {i ∈ N|Bi = maxj ∈N Bj }.Wefirst claim that any agent j ∈ N is indifferent
between any agent in M being chosen as the proposer. Indeed, if j would strictly prefer
some particular agent, say i ∈ M to win, agent j would slightly increase his bid to agent
i and decrease his bid to the other agents in M so that agent i is chosen as the proposer
for sure. Following the change, by Lemma 1 (b), agent j would be better off.

If M = N, Claim 3 is proven. Suppose, by way of contradiction, that M 6= N and
denote by m (< n) the cardinality of M. Assume, for convenience, 1 ∈ M. We now show
that agent 1 can achieve a better outcome by changing the bids he submitted in stage 1.
Consider the following change in the bids by agent 1: bɪ⁰ = bɪ - e, bl⁰ = bl — 2' for any
i ∈ M∖{1}, b1⁰ = bl + (2m - 1)e for a particular j / M, and b1⁰ = b¹1 otherwise, with e > 0
and small enough. Then, B₁ >B₁⁰ >B_i⁰ for all i 6= 1.Inparticular,1 is chosen as the
proposer for sure. We claim that 1 is strictly better following this change in bids. To see
this, note first that bⁱ_l did not change for any i 6=1and B_l⁰ <Bl . Hence, in the economy
with agents N ∖{1}, after the change in the bids the “initial endowments” change from
(w^j - (bj - B₁∕n)e)j∈N∖{₁} to (w^j - (b^j₁ - B^r₁ /n)e)j∈N∖{₁}. Given B₁ > B'₁, by Lemma 1 (c),

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