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 ,wj - (bα - BαMe)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 wα - (bα - Bα∕n)e) which if
preferred to xα 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 >Bi0 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 bil did not change for any i 6=1and Bl0 <Bl . Hence, in the economy
with agents N ∖{1}, after the change in the bids the “initial endowments” change from
(wj - (bj - B1∕n)e)j∈N∖{1} to (wj - (bj1 - Br1 /n)e)j∈N∖{1}. Given B1 > B'1, by Lemma 1 (c),