A theory of fairness, competition, and cooperation

Proof of Proposition 2
We Žrst show that it is indeed a subgame perfect equilibrium if at least two proposers offer s 5 1 which is accepted by the responder. Note Žrst that the responder will accept any offer s $

0.5, because
1


n 2 2

(A1) s 2 _{n 2 1} b_i(s 2 1 1 s) 2 _{n 2 1} b_i(s 2 0) $ 0.
To see this, note that (A1) is equivalent to (A2) (n 2 1) s $ b_i(ns 2 1). Since b_i # 1, this inequality clearly holds if (A3) (n 2 1) s $ ns 2 1,
which must be the case since s # 1. Hence, the buyer will accept s 5 1. Given that there is at least one other proposer who offers s 5 1 and given that this offer will be accepted, each proposer gets a monetary payoff of 0 anyway, and no proposer can affect this outcome. Hence, it is indeed optimal for at least one other proposer to offer s 5 1, too.
Next, we show that this is the unique equilibrium outcome. Suppose that there is another equilibrium in which s , 1 with positive probability. This is only possible if each proposer offers s , 1 with positive probability. Let s_i be the lowest offer of proposer i that has positive probability. It cannot be the case that player i puts strictly positive probability on offers s_i [ [s_i, s_j) because the probability that he wins with such an offer is zero. To see this, note that in this case player i would get
(A4) U (s ) 5 2 ^ai s 2 ^ai (1 2 s) 5 2 ^ai .

^{i i} n 2 1
n 2 1
n 2 1

On the other hand, if proposer i chooses s_i [ (max_jÞi s_j,0.5 ,1), then there is a positive probability that he will win—in which case he gets
a_i n 2 2
(A5) 1 2 s_i 2 _{n 2 1} (2s_i 2 1) 2 _{n 2 1} b_i(1 2 s_i)
n 2 2 a_i a_i
> (1 2 s_i)[1 2 _{n 2 1} b_i] 2 _{n 2 1} . 2 _{n 2 1} .
Of course, there may also be a positive probability that proposer i does not win, but in this case he again gets 2 a_i/(n 2 1). Thus, proposer i would deviate. It follows that it must be the case that s_i 5 s for all i.
Suppose that proposer i changes his strategy and offers s 1 e , 1 in all states when his strategy would have required him to

choose s. The cost of this change is that whenever proposer i would have won with the offer s he now receives only 1 2 s 2 e. However, by making e arbitrarily small, this cost becomes arbitrarily small. The beneŽt is that there are now some states of the world which have strictly positive probability in which proposer i does win with the offer s 1 e but in which he would not have won with the offer s. This beneŽt is strictly positive and does not go to zero as e becomes small. Hence, s , 1 cannot be part of an equilibrium outcome.

QED