A theory of fairness, competition, and cooperation

Proof of Proposition 3
We Žrst show that s 5 1, which is accepted by all responders, is indeed a subgame perfect equilibrium. Note that any offer s $

5. 
   will be accepted by all responders. The argument is exactly the same as the one in the beginning of the proof of Proposition 1. The following Lemma will be useful.
   

LEMMA 1. For any s , 0.5 there exists a continuation equilibrium in which everybody accepts s.
Given that all other players accept s player i prefers to accept as well if and only if
1 n 2 2
(A6) s 2 _{n 2 1} a_i(1 2 s 2 s) 2 _{n 2 1} b_i(s 2 0)
1 1

which is equivalent to

$ 0 2 _{n 2 1} a_i(1 2 s) 2 _{n 2 1} a_is,

(A7) (1 2 b_i)(n 2 1) 1 2a_i 1 b_i $ 0.
Since we assume that b_i , 1, this inequality must hold. h
Consider now the proposer. Clearly, it is never optimal to offer s . 0.5. Such an offer is always dominated by s 5 0.5 which yields a higher monetary payoff and less inequality. On the other hand, we know by Lemma 1 that for any s # 0.5 there exists a continuation equilibrium in which this offer is accepted by every- body. Thus, we only have to look for the optimal s from the point of view of the proposer given that s will be accepted. His payoff function is
1 n 2 2
(A8) U₁(s) 5 1 2 s 2 _{n 2 1} b₁(1 2 s 2 s) 2 _{n 2 1} b₁(1 2 s).

Differentiating with respect to s yields

dU₁
2 n 2 2

(A9)
_ds 5 2 1 1 _{n 2 1} b₁ 1 _{n 2 1} b₁,

which is independent of s and is smaller than 0 if and only if (A10) b₁ # (n 2 1)/n.
Hence, if this condition holds, it is an equilibrium that the proposer offers s 5 0 which is accepted by all responders. We now show that the highest offer that can be sustained in a subgame perfect equilibrium is given by (8).
LEMMA 2. Suppose that s , 0.5 has been offered. There exists a continuation equilibrium in which this offer is rejected by all responders if and only if

(A11) s # ^ai
(1 2 b_i)(n 2 1) 1 2a_i 1 b_i


;i [ 2, . . . , n .

Given that all other responders reject s, responder i will reject s as well if and only if

a_i n 2 2
(A12) 0 $ s 2 _{n 2 1} (1 2 2s) 2 _{n 2 1} b_is,
which is equivalent to (A11). Thus, (A11) is a sufficient condition for a continuation equilibrium in which s is rejected by everybody. Suppose now that (A11) is violated for at least one i [
2, . . . , n . We want to show that in this case there is no continua- tion equilibrium in which s is rejected by everybody. Note Žrst that in this case responder i prefers to accept s if all other responders reject it. Suppose now that at least one other responder accepts s. In this case responder i prefers to accept s as well if and only if
(A13)