A theory of fairness, competition, and cooperation

a
# ₂ Û a 1 b_j21 # b_j a Û a(1 2 b_j) # 1 2 b_j Û a # 1,

which proves our claim.
QED


Proof of Proposition 5
Suppose that one of the players i [ n8 1 1, . . . , n chooses g_i , g. If all players stick to the punishment strategies in stage 2, then deviator i gets the same monetary payoff as each enforcer j [ 1, . . . , n8 . In this case monetary payoffs of i and j are given by

(A21) x

5 y 2 g

g 2 g_i
1 a[(n 2 1) g 1 g ] 2 n8

i i i
n8 2 c

(A22) x 5 y 2 g 1 a[(n 2 1) g 1 g ] 2 c ^{g 2 g}i 2 ^{n8 2 c} ( g 2 g )
^{j i} n8 2 c n8 2 c ^{i i}
g 2 g_i
5 y 2 g_j 1 a[(n 2 1) g 1 g_i] 2 (n8 2 c 1 c) _{n8 2 c} 5 x_i.
Thus, given the punishment strategy of the enforcers, devia- tors cannot get a payoff higher than what the enforcers get. However, they get a strictly lower payoff than the nonenforcers who did not deviate. We now have to check that the punishment strategies are credible; i.e., that an enforcer cannot gain from reducing his p_ij. If an enforcer reduces p_ij by e, he saves ce and experiences less disadvantageous inequality relative to those (n 2 n8 2 1) players who chose g but do not punish. This creates a nonpecuniary utility gain of [a_i(n 2 n8 2 1) ce]/(n 2 1). On the other hand, the enforcer also has nonpecuniary costs because he experiences now disadvantageous inequality relative to the defec- tor and a distributional advantage relative to the other (n8 2 1) enforcers who punish fully. The latter generates a utility loss of b_i (n8 2 1) ce/(n 2 1), whereas the former reduces utility by a_i(1 2 c)e/(n 2 1). Thus, the loss from a reduction in p_ij is greater

than the gain if (A23)

1 ce
_{n 2 1} [a_i(1 2 c)e 1 b_i(n8 2 1) ce] . ce 1 a_i(n 2 n8 2 1) _{n 2 1}
holds. Some simple algebraic manipulations show that condition (A23) is equivalent to condition (13). Hence, the punishment is credible.
Consider now the incentives of one of the enforcers to deviate in the Žrst stage. Suppose that he reduces his contribution by e .
0. Ignoring possible punishments in the second stage for a moment, player i gains (1 2 a)e in monetary terms but incurs a nonpecuniary loss of b₁e by creating inequality to all other players. Since 1 2 a , b_i by assumption, this deviation does not pay. If his defection triggers punishments in the second stage, then this reduces his monetary payoff which cannot make him better off than he would have been if he had chosen g_i 5 g. Hence, the enforcers are not going to deviate at stage 1 either. It is easy to see that choosing g_i . g cannot be proŽtable for any player either, since it reduces the monetary payoff and increases inequality.
QED