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The Drama of the Commons
It remains to show that there is no equilibrium with appropriation decisions above We fix The critical condition is Because a decrease of the appropriation level now generates inequity in favor of player i, we get the following condition:
Because βi < 1, the last term is negative if xi is close to . Hence, there are no equilibria with
Proof of Proposition 3 (Asymmetric Equilibria with Inequity-Averse Subjects)
We first show (ii): Let us assume there is an equilibrium with some By reordering the players, we can assume that we have Furthermore, let k be the highest index for which Now let’s consider i ≤ k. Because we are in an equilibrium, we have Remember that πj– πi= (a – bΣxk) (xj- xi). So
We fix 
The critical condition is 
Because a decrease of the appropriation level now generates inequity in favor of player i, we get the following condition:
. Hence, there are no equilibria with 
By reordering the players, we can assume that we have 
Furthermore, let k be the highest index for which 
Now let’s consider i ≤ k. Because we are in an equilibrium, we have 
Remember that πj – πi = (a – bΣxk) (xj - xi). So