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• #### Index 499-521

which subjects appropriate more than in the SNE. If all players in a group are inequity averse and given the parameters of the Ostrom et al. experiment, the range of possible Nash equilibria is where is always an equilibrium independent of αi and βi .7

So far we have concentrated on symmetric equilibria. However, there are also asymmetric equilibria. The following proposition provides the details.8

###### Proposition 3 (Asymmetric Equilibria with Inequity-Averse Subjects)

(i) If there are at least k players with then there is an equilibrium with less appropriation than in the SNE. In this equilibrium at least k players choose the same appropriation the other players j choose higher appropriation levels. (ii) If there is no k such that there are at least k players with then there is no equilibrium with less appropriation than in the SNE.

Corollary 1: If there are or more selfish players, then there is no equilibrium with less appropriation than in the SNE.

The intuition of Proposition 3 is straightforward. To get more efficient equilibria than the SNE requires that a relatively large fraction of subjects have rather high combinations. Notice that because 0 < βi < 1 and αi > βi , the expression is between zero and one. This means that for the only equilibrium is the SNE. Only if k is higher than n/2 there are combinations to ensure a more efficient equilibrium. For example, if the group size is 8, it takes at least 5 nonegoistic players to reach such an equilibrium. In this case the combinations of these 5 subjects must be at least The more people who are nonselfish, the weaker is the requirement for Thus, to reach a more efficient outcome, it takes either many subjects with moderate combinations or it takes fewer subjects (but still more than n/2) with very high combinations. Notice that the expression rises in βi and decreases in αi. It is therefore more likely to reach more efficient outcomes if subjects have a rather large utility loss from advantageous inequality, and a rather small utility loss from disadvantageous inequality.

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