Masuda et al. (Games Econ Behav 83:73–85, 2014) showed that the minimum approval mechanism (AM) implements the efficient level of public good theoretically and experimentally in a linear public good game. We extent this result to a two-players common pool resource (CPR) game. The AM adds a second stage into the extraction game. In the first stage, each group member proposes his level of extraction. In the second stage, the proposed extractions and associated payoffs are displayed and each player is asked to approve or to disapprove both proposed extractions. If both players approve, the proposals are implemented. Otherwise, a uniform level of extraction, the disapproval benchmark (DB), is imposed onto each player. We consider three different DBs: the minimum proposal (MIN), the maximum proposal (MAX) and the Nash extraction level (NASH). We derive theoretical predictions for each DB following backward elimination of weakly dominated strategies (BEWDS). We first underline the strength of the AM, by showing that the MIN implements the optimum theoretically and experimentally. The sub-games predicted under the NASH are Pareto improving with respect to the Nash equilibrium. The MAX leads, either to Pareto improving outcomes with respect to the free access extractions, or to a Pareto degradation. Our experimental results show that the MAX and the NASH reduce the level of over-extraction of the CPR. The MAX leads above all to larger reductions of (proposed and realized) extractions than the NASH.
A pro-social player aims at maximizing the joint payoff of himself and the other party, as the benevolent social planner. On the other hand, a selfish rational player maximizes only his own payoff without regard to the others payoff. Therefore if the AM induces a social optimum level of extraction for a selfish player, it also does so for any player whose preferences support pro-social behavior.
Under the MIN, voluntariness is satisfied only if total group extraction is above the threshold \(X=(a-p)/b\). However, it is not satisfied under the MAX as mentionned in appendix 6.7.
Secondly, we show that Pareto-efficient sub-game \(({\hat{x}}, {\hat{x}})\) weakly dominates other symmetrical sub-games. The two players determine the outcome of the DB in the set of a uniform extraction vectors. To do so, player i maximizes his payoff under the constraint of uniform extraction vector: maximize\(\pi _{i}(x_{i},x_{j})\)wrt\(x_{i}=x_{j}\). The solution of this problem is the Pareto-efficient extraction level \({\hat{x}}=\frac{a-p}{4b}=6\) tokens, \({\hat{\pi }}=\frac{(a-p)^{2}}{8b}+pw=312\) ecus and \({\hat{X}}=\frac{a-p}{2b}=12\) tokens.
the comparison between NASH and MAX versus MIN, as well as NASH, and MIN versus NASH is not of interest; that is why they are alternatively removed from the sample.