Free Access to the Commons: Random Priority versus Average Cost

A technology with increasing marginal costs produces indivisible units of service. Each agent wants at most one unit of service. We compare two natural mechanisms where users have free access to the technology. In one mechanism, Average Cost, each user pays the same average cost, in the other, Random Priority, users are randomly ordered (without bias) and successively offered to buy at the true marginal cost. Both mechanisms, AC and RP, lead to inefficient overproductions. Which one leads to a less severe tragedy ? We show that RP tends to overproduce less but that which game collects more social surplus depends much on the configuration of the demand (namely the social value of the goods produced). We give general results as well as canonical examples, describing demand and cost profiles for which one mechanism outperforms the other. We find that the key element for the comparison is the crowding ratio, i.e., the number of potential users over the number of units of output users can afford: (1) we prove that the more crowded the commons, the more RP outperforms AC, and (2) we exhibit a threshold value of 2.4 for the crowding ratio beyond which RP strongly outperforms AC, and beneath which AC only mildly outperforms RP. Thus the Random Priority mechanism offers a (partial) probabilistic resolution of the tragedy of crowded commons.