  The productivity of a common pool of resources may degrade when overly
exploited by a number of selfish investors, a situation known as the tragedy of
the commons (TOC). Without regulations, agents optimize the size of their
individual investments into the commons by balancing incurring costs with the
returns received. The resulting Nash equilibrium involves a self-consistency
loop between individual investment decisions and the state of the commons. As a
consequence, several non-trivial properties emerge. For $N$ investing actors we
prove rigorously that typical payoffs do not scale as $1/N$, the expected
result for cooperating agents, but as $(1/N)^2$. Payoffs are hence reduced with
regard to the functional dependence on $N$, a situation denoted catastrophic
poverty. We show that catastrophic poverty results from a fine-tuned balance
between returns and costs. Additionally, a finite number of oligarchs may be
present. Oligarchs are characterized by payoffs that are finite and not
decreasing when $N$ increases. Our results hold for generic classes of models,
including convex and moderately concave cost functions. For strongly concave
cost functions the Nash equilibrium undergoes a collective reorganization,
being characterized instead by entry barriers and sudden death forced market
exits.
