a) Use more resources: +1 for them, -1.5 for everyone else b) Use same resources: +0.5 for them, +0 for everyone else
There is only one Nash equilibrium: (a,a) with payout (-0.5,-0.5).
Repeat it again and again without cooperation and you have the tragedy of the commons.
Nash equilibrium isn't about a repeating game reaching equilibrium over time, but about the decisions multiple agents face in one specific round.
That outcome cannot be NE by your own given inputs, since both actors can improve their outcomes by switching to a different strategy.
The payout matrix is:
(-0.5, -0.5) (+1, -1)
(-1, +1) (+0.5, +0.5)
As you can see, from the [a,a] option, no agent can improve their situation by UNILATERALLY changing. In the case of [a,b], [b,a] and [b,b], one (or both) of the agents can always improve their own payout by switching from b to a.
So there's only one Nash equilibrium, at [a,a].
Tragedy of the commons in the classical sense does not have a Nash equilibrium, because an actor can always chose to consume the extra unit of resources to benefit themselves more.
This lack of equilibrium is precisely why it's called a tragedy, because it inevitably leads to the over-exploitation (usually destruction) of the common good.