That's not true. As with all game theory payouts, it's not saying that a Nash equilibrium is inevitable, it's saying that to avoid it you need incentives outside the game, it's saying that the incentives of the game itself don't suffice.

In terms of "the tragedy of the commons", that outside incentive is usually public shaming in smaller communities, or rules and laws in larger and more anonymous communities. If you look at shared resources across history and cultures, you usually find communities enforce very strong outside incentives against spoiling the commons instead of just relying on the kindness of strangers.

That isn’t what the tragedy of the commons is. It isn’t about something inherent in having a shared resource that dooms it.

It is about the idea that when there is a limited, shared public resource, individuals making rational self-interested decisions will often lead to a situation which no individual would choose.

That doesn’t mean it always has to end in tragedy, though. It just means you can’t just go fully “laissez faire” and let everyone do whatever they want, and expect the common to be preserved. The only way to protect the common is to take some sort of collective action; pass a law that limits how much an individual can take from the common, or set a quota and allow people to bid, etc.

The collective action doesn’t even always have to be a legislative action; you could create a taboo around using too much of a common, or use social shaming to get people to only use their share.

The point is that you have to take SOME collective action to prevent the tragedy.