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Two technologies could each become the standard. One is better if everyone adopts it; the other is safer if you are unsure what others will do. Over the long run, with the occasional random switch, the safe one tends to take over, even though the world would be better off with the other.

Stochastic stability and conventions

A coordination game has several equilibria, and the ESS cannot say which a society lands on. In the finite-population model below, players best-respond to the previous population and make rare independent mistakes. That tie-break selects the risk-dominant convention, the one with the larger basin, even when it pays less.

From earlier lessons

Two quick ones before we start.

Stag Hunt has two pure equilibria: all-Stag and...

Risk-dominance favours the equilibrium with the larger...

Predict, then play

In the Stag Hunt, all-Stag pays 4 each and all-Hare pays 3, but Stag is only worth it if you are confident others hunt. Add rare mutations and watch which convention the population settles into over the long run.

Why did that happen?

Take a guess, then reveal the reason

Both all-Stag and all-Hare are stable against a single defector. In this best-response-with-mutation model, what separates them is how much noise it takes to escape each. Escaping all-Hare toward Stag requires a lot of players to mistakenly hunt at once, because Stag only pays when most others do. Escaping all-Stag toward Hare needs only a few mistakes, since Hare is safe. So random shocks tip the population out of Stag easily and out of Hare rarely, and over a long horizon it lingers at Hare, even though everyone would prefer Stag.

The math

Definition, stochastic stability and risk-dominance

In the Kandori-Mailath-Rob model, a finite population best-responds to the previous action frequencies but each player mutates independently with a small probability. A convention is stochastically stable if its stationary probability stays positive as that mutation rate goes to zero. For a sufficiently large population in a symmetric $2\times2$ game with two strict equilibria, the risk-dominant equilibrium, the one with the larger basin, is selected. Other learning and mutation models can select differently.

In the Stag Hunt, Stag pays $4$ against Stag and $0$ against Hare; Hare pays $3$ regardless. Playing Stag is a best response only if you believe the partner hunts with probability $q$ satisfying $4q \ge 3$ , that is $q \ge \tfrac34$ . So Stag’s basin (the beliefs for which it is optimal) is only $\tfrac14$ , while Hare’s is $\tfrac34$ .

Since Hare’s basin exceeds one-half, Hare is risk-dominant and stochastically stable in this model. Lowering the mutation rate does not change which convention wins; it only lengthens the time the population waits before drifting back to Hare. The long-run stationary outcome is start-independent.

Work it out

Compare the basins

You should hunt Stag only if the expected payoff beats Hare. Stag pays $4q$ when the partner hunts with probability $q$ (and $0$ otherwise); Hare pays $3$ no matter what. So Stag is a best response iff $4q \ge 3$ , i.e., $q \ge \tfrac34$ . That leaves only a $\tfrac14$ slice of beliefs favouring Stag, and $\tfrac34$ favouring Hare.

The larger basin belongs to Hare, so Hare is risk-dominant. In the KMR noisy dynamics, the population is pushed out of the small Stag basin by even modest mutation, but escaping the large Hare basin requires an improbable coincidence of mistakes. So it settles at Hare in the long run, despite Stag being payoff-dominant. Lowering the mutation rate only makes the rare escapes rarer; it does not favour Stag.

Key idea

In the KMR best-response-with-mutation model, the long-run winner among conventions is the risk-dominant one, with the larger basin, not the payoff-dominant one. Lowering the noise changes only the waiting time, not the outcome.

Check yourself

Check yourselfapply

In the Stag Hunt, why is all-Hare stochastically stable even though all-Stag pays more?

Check yourselfanalyse

What happens if you lower the mutation rate toward zero?

Check yourselfbrings back: classic 2x2 games

How does this refine the Stag Hunt's two equilibria from the earlier zoo lesson?

Sources

Kandori, M., Mailath, G. J., and Rob, R. (1993). Learning, mutation, and long run equilibria in games. Econometrica, 61(1), 29-56.
Young, H. P. (1993). The evolution of conventions. Econometrica, 61(1), 57-84.