Evolutionarily Stable Strategies

Watch any animal population long enough and you notice something strange: the mix of behaviours holds steady. A certain fraction of the males fight for territory and a certain fraction sneak; a certain share of a species bluffs and the rest back down; the sex ratio sits stubbornly at one-to-one. Nobody voted on these proportions. No committee balances the population. And yet, left alone, it returns to the same blend again and again, as if some invisible hand were tuning it. That invisible hand has a name, and it is the subject of this course: an evolutionarily stable strategy — an ESS — is a strategy that, once (almost) everyone in a population is playing it, cannot be invaded by any rare alternative. Try to introduce a mutant that does something different, and it does worse, so selection stamps it out and the population snaps back. That is the equilibrium biology settles into — reached not by anyone reasoning, but by the blind bookkeeping of who leaves more offspring.

This is an expert course, and it sits at the exact seam where two of the platform’s biggest ideas meet. From game theory it takes the machinery — players, payoffs, best responses, and the Nash equilibrium where no one can do better by switching. From natural selection it takes the engine — variation, differential reproduction, and the cold rule that whatever out-breeds spreads. The fusion, worked out by John Maynard Smith and George Price in 1973, is one of the most beautiful moves in modern science: replace the rational chooser of classical game theory with a whole population where fitter strategies simply reproduce more. The equilibrium is no longer something clever agents compute — it is something a mindless population arrives at. Because it also assumes cooperation’s home turf (repeated games, replicator dynamics), this course leans on the evolution of cooperation too; if those three prerequisites aren’t solid, take them first.

From that base we build the model floor to ceiling. We’ll define the ESS precisely and learn its invasion test — the two-line inequality that decides, for any candidate strategy, whether a rare mutant could get a foothold. We’ll spend real time on the model’s flagship, the Hawk–Dove game: two animals contesting a resource worth V, where a lost fight costs C in injury. You’ll prove with your own hands why pure aggression isn’t stable when fighting is dangerous, and watch the population settle instead on a mixed equilibrium of exactly V/C hawks — a stable blend of bullies and pacifists that no one designed. We’ll pin down how an ESS relates to the Nash equilibrium it refines (every ESS is a Nash equilibrium, but not every Nash equilibrium survives the extra test of being uninvadable), and untangle the two faces of a mixed strategy: one animal flipping a coin, versus a stable polymorphism of pure types. We’ll meet frequency-dependent selection — the deep reason these equilibria exist at all, where a strategy’s payoff depends on how common it is, giving rare types an advantage (the one-to-one sex ratio, left-mouthed and right-mouthed scale-eating fish, the rock–paper–scissors side-blotched lizards). We’ll tie it back to cooperation, seeing why Tit-for-Tat is almost an ESS in the repeated dilemma and why Always Defect is stable too — a world with two stable states. And we’ll be ruthlessly honest about where the model lies: an ESS is about stability, not optimality — populations get stuck at wasteful equilibria the Hawk–Dove fighting proves; not every game even has a pure ESS; and “evolutionarily stable” is dangerously easy to narrate loosely without ever checking the actual invasion inequality.

Throughout, you’ll run a live Hawk–Dove invasion lab — sliding the value and cost dials, dropping the population at any starting mix, and pressing let selection run to watch it climb or fall to the stable proportion on its own. By the end you’ll be able to look at any population of competing behaviours — animals, firms, strategies, norms — and ask the one question that matters: if almost everyone did this, could anything rare do better? If the answer is no, you’ve found an ESS. If the answer is yes, you’ve found the mutant that will take over next.