Taylor & Jonker, 1978 — writing "what catches on" as an equation
Classical game theory assumes rational players: they know all the options, can compute the payoffs, and can reason about what the other side is thinking. That is a workable assumption for analysing a negotiation. It is an extravagant one for a colony of bacteria, the trees in a forest, or ten thousand retail traders who mostly copy whatever is working.
The latter is exactly what biologists ran into in the 1970s: animals cannot calculate, and yet the distribution of their behaviour looks suspiciously like the output of a calculation. Is there a mechanism that pushes a population towards the "computed" outcome without anyone in it understanding anything?
Replicator dynamics answers yes, and it needs only two ingredients: what does well gets copied more, and how well something does depends on what everyone else is doing. The copying can be reproduction, imitation, or capital flowing in; the mechanisms differ and the equation is the same.
That leading xi factor in rule 4 is easy to skip past, and it sets the whole character of the model: however excellent a strategy is, if few are playing it right now it also spreads slowly. Good things do not roll out overnight — they have to survive the phase where the base is too small. The same factor explains rule 5: zero times anything is zero.
One more thing worth stating plainly: this equation describes no individual at all. It only tracks shares. Ask it whether a particular person will switch strategies and it has no answer, and no ambition to have one.
With two strategies you can draw everything on a single line segment: the horizontal axis is the share of strategy A, from 0 to 1, and the arrows show which way it travels. Different payoff structures produce only four possible pictures.
Add strategies and it stops being a line. Three strategies live inside a triangle (each corner is "everyone plays this one"), and the usual endings include converging to a corner, converging to a point on an edge, or circling forever. Rock-paper-scissors is the circling kind — it has no resting place, only a wheel that keeps turning. The three male morphs of the Californian side-blotched lizard (Uta stansburiana) run exactly such a cycle, with a period of roughly six years (Sinervo and Lively, 1996).
With four or more strategies, replicator dynamics can even become chaotic: trajectories never repeat and long-run behaviour is unpredictable. So "evolution tends towards stability" is not a general truth — it is a special case of certain payoff structures.
The most direct application is why the proportion of a behaviour sits at some peculiar value. Why an animal population always contains a small band of cheats rather than all of them or none; why a low-price strategy and a high-service strategy coexist in an industry for decades; why some bad habit that hurts everybody cannot be swept away. That is what the first phase line is about.
It also explains why a head start is decisive in some settings and worthless in others. In the third phase line (bistability), which side of the watershed you start on determines everything: keyboard layouts, charging connectors, the early users of a social platform all have this structure. In the second (coexistence), the starting point is irrelevant — wherever you begin you return to the same mix, and spending money to win the opening is simply wasted.
There is a less obvious contribution too: it demotes rationality from an assumption to a result. Nobody in a replicator model computes anything, and yet the final mix frequently lands exactly on the point game theory derives from rational play. That means the reach of the Nash equilibrium is wider than its derivation suggests — you do not need the players to be clever, you only need a selection mechanism to exist.