I should pay more attention to versus . In Marc Harper’s theorem I was content to have a non-strict inequality, so I used a non-strict inequality in the definition, but a strict inequality in the definition should give a strict inequality in the theorem, as long as . Thanks!

In the replicator dynamics, Maynard Smith’s ESS definition implies local asymptotic stability.

Bernhard Thomas’s definition, on the other hand, says that strategy q beats or ties any other strategy distribution it is competing with, whether rare or common. It is equivalent to saying that q is an ESS if for any other distribution p and any proportion x of strategy p,

q.[(1-x)q+xp] >= p.[(1-x)q+xp]

So, no matter what proportions q and p are blended in, q wins or ties.

To me, this doesn’t fit the name “evolutionarily stable”. I think it should instead be something like “evolutionarily dominant”. In this view “evolutionarily stable” means you’re safe against small group of invaders, but another strategy could still displace you if it arrives in sufficiently large numbers. “Evolutionarily dominant” means that nothing can beat you. (This distinction is relevant in many biological situations.)

Also, it’s not true that Thomas’s definition is strictly stronger than Maynard Smiths, at least not in the way you’ve written them. This is because the inequality in (3) is strict while the inequality in (1) is not.

If we strengthen Thomas’s definition by making the inequality in (1) strict for all p != q, then this strengthened version of (1) implies that q is a global attractor for the replicator dynamics. So in some sense, Maynard Smith’s definition is about local stability, and Thomas’s definition is about global stability.

The derivative of I being zero does not imply that the dynamic is stationary. Sometimes I is a constant of motion (e.g. for the rock-scissors-paper game), which has concentric orbits about the center of the simplex in dimension three. These orbits are not attractive (there is no limit cycle), so while the population is stuck on a particular cycle based on the initial point and is in some sense stable, it is not stable in the intuitive sense of evolutionary/selective stability. It makes sense that the relative entropy (with q being the center of all the cycles, which is a stable point, but not asymptotically so) is constant on these cycles — there is no information gain for otherwise the dynamic would not cycle.

Another situation is that one can have an evolutionarily stable set for a particular game matrix A, for instance, a connected line of stable points such that the set is locally asymptotically stable, but that there is no motion along the line, so that no point in the line is distinguished as locally asymptotically stable. In three dimensions it would look something like this (http://ars.els-cdn.com/content/image/1-s2.0-S0370157307001810-gr4.jpg) if all those points were connected in a line across the ternary plot. The matrix given by equation (7.20) in Hofbauer and Sigmond produces such a set of equilibria.

In this case, we don’t have have evolutionary stability in the sense of Maynard Smith because points q on the line are not resistant to invasion. In other words, condition (3) would not be satisfied because with the influx of a particular distribution of mutants (shifting along the line of equilibria from q to p), the population would not tend back to the distribution before the influx, and so the point q is not selectively stable, and this holds for every point on the line. If an evolutionarily stable set is a single point, then it would typically be an evolutionarily stable state.

Finally, some more mathematical reasons: there are different conclusions from the Lyapunov stability theorem if the derivative of the relative entropy I is always negative versus just non-positive, namely that the equilibrium is locally asymptotically stable rather than just stable. Also, Ross Cresmann showed that evolutionary stability is equivalent to the dynamical system notion of strong stability, which many find to be intuitively satistifying.