Lecture 12: Evolutionary stability: social convention, aggression, and cycles

What this lecture covers

This lecture puts the evolutionary-stability definition from the previous lecture to work on a series of increasingly rich examples, moving from a trivial check to social conventions, to mixed (polymorphic) populations, to a general model of aggression, and finally to a game with no evolutionarily stable strategy at all. It answers a natural next question: what real biological and social patterns can this machinery actually explain or predict?

This is the second of two lectures on evolutionary game theory, following directly from the formal definitions given last time, and it closes out the unit before the course's midterm. After watching, you should be able to check whether a pure or mixed strategy is evolutionarily stable using the Nash-equilibrium-based definition, and explain what a polymorphic evolutionarily stable population means in biological terms.

Key ideas

• Quick-check procedure: to test if S is evolutionarily stable, first check (S,S) is a symmetric Nash equilibrium; if it's strict, you're done; if there's a tie, check that S beats the tying strategy more than that strategy beats itself.
• Multiple stable conventions: symmetric coordination games like "drive on the left" versus "drive on the right" can have more than one evolutionarily stable strategy, and they need not be equally efficient.
• Monomorphic vs. polymorphic populations: evolutionary stability can describe a population where everyone plays the same pure strategy (monomorphic) or a stable mix of types coexisting (polymorphic), extending the definition from pure to mixed strategies.
• Chicken/symmetric Battle of the Sexes: a symmetric aggression-versus-passivity game has no symmetric pure-strategy equilibrium, so the only evolutionarily stable outcome is a mixed, polymorphic population.
• Hawk-Dove game: with prize V and fighting cost C, all-dove is never stable; all-hawk is stable only if V ≥ C; if C > V, the unique evolutionarily stable outcome is a mix with hawk proportion V/C.
• Comparative statics and identification: as the cost of fighting rises, the stable hawk proportion falls but the population's average payoff actually rises, and observing real fight proportions lets researchers back out the underlying ratio V/C.
• No stable strategy is a real prediction: in a rock-paper-scissors-type three-strategy game, no strategy or mix is evolutionarily stable, predicting cyclical dynamics — confirmed by the mating-strategy cycle observed in side-blotched lizards.

Walkthrough

A quick worked check of evolutionary stability (0:01)

The lecture restates the economic definition from last time and applies it to a small example: find the symmetric Nash equilibrium (A,A), check whether it's strict (it isn't, since B ties against A), then compare A's payoff against B to B's payoff against itself. Since A beats B by more than B beats itself, A is confirmed evolutionarily stable — illustrating how fast the definition is to apply once stated.

Social conventions: driving on the left or right (3:09)

Using a simple symmetric game where matching your opponent's side of the road pays off and mismatching doesn't, the lecture shows both "everyone drives left" and "everyone drives right" are strict Nash equilibria and therefore both evolutionarily stable, even though the payoffs to the two conventions aren't equal. This illustrates two lessons: a game can have multiple evolutionarily stable conventions (matching real-world variation in driving side by country), and evolutionary stability does not guarantee efficiency.

Polymorphic populations: chicken and elephant seals (10:25)

A symmetric version of Battle of the Sexes, known as the game of chicken (aggressive versus benevolent), has no symmetric pure-strategy Nash equilibrium, so the only stable outcome is the mixed strategy (2/3, 1/3) seen earlier in the course. The lecture generalizes the evolutionary-stability definition from pure strategies to mixed ones, calling a stable mixed population "polymorphic" as opposed to a uniform "monomorphic" population, and gives a heuristic argument for why more-aggressive or more-passive mutations both fail to invade this mix. The elephant-seal mating example (large dominant males versus small "sneaky" males that mimic females) illustrates a real polymorphic population.

The hawk-dove game (32:10)

A more general aggression game is introduced with a prize V for winning a contest and a cost C for fighting. The lecture shows an all-dove population is never evolutionarily stable (a hawk mutation always invades), while an all-hawk population is stable only when V is at least as large as C. When C exceeds V, solving for the symmetric mixed equilibrium (by equating hawk's and dove's expected payoffs) gives a hawk proportion of V/C, which the lecture argues heuristically is evolutionarily stable because both more-aggressive and more-passive mutations do poorly.

Comparative statics and identifying V and C from data (50:03)

The lecture works out how the stable mix responds to changes in the game: a larger prize V increases the proportion of hawks, and a larger fighting cost C increases the proportion of doves. Counterintuitively, the population's average payoff actually rises as C increases, because fewer fights occur in equilibrium even though each fight becomes costlier. This lets researchers use the observed proportion of aggressive behavior in a real population to infer the underlying ratio V/C, an example of what the lecture calls identification.

No evolutionarily stable strategy: rock-paper-scissors and lizard cycles (58:19)

A three-strategy game shaped like rock-paper-scissors (called "scratch, bite, trample") is analyzed, and the only candidate stable mix, (1/3, 1/3, 1/3), fails the stability check: a pure "scratch" mutation does better against itself than the mix does against scratch. Since nothing in this game is evolutionarily stable, the model predicts persistent cycling between strategies rather than settling down. The lecture connects this to a documented cycle among three color morphs of side-blotched lizards (orange harem-keepers invaded by yellow "sneaker" males, invaded by blue near-monogamous males, invaded by orange again), presented as a genuine out-of-sample prediction of the theory.

Before you watch

• Have the formal definition of evolutionary stability and its connection to Nash equilibrium fresh from the previous lecture in this course.
• Recall the Battle of the Sexes game and its mixed-strategy equilibrium (2/3, 1/3) from earlier lectures on mixed strategies.
• Be comfortable solving for a symmetric mixed-strategy Nash equilibrium by equating two strategies' expected payoffs.

Check your understanding

1. Why can a social convention like "drive on the left" be evolutionarily stable even though a different convention would be more efficient?
2. What distinguishes a monomorphic evolutionarily stable population from a polymorphic one, and which one applies to the symmetric chicken game?
3. In the hawk-dove game, why is an all-dove population never evolutionarily stable, regardless of the values of V and C?
4. As the cost of fighting C increases in the hawk-dove model, why does the population's average payoff go up even though fights become more costly?
5. Why does the "scratch, bite, trample" game have no evolutionarily stable strategy, and what real-world pattern does this predict?