Frequency-Dependent Selection

So far this course has kept promising you stable mixes — a population that settles at $p^* = V/C$ Hawks, a 50:50 split of two morphs, a population that refuses to collapse onto a single winning move. But we never opened the hood. Why does a mixture hold? What force reaches in and pushes a runaway type back down before it takes over?

The answer is one of the most important ideas in all of evolutionary game theory: frequency-dependent selection. It’s the engine. Every mixed ESS you’ve met runs on it. And once you see it, you’ll spot it everywhere — in traffic, in markets, in the sex ratio of nearly every animal on Earth.

The idea: your payoff depends on how common you are

Picture two ways the world can score a strategy.

In the first, frequency-independent selection, each type carries a fixed fitness number — a taller giraffe reaches more leaves regardless of how many tall giraffes there are. Whichever type has the biggest number simply wins, its share climbs to 100%, and the loser disappears. No mixture survives. This is the classic “survival of the fittest” cartoon, and for many traits it’s roughly right.

But an enormous class of traits doesn’t work that way. Under frequency-dependent selection, a type’s fitness depends on its own frequency in the population. The payoff isn’t a number written on the strategy; it’s a number that moves as the strategy spreads.

The everyday analogy is the commuter picking a route. Two roads run from home to work. If the fitness of “take Route A” were fixed, everyone would learn it’s faster and pile onto it — but the moment everyone piles on, Route A jams and Route B is suddenly the quick one. The payoff of your choice depends on how many others made the same choice. That’s frequency dependence in one sentence: the emptier road is the faster road, so no route can win outright.

Two flavours: rare-type advantage vs common-type advantage

Frequency dependence comes in two opposite signs, and telling them apart is the whole game.

Negative frequency dependence (rare-type advantage). A type does better when it’s rare and worse when it’s common. As it spreads, its own success erodes. This is a self-limiting, stabilising force — it pushes the population toward an interior mix and holds it there. The emptier road, the parking spot, the Hawk in a sea of Hawks: all negative FD.

Positive frequency dependence (common-type advantage). A type does better when it’s common. Success breeds more success — think a language, a currency, a keyboard layout, a social-media platform. The more people already use it, the more valuable it is to you. This doesn’t make a stable mix; it makes a tipping point that runs to all-one-type. It’s the same increasing-returns logic behind the critical-mass and lock-in models: whichever type gets an early lead snowballs to fixation, and often the “winner” isn’t even the best option.

Here’s the crucial bridge back to Lesson 2. Remember the Hawk–Dove mixed ESS at $p^* = V/C$? That number is pure negative frequency dependence. A lone Hawk in a Dove world grabs the whole resource $V$ every time — enormous payoff. But as Hawks get common, Hawks increasingly meet other Hawks, pay the injury cost $C$, and their average payoff falls. It keeps falling until, at exactly $p^* = V/C$, a Hawk does no better than a Dove and the pressure to be more Hawkish vanishes. The mixed ESS isn’t a coincidence — it’s the resting point where the rare-type advantage has been fully used up.

Worked example 1 — Fisher’s 1:1 sex ratio

Why are there roughly equal numbers of males and females in almost every sexually reproducing species? It looks wasteful — a herd of 100 cows and 1 bull would produce far more calves per year than 50 cows and 50 bulls. Yet nature stubbornly builds 50:50. Ronald Fisher explained it in 1930 with what is, in hindsight, a pure ESS argument (later framed game-theoretically by Hamilton).

The key accounting fact: every offspring has exactly one mother and one father. So across a whole generation, the total reproductive contribution of all males combined equals that of all females combined — the two sexes split the next generation’s parentage 50/50 no matter how many of each there are. Now suppose one sex is rare. That fixed pot of “male parentage” is divided among fewer males, so each rare-sex individual scoops up a bigger share.

Watch the arithmetic when a population runs male-poor:

A parent that “invests” in the rare sex out-reproduces a parent that invests in the common sex, so genes for making the rare sex spread — which drags the ratio back toward the middle. The advantage only disappears at 50:50, where a son and a daughter have identical expected payoff. That balance point is an ESS, held in place by negative frequency dependence: being the rare sex is good, so no sex can stay rare.

Worked example 2 — Left- and right-mouthed scale-eating fish

Some models live in textbooks; this one lives in Lake Tanganyika. The cichlid Perissodus microlepis makes a living tearing scales off other fish, darting in from behind and biting a mouthful off a flank. Its mouth is asymmetric — an inherited polymorphism twists the jaw permanently to one side:

• “Lefty” mouths bend to one side, so the fish attacks its prey’s right flank.
• “Righty” mouths bend the other way, attacking the left flank.

Here’s the frequency dependence. Prey fish aren’t passive — they learn to guard the side they get bitten on most. If lefties are common, prey watch their right flanks, and lefty attacks start missing. That makes the rare morph the successful one: whichever mouth is rarer catches prey off-guard and lands more bites, so it has higher fitness precisely because it’s rare. Classic negative frequency dependence.

The prediction is a population that hovers at 50:50 lefties to righties — and that’s what Michio Hori found by tracking the lake across years (Hori 1993). The ratio didn’t sit frozen at 50%; it oscillated around it, drifting above 50% lefty, then getting punished for being common and sliding back below, generation after generation. A stable mixed ESS you can literally net out of a lake and count.

Worked example 3 — Rock–paper–scissors lizards that never settle

Now the plot twist, and it’s the reason this lesson is the perfect on-ramp to “not every game has an ESS.”

The side-blotched lizard Uta stansburiana comes in three male types, each flagged by a throat colour, and each playing a different mating strategy (Sinervo & Lively 1996):

• Orange — hyper-aggressive, holds a big territory, monopolises many females.
• Blue — cooperative mate-guarder, defends a single female closely.
• Yellow — “sneaker” males that mimic females and slip in to mate on the sly.

The magic is in who beats whom:

Read the last column: orange beats blue, blue beats yellow, yellow beats orange. It’s a loop — literal rock–paper–scissors. And rock–paper–scissors has no single best move.

So what happens? Each morph invades and booms whenever the morph it beats is common. Yellow sneakers surge when orange dominates; then blue rises to police the sneakers; then orange rises to bully the guarders; then yellow again. The three frequencies cycle, and Sinervo tracked them chasing each other around a roughly 5–6 year loop, never settling down.

The pitfall: frequency dependence does not always make a stable mix

This is the misconception to burn out of your head. It’s tempting to reason “payoffs depend on frequency, therefore the population balances into a nice stable mixture.” Wrong — twice over.

• Positive frequency dependence (common-type advantage) doesn’t balance at all; it tips to one type. Lock-in, not a mix.
• Even negative frequency dependence only gives a stable mix when there’s a rest point with restoring force (two strategies pulling against each other: Hawk–Dove, sex ratio, scale-eaters). Arrange three or more strategies in a beats-loop and negative-ish feedback produces perpetual cycles with no ESS.

Sort your intuition into three boxes, not one.

When to reach for this model

Whenever you catch yourself saying “the best move depends on what everyone else is doing,” you’re looking at frequency-dependent selection. Reach for it when:

• Rare types get an edge. Minority games, contrarian trades, niche differentiation, novelty-seeking consumers — being unusual pays because it’s unusual.
• Congestion punishes the crowd. Traffic routes, checkout lines, fishing grounds, submarket entry — the emptier option is the better one, so no option wins outright.
• Predators over-target the common form. Apostatic selection: predators form a “search image” for whatever prey type is most abundant and hammer it, so rare colour-morphs of a prey species survive better — negative FD maintaining diversity.
• Success breeds success. Network effects, standards, currencies, platforms — here it flips positive, and you should expect a tipping point and lock-in rather than a mix.
• Strategies chase each other in a loop. Any orange-beats-blue-beats-yellow-beats-orange structure — expect cycles, and don’t waste time hunting for an ESS that isn’t there.

The diagnostic is always the same single question: as a strategy spreads, does its payoff fall, rise, or chase its own tail?

Recap

Frequency-dependent selection is the engine under every stable mix in this course — and, in one memorable case, the engine that refuses to produce one at all. A strategy’s fitness isn’t a fixed number; it’s a function of how common the strategy already is. Read the sign of that function and you know where the population is headed.

You now know why stable mixes exist — and why, sometimes, they stubbornly don’t. The lizards handed us a game with strong frequency dependence and no ESS. That’s not a curiosity; it’s the door into the next lesson, where we make it precise: which games have an evolutionarily stable strategy, and which never will.