1
Fitness is the average payoff
A cooperator meets another individual at random and receives the payoff for that pairing. Its average, over the chance of meeting a cooperator, is the cooperators' fitness, and the defectors have their own.
Lesson · Games and the Social Contract · Many players
The Evolution of Cooperation
Instead of choosing, the players are born cooperators or defectors, and each generation a type grows in proportion to its average payoff. In the prisoner's dilemma the defectors out-earn the cooperators at every share and take over; in the stag hunt, cooperators win from above a threshold and lose from below it; and when cooperators meet their own kind more often than chance, the dilemma payoffs can favour them.
new share = share · payoff / mean payoff
dilemma: defectors out-earn at every share
stag hunt: the winner depends on the start
selection compares types, not populations
One exact generation at a time
Core ideas
What this lesson establishes
The replicator step replaces choice with inheritance. The three ideas below are what the lab computes.
2
A share grows by its relative fitness
The next generation's share of cooperators is the old share times the cooperators' fitness, divided by the population's mean fitness. If cooperators earn more than the mean their share rises, and if less, it falls.
3
Rest points and which ones hold
A share that does not move is a rest point. A rest point is stable if shares nearby move toward it, and in the stag hunt the interior one is not: it divides the starts that end in cooperation from the starts that end in defection.
The material
Shares, generation by generation
The step, applied to the dilemma, the stag hunt and a population that sorts itself.
Everything so far has had players who choose. Here each individual is born a cooperator or a defector and plays that type against one other individual drawn at random. A share x of the population cooperates, so a cooperator meets a cooperator with chance x and a defector with chance 1 − x. The payoffs are fitness: the number of offspring, who inherit the type.
Definition · Replicator step
With cooperators' average payoff fC and defectors' fD, the next generation's cooperating share is x′ = x·fC / (x·fC + (1 − x)·fD): the old share times its payoff, over the population's mean payoff.
Take the dilemma of “The Prisoner's Dilemma”, with R = 3, S = 0, T = 5 and P = 1. A cooperator earns 3 against a cooperator and 0 against a defector, so on average 3·x. A defector earns 5 and 1, so on average 1 + 4·x. Start with half of the population cooperating, x = 1/2.
fC = 3·(1/2) = 3/2
fD = 1 + 4·(1/2) = 3
x′ = (1/2)·(3/2) / ((1/2)·(3/2) + (1/2)·3) = 1/3
After one generation a third of the population cooperates. The defectors out-earn the cooperators by 1 + x at every share, so the cooperating share falls each generation, and the lab's longer run, five generations from the same start, ends close to zero.
Compare what the two populations earn. A population of cooperators averages 3 each and a population of defectors averages 1, and selection still moves toward the defectors. Selection compares types inside a population, one against the other, and the mean of the whole is nothing it looks at.
The stag hunt
In the stag hunt of “Coordination, Conventions and the Stag Hunt”, stag pays 4 against stag and 0 against hare, and hare pays 3 whatever the other does. A stag hunter earns 4·x and a hare hunter earns 3. They tie at x = 3/4, the very number that was the belief threshold for hunting stag. Above it stag hunters out-earn hare hunters and grow; below it they shrink.
From x = 4/5 the stag hunters' share rises, to 64/79 in one generation. From x = 1/2 it falls, to 2/5. Both populations are in equilibrium at their ends, all stag and all hare, and which end is reached depends on the start. History, in the sense of where the population began, settles a stag hunt in a way it cannot settle the dilemma.
Meeting your own kind
Suppose the dilemma's payoffs are unchanged but half of every individual's meetings are with its own kind and half are at random. A cooperator's average is then 3/2 + (3/2)·x and a defector's is 1 + 2·x. At x = 1/2 that is 9/4 against 2, and the cooperating share rises, to 9/17 in one generation.
Nothing in the payoffs has become kinder. The change is in who meets whom, and with enough of it the dilemma's cooperators do better than its defectors. This is why the repeated games of earlier lessons and the structured populations of biology can sustain cooperation that a well-mixed population cannot. What the replicator step does not show is how any of this is inherited, or what happens when a rare new type appears.
Interactive
The many-player dilemma
payoffs as exact expressions in k, and what follows from them
Method
Running the replicator step
One generation by hand, and then as many as the lab will run.
1
Write each type's average payoff as a function of the share
A cooperator's is its payoff against a cooperator times x plus its payoff against a defector times 1 − x. A defector's is the same with its own payoffs.
2
Evaluate both at the starting share
Use exact fractions. The two numbers are fC and fD.
3
Form the mean payoff
It is x·fC + (1 − x)·fD, the average over the whole population. It must be positive.
4
Divide to get the new share
The cooperators' part of the mean over the whole mean is x′. If it is above x the cooperators have grown.
5
Repeat, and look for where the share stops
A share that returns to itself is a rest point. Check which way the neighbouring shares move.
Where a population goes
Score 0 / 0Choose the answer you can justify.
One generation of the dilemma
Payoffs 3, 0, 5, 1 and a starting share of one half.
Cooperator: 3 x 1/2 + 0 x 1/2 = 3/2
Defector: 5 x 1/2 + 1 x 1/2 = 3
Mean: 1/2 x 3/2 + 1/2 x 3 = 9/4
New share: (1/2 x 3/2) / (9/4) = 1/3
Stag hunt from 4/5: stag 16/5, hare 3: share rises
Stag hunt from 1/2: stag 2, hare 3: share falls to 2/5
The mean payoff of the dilemma's starting population is 9/4, which is well below the 3 of a population of cooperators. The step ignores that: it compares 3/2 with 3 and the defectors grow.
Common mistakes
1
Thinking natural selection favours what is good for the groupA population of cooperators averages 3 and one of defectors averages 1, and the dilemma population moves to the defectors anyway: from half cooperating to a third in one generation. The step compares the two types' earnings within the population, so a type that is bad for the group but better for the individual spreads.
2
Reading a rising share as a verdict that the type is goodIn the stag hunt the stag hunters grow from 4/5 and shrink from 1/2, under the same payoffs. A share rises because it out-earns the other type at the share it has, not because it is better in itself.
3
Treating the step as a model of choiceNo one in the population decides anything. Types are inherited, and the step is a rule for how shares change. That is why the lab's results are about which types persist and not about what a rational player does.
Check your understanding
✓
Finish when you can say where a population goes from a stated start.
Given a two-player game and a starting share, compute each type's average payoff, apply the replicator step for one generation by hand, and say whether the cooperating share rises or falls and where it ends.
Note: The step is a rule given here and not derived, and it assumes an infinite population, no mutation, and payoffs that are never negative. A different rule for how shares change gives different paths, though for these games the same ends.