Evolutionary Game Theory

William Hamilton addressed the harder problem in 1964. Sterile workers, alarm calls that attract predators, and animals that forgo breeding to help others raise young all appear to be selected against. Hamilton's resolution was to change the unit being counted. A gene that causes helping is also present in relatives, with probability $r$, so the behaviour spreads when the benefit to the recipient, discounted by relatedness, exceeds the cost to the actor: $rb > c$. For full siblings $r = 1/2$, so helping must do a sibling more than twice as much good as it does the helper harm.

The rule explains a pattern that had looked arbitrary. Sterile worker castes have evolved independently more than a dozen times in ants, bees and wasps, and almost nowhere else among animals, and these insects share an unusual genetics: males are haploid, so full sisters share three-quarters of their genes rather than a half. A worker is therefore more closely related to her sisters than she would be to her own daughters, which makes raising sisters a better genetic investment than reproducing. Hamilton pointed this out, and it remains the most-cited application of the rule — though termites are eusocial without haplodiploidy, and the modern account leans more on the benefits of staying in a defensible nest.

The accounting also makes quantitative predictions that have been tested in the field. Birds that help at the nest are overwhelmingly helping relatives, and the amount of help tracks relatedness; ground squirrels give alarm calls more readily when kin are within earshot; and in social insects, conflicts over the sex ratio of the brood come out close to where the inclusive-fitness arithmetic of workers and queen predicts, which is not where either party alone would want it.

Whether inclusive fitness is the right formalism, or one correct way of keeping the books among several, has been argued since 2010, when Martin Nowak, Corina Tarnita and E. O. Wilson argued that ordinary models of natural selection handle the same cases without it. More than a hundred biologists signed replies. The empirical claims are not what is in dispute; the argument is about which quantity is doing the explanatory work, and it is a good example of a field disagreeing about a formalism while agreeing about every observation.

John Maynard Smith and George Price asked in 1973 why animals contesting a resource usually posture instead of fighting. The received explanation was restraint for the good of the species, which is not a mechanism natural selection can supply.

Their model has two strategies. A Hawk escalates until it wins or is injured; a Dove displays and retreats if the opponent escalates. Let the resource be worth $V$ and an injury cost $C$. The payoffs to the row player are:

	vs Hawk	vs Dove
Hawk	$(V-C)/2$	$V$
Dove	$0$	$V/2$

Two Hawks fight: each wins half the time and is injured half the time. A Hawk against a Dove takes the resource unopposed. Two Doves share, or settle it by display.

Take $V = 50$ and $C = 100$: injury costs twice what the resource is worth.

Is all-Dove stable? The population average is $V/2 = 25$. A rare Hawk meets only Doves and scores $V = 50$. It invades. Is all-Hawk stable? The average is $(V-C)/2 = -25$ — worse than nothing. A rare Dove meets only Hawks and scores 0, which is better. It invades too. Neither pure strategy is an ESS.

So let a fraction $p$ of the population play Hawk. The expected payoffs are

$$W_H = p\,\frac{V-C}{2} + (1-p)V, \qquad W_D = (1-p)\frac{V}{2}.$$

The mixture is stable when the two are equal, since then neither type is gaining:

$$W_H - W_D = \frac{p(V-C) + (1-p)V}{2} = \frac{V - pC}{2} = 0 \;\Longrightarrow\; p^{*} = \frac{V}{C}.$$

With the numbers above, $p^{*} = 0.5$. Checking: $W_H = 0.5(-25) + 0.5(50) = 12.5$ and $W_D = 0.5(25) = 12.5$. Equal, as required.

The dynamics make the stability explicit. The replicator equation for this game is

$$\dot{p} = p(1-p)\,(W_H - W_D) = \frac{p(1-p)(V - pC)}{2},$$

which is positive for $p < V/C$ and negative above it: the population is pushed back towards $p^{*}$ from either side. This is an ordinary one-dimensional dynamical system with an attracting fixed point, and the entire apparatus of dynamical systems applies — which matters because games with three or more strategies produce cycles, and some produce chaos.

Now the uncomfortable part. At the stable mixture the average payoff is 12.5. In an all-Dove population it would be 25. Selection has driven the population to a state in which every individual does half as well as they would under universal restraint, and no individual can do anything about it — the structure of the prisoner's dilemma, derived from nothing but the costs of fighting. This is why "for the good of the species" is not available as an explanation: the good of the species is not what selection maximises.

One more reading of $p^{*} = V/C$ deserves notice. The stable proportion of aggressors depends only on how the resource compares with the injury, and says nothing about the species, the weapons or the context. Where injury is cheap relative to the prize — a mating opportunity that will not recur, a contest between animals without dangerous weapons — the model predicts escalation, and that is where fights to the death are in fact observed. The formula also does not care whether the mixture is achieved by half the individuals being aggressive or by every individual escalating half the time. Those are biologically very different and mathematically identical.

The hardest case is cooperation that is not explained by relatedness. Robert Axelrod attacked it empirically in 1980 by inviting people to submit programs to play the repeated prisoner's dilemma and running them against each other. The winner was the shortest entry: cooperate first, then do whatever the opponent did last. Tit-for-tat won the second tournament too, against entrants who knew the result of the first, and won an evolutionary version in which programs reproduced in proportion to their scores.

The lesson drawn — that cooperation emerges when interactions repeat — is correct, and the specific claim about tit-for-tat was oversold. It is not evolutionarily stable: once everyone cooperates, unconditional cooperators are neutral and drift in, which lets defectors back. It also handles mistakes badly, since one accidental defection locks two tit-for-tat players into permanent retaliation, and strategies that forgive occasionally beat it. The 2012 tournament was won by a team of colluding entries that identified each other by an opening signature and sacrificed themselves to feed a designated winner, which is a fact about tournaments rather than about cooperation.

Martin Nowak and Robert May found a mechanism that needs neither memory nor recognition. Put players on a lattice so that each interacts only with neighbours and imitates whichever neighbour scored best. Cooperators persist indefinitely, in shifting clusters, in a game where the well-mixed model says they must disappear — because a cooperator's neighbours are disproportionately cooperators. Space does the work that reciprocity does. Nowak later reduced the known mechanisms to a short list, each with a quantitative condition: kin selection needs $r > c/b$, network reciprocity needs the benefit-to-cost ratio to exceed the average number of neighbours, and so on.

What none of them explains well is people, which is the open problem above. Humans cooperate with strangers in one-shot anonymous encounters and pay to punish free-riders they will never meet again, at rates that none of the mechanisms predicts. The possibilities — cultural group selection, internalised norms applied outside the conditions they evolved for, selection for being the kind of partner others seek — are hard to separate, and the laboratory cannot easily distinguish a genuine preference from a psychology built for a world where nothing was ever truly anonymous.