Evolutionary and behavioural game theory

Picture a single working borehole at the end of a long dry season, and two herding families arriving at once. Neither has read a textbook; neither computes a best response. Yet across a whole rangeland, over thousands of such encounters, a stable pattern emerges — some fraction of meetings end in an armed standoff, the rest in one side watering elsewhere. Classical game theory asks what rational players would choose. This module asks a different and often more useful question: which behaviours survive? When strategies are carried by habit, culture, imitation or sheer repetition rather than by deliberation, the tools you need are evolutionary and behavioural game theory — and they will carry you from animal contests to social conventions to the stubborn persistence of corruption.

From rational players to evolving behaviours

Evolutionary game theory, introduced by John Maynard Smith and George Price in 1973, drops the rationality assumption entirely. A strategy is not a choice but a behaviour — a hard-wired disposition, a cultural script, a rule of thumb, a routine inside a firm. Payoffs are not utilities to be maximised but fitness: the rate at which a behaviour reproduces itself, whether through offspring, imitation of the successful, or the survival of the organisations that use it. You reason about a population, not a person. The state of the system is the frequency of each behaviour, and those frequencies change over time as more successful behaviours are copied and less successful ones die out. Nothing here requires anyone to be clever; it only requires that success breeds imitation.

The evolutionarily stable strategy (ESS)

The central solution concept is the evolutionarily stable strategy, or ESS: a strategy that, once adopted by almost the whole population, cannot be invaded by any rare alternative. Write E(σ, τ) for the payoff to an individual playing σ against an opponent playing τ. Suppose incumbents play σ* and a small fraction ε of mutants plays some σ ≠ σ*. A random opponent is an incumbent with probability 1 − ε and a mutant with probability ε, so the incumbent out-earns the mutant precisely when (1 − ε)·E(σ*, σ*) + ε·E(σ*, σ) exceeds (1 − ε)·E(σ, σ*) + ε·E(σ, σ). Letting ε shrink to zero and comparing terms yields the two ESS conditions.

1. First-order (equilibrium) condition: E(σ*, σ*) ≥ E(σ, σ*) for every alternative σ. The incumbent is a best reply to itself — that is, σ* is a Nash strategy.
2. Second-order (stability) condition: where the first holds with equality, E(σ*, σ*) = E(σ, σ*), it must be that E(σ*, σ) > E(σ, σ). When the mutant does equally well against the incumbent, the incumbent must do strictly better against the mutant than the mutant does against itself.

The Hawk–Dove game

Hawk–Dove is the canonical model of conflict over a winner-take-all resource. Two individuals contest a prize of value V. A Hawk escalates and is willing to fight; a Dove displays and retreats the moment the other escalates. When two Hawks meet they fight: each wins half the time, taking V, and loses half the time, paying an injury cost C, for an expected (V − C)/2. A Hawk facing a Dove takes the whole prize V while the Dove, having fled, gets nothing. Two Doves settle without injury and split the value, V/2 each.

Payoffs to the ROW player (resource value V, injury cost C):

                        OPPONENT
                     Hawk          Dove
           Hawk    (V−C)/2           V
  ROW
           Dove       0            V/2

  Hawk vs Hawk : fight; win ½ the time for V, lose ½ paying C  ⇒  (V−C)/2
  Hawk vs Dove : the Dove flees, the Hawk takes everything     ⇒  V   (Dove gets 0)
  Dove vs Dove : share, or settle the prize by display         ⇒  V/2 each

Everything turns on the sign of V − C. Assume fighting is expensive relative to the prize, V < C — the case that makes the game interesting and describes most real contests, where losing a fight costs far more than the resource is worth. Then neither pure strategy is an ESS. In an all-Hawk population each earns (V − C)/2 < 0, so a Dove mutant, earning 0 against Hawks, invades. In an all-Dove population each earns V/2, but a Hawk mutant takes V > V/2 from its Dovish neighbours and invades. Each pure strategy is undone by the other; stability must lie in between.

Solving for the polymorphic ESS

The interior ESS can be read two ways that are mathematically identical: a monomorphic population in which every individual escalates with probability p*, or a polymorphic population in which a fraction p* are pure Hawks and the rest pure Doves. Either way, the defining property of an interior equilibrium is indifference — if Hawks earned more than Doves the Hawk frequency would rise, and vice versa — so at p* the two behaviours must earn exactly the same fitness. Impose that condition and solve.

Let p = fraction of the population playing Hawk
        (equivalently, the chance a random opponent is a Hawk).

  Fitness of a Hawk :  W(H) = p·(V−C)/2 + (1−p)·V
  Fitness of a Dove :  W(D) = p·0        + (1−p)·(V/2)

  Set W(H) = W(D):
        p·(V−C)/2 + (1−p)·V = (1−p)·(V/2)

  Multiply every term by 2:
        p(V−C) + 2(1−p)V = (1−p)V
  Move the right-hand side across:
        p(V−C) + (1−p)V = 0
  Expand and cancel pV:
        pV − pC + V − pV = 0
        V − pC = 0

        ⇒   p* = V / C      (interior when V < C, so 0 < p* < 1)

The equilibrium fraction of Hawks is the ratio of the prize to the cost of fighting, p* = V/C. The reading is intuitive and sharp: the more valuable the contested resource, the more escalation the population sustains; the more dangerous a fight, the less. When V ≥ C the formula returns p* ≥ 1, which the unit interval cannot hold — the interior solution has left the building, and pure Hawk becomes the ESS, because now winning is worth the expected cost and everyone should fight. For V < C the population settles at a genuine mix: aggression is common enough to be worth deterring, but rare enough that most contests stay peaceful.

How a peace convention locks in

Real populations rarely sit at the bloody mixed equilibrium, because contests are almost never truly symmetric. One party arrived first; one is the customary owner; one is on home ground. Even a payoff-irrelevant asymmetry — an uncorrelated asymmetry, in the jargon — can anchor a convention: play Hawk if you are the owner, Dove if you are the intruder. This Bourgeois strategy is itself an ESS. If everyone follows it, ownership is settled without a fight, and a mutant who ignores the marker does strictly worse. Across much of Africa this is the game-theoretic skeleton of customary land-and-water tenure, first-comer rights, and elders' seasonal allocation of grazing: a shared signal both sides read the same way, sparing them the dissipation of the mixed ESS. Two features matter. The convention is self-enforcing — no external police are needed once it is common knowledge — and its content is arbitrary: the opposite rule, Dove-if-owner and Hawk-if-intruder, is equally stable. That arbitrariness is why conventions, once locked in, are so persistent and so hard to reform: they are held in place not by their fairness but by everyone's expectation that everyone else will follow them.

Corruption as an evolutionarily stable equilibrium

The same logic explains one of the continent's most corrosive traps. Treat the decision to pay or take a bribe as a behaviour matched at random across a population of officials and citizens. Paying is a best response when almost everyone else pays: refuse the bribe and you are obstructed, delayed, passed over, sometimes reported by the very system you were trying to keep clean. Honesty pays only when honesty is common, because then the clean, efficient transaction is actually available and it is the briber who stands out. This is a coordination game with two evolutionarily stable equilibria — a clean one and a corrupt one — separated by a tipping point. Above that threshold an isolated honest official earns less than a corrupt neighbour and is selected against; her integrity does not spread because it does not pay. The corrupt equilibrium is not held together by anyone preferring corruption in the abstract; it is held together, exactly like the arbitrary convention above, by each person's expectation that everyone else will take the bribe. The fully worked model is Exercise 2; the qualitative lesson comes first.

Behavioural game theory: where humans depart from Nash

The Ultimatum Game

Evolutionary game theory removes rationality from below; behavioural game theory studies how actual humans deviate from it in the laboratory. The sharpest instrument is the Ultimatum Game. A Proposer is handed a sum of money and offers some split to a Responder; if the Responder accepts, the split stands, and if she rejects, both get nothing. Solve it by backward induction under pure self-interest: the Responder should accept any positive amount, since something beats nothing, so the Proposer should offer the smallest positive sum and keep almost everything. That is the unique subgame-perfect prediction — offer ε, accept anything.

Human beings do no such thing. Across thousands of runs Proposers typically offer close to an even split, and Responders routinely reject offers they regard as unfair — commonly those below roughly a fifth to a quarter of the pie — even though rejection means walking away with nothing. Rejecting a positive offer is costly punishment of another's unfairness: negative reciprocity, the willingness to pay in order to sanction those who treat you badly. The behaviour flatly refutes the self-interest model and reveals genuine fairness preferences — inequity aversion — inside the payoff function itself. And because Proposers anticipate rejection, even a purely selfish Proposer facing fair-minded Responders offers more; fairness in one player disciplines the other.

Fairness is calibrated, not constant

Is fairness a human universal? The decisive evidence is the fifteen-society study led by Joseph Henrich and colleagues (2001), who ran the Ultimatum Game in small-scale societies across several continents, including African groups such as the Orma pastoralists of Kenya and the Hadza foragers of Tanzania. Behaviour varied enormously — and it varied with local institutions, not with anything like a fixed human constant. Two variables predicted generous offers: the degree of market integration and the payoffs to cooperation in everyday life. The Orma recognised the game at once as a version of harambee, their institution of collective contribution, and made notably generous offers; the Hadza, a foraging society with tolerated scrounging and weaker cooperative production, made lower offers and rejected more often. The lesson is neither that fairness is universal nor that it is absent, but that it is a learned norm calibrated to the economic life a society actually leads — which is why the same game elicits open-handed sharing in one place and hard bargaining in another.

Bounded rationality: level-k and cognitive hierarchy

A second, complementary departure is that people simply do not run the infinite regress that Nash equilibrium assumes — I think that you think that I think, without end. Level-k and cognitive-hierarchy models replace it with finite depth. A level-0 player acts naively or at random; a level-1 player best-responds believing everyone else is level-0; a level-2 player best-responds to level-1s, and so on, with most people clustering at one or two steps. In the beauty-contest game — choose a number from 0 to 100, closest to two-thirds of the group average wins — the unique Nash equilibrium is 0, reached only by infinite iteration, yet real players choose numbers like 33 or 22, the fingerprints of one or two rounds of reasoning. The model does not call people irrational; it calls them boundedly rational, reasoning a few steps rather than infinitely many.