Evolutionary game theory

Evolutionary game theory studies frequency-dependent selection: the reproductive success of a strategy depends on the strategies it encounters in the population.

Core idea. In ordinary selection models, fitness can be assigned directly to each type. In an evolutionary game, fitness emerges from interactions, so it can change as population composition changes.

Strategies and frequencies

Suppose a population contains \(m\) strategies with frequencies

\[x_1,\ldots,x_m,\]

where

\[\boxed{x_i\ge0,\qquad \sum_{i=1}^m x_i=1}.\]

Write the frequency vector as

\[\mathbf x=(x_1,\ldots,x_m)^T.\]

The payoff matrix

Let

\[A=(a_{ij}).\]

The entry \(a_{ij}\) is the payoff obtained by a focal individual using strategy \(i\) when it interacts with strategy \(j\).

Rows therefore describe the focal strategy and columns describe the strategy encountered.

A two-strategy game

For strategies \(A\) and \(B\), write

\[A=\begin{pmatrix}a&b\\c&d\end{pmatrix}.\]

If the frequency of strategy \(A\) is \(x\), then strategy \(B\) has frequency \(1-x\).

The expected payoff to strategy \(A\) is

\[\boxed{f_A=ax+b(1-x)},\]

while the expected payoff to strategy \(B\) is

\[\boxed{f_B=cx+d(1-x)}.\]

The payoffs change with \(x\), which is the source of frequency dependence.

General expected payoff

For the frequency vector \(\mathbf x\), the expected payoff to strategy \(i\) is

\[\boxed{f_i=(A\mathbf x)_i}.\]

The population mean payoff is

\[\boxed{\bar f=\mathbf x^TA\mathbf x}.\]

This is the frequency-weighted average of the strategy payoffs.

Payoff and fitness

In evolutionary applications, payoff is usually interpreted as a contribution to reproductive success, survival, transmission or another fitness-related quantity.

The mapping between payoff and biological fitness should be stated rather than assumed. In some models payoff is fitness itself; in others one might use a transformation such as

\[w_i=1+s f_i,\]

where \(s\) controls the strength of selection.

Payoff is model-dependent. It need not represent money or a conscious decision. It is a mathematical score describing the biological consequence of an interaction.

Invasion when rare

A central evolutionary question is whether a rare strategy can increase in a population dominated by another strategy.

If resident strategy \(R\) has payoff \(f_R\) and a rare mutant \(M\) has payoff \(f_M\), then the mutant is favoured when

\[\boxed{f_M>f_R}.\]

Because payoffs depend on population composition, invasion must be evaluated in the resident environment.

Pure-strategy stability

Consider a population entirely composed of strategy \(A\). A rare \(B\) mutant mainly interacts with \(A\) residents.

The resident payoff is approximately \(a\), while the mutant payoff is approximately \(c\). Thus \(A\) resists invasion if

\[\boxed{a>c}.\]

If \(a

Mixed populations

An internal frequency \(x^*\) at which both strategies receive the same payoff satisfies

\[f_A(x^*)=f_B(x^*).\]

For the two-strategy matrix above,

\[ax^*+b(1-x^*)=cx^*+d(1-x^*).\]

When the denominator is non-zero,

\[\boxed{x^*=\frac{d-b}{a-b-c+d}}.\]

This is biologically relevant only if

\[0Whether this mixed state is stable depends on how payoff differences behave on either side of \(x^*\).

Nash equilibrium

A strategy distribution is a Nash equilibrium if no individual can obtain a greater payoff by unilaterally switching strategy while the population environment is held fixed.

For a symmetric evolutionary game, a pure strategy \(A\) is a Nash equilibrium if

\[a\ge c.\]

A Nash equilibrium describes resistance to an immediate unilateral payoff improvement, but evolutionary stability asks for a stronger invasion property.

Evolutionarily stable strategy

A strategy \(S\) is an evolutionarily stable strategy, or ESS, if a population using \(S\) cannot be invaded by a sufficiently rare alternative strategy \(T\).

For a symmetric game, one standard condition is

\[u(S,S)>u(T,S),\]

for every alternative \(T\). If equality holds for some \(T\), then one additionally requires

\[u(S,T)>u(T,T).\]

These conditions say that the resident must either outperform the mutant directly in the resident environment, or—when they tie there—perform better in encounters involving the mutant.

ESS and Nash equilibrium are related but not identical. Every ESS is a symmetric Nash equilibrium under the standard definition, but a symmetric Nash equilibrium need not be evolutionarily stable.

Hawk–Dove example

Suppose individuals compete over a resource of value \(V\), and escalated conflict carries cost \(C\), with \(C>V>0\).

A common payoff matrix is

\[A=\begin{pmatrix}\frac{V-C}{2}&V\\0&\frac V2\end{pmatrix},\]

where the rows and columns correspond to Hawk and Dove.

If the frequency of Hawk is \(x\),

\[f_H=x\frac{V-C}{2}+(1-x)V,\]\[f_D=(1-x)\frac V2.\]

Setting \(f_H=f_D\) gives

\[\boxed{x^*=\frac VC}.\]

Since \(C>V\), this lies between \(0\) and \(1\). The game therefore supports a mixed population rather than universal escalation.

Coordination games

In some games, a strategy performs best when it is already common. Then both pure strategies may resist invasion and an internal threshold separates their basins of attraction.

This creates positive frequency dependence: common strategies are favoured because they are common.

Anti-coordination games

In other games, a strategy performs better when it is rare. This negative frequency dependence can maintain coexistence of strategies.

The Hawk–Dove game is a standard example.

Biological interpretations

Evolutionary games can model aggressive versus non-aggressive behaviour, cooperation and defection, host–pathogen interactions, microbial public goods, mating strategies, resource competition and many other frequency-dependent biological processes.

Games versus ordinary fixed-fitness selection

With fixed fitnesses, one might write

\[w_A=\text{constant},\qquad w_B=\text{constant}.\]

In a game, the corresponding fitnesses depend on composition:

\[w_A=w_A(x),\qquad w_B=w_B(x).\]

This feedback between frequency and fitness can create internal equilibria, bistability or cyclic behaviour in multi-strategy systems.

Finite populations

Payoff calculations are often first developed for effectively infinite well-mixed populations. In finite populations, demographic sampling adds genetic drift.

Game payoffs can then be incorporated into stochastic processes such as the Moran model by making reproduction probabilities depend on game-derived fitness.

Structured populations

If interactions occur on networks or spatial domains, an individual's opponents may not be sampled from the whole population. Local structure can therefore change expected payoffs and evolutionary outcomes.

What this page does not yet specify

A payoff matrix tells us how strategy success depends on population composition, but it does not by itself specify how frequencies change through time.

To obtain dynamics, we need an evolutionary update rule.

Transition to replicator dynamics

The next lesson introduces the classical deterministic rule in which strategies grow according to how their payoff compares with the population mean:

\[\frac{dx_i}{dt}=x_i(f_i-\bar f).\]

This turns the static payoff structure developed here into a dynamical system.

Key idea. Evolutionary game theory makes fitness frequency-dependent. The payoff matrix determines how well each strategy performs in the current population, while invasion and ESS concepts describe evolutionary resistance. The next step is to specify how those payoff differences drive changes in strategy frequencies.