5.3: Frequency-Dependent Selection
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A polymorphism may also result from frequency-dependent selection. A wellknown model of frequency-dependent selection is the Hawk-Dove game. Most commonly, frequency-dependent selection is studied using game theory, and following John Maynard Smith, one looks for an evolutionarily stable strategy (ESS).
We consider two phenotypes: Hawk and Dove, with no mating between different phenotypes (for example, different phenotypes may correspond to different
| player \(\backslash\) opponent | \(\mathrm{H}\) | \(\mathrm{D}\) |
|---|---|---|
| \(\mathrm{H}\) | \(E_{H H}=-2\) | \(E_{H D}=2\) |
| \(\mathrm{D}\) | \(E_{D H}=0\) | \(E_{D D}=1\) |
Table 5.12: General payoff matrix for the Hawk-Dove game, and the usually assumed values. The payoffs are payed to the player (first column) when playing against the opponent (first row).
species, such as hawks and doves). We describe the Hawk-Dove game as follows: (i) when Hawk meets Dove, Hawk gets the resource and Dove retreats before injury; (ii) when two Hawks meet, they engage in an escalating fight, seriously risking injury, and; (iii) when two Doves meet, they share the resource.
The Hawk-Dove game is modeled by a payoff matrix, as shown in Table \(5.12\). The player in the first column receives the payoff when playing the opponent in the first row. For instance, Hawk playing Dove gets the payoff \(E_{H D}\). The numerical values are commonly chosen such that \(E_{H H}<E_{D H}<E_{D D}<E_{H D}\), that is, Hawk playing Dove does better than Dove playing Dove does better than Dove playing Hawk does better than Hawk playing Hawk.
Frequency-dependent selection occurs because the expected payoff to a Hawk or a Dove depends on the frequency of Hawks and Doves in the population. For example, a Hawk in a population of Doves does well, but a Hawk in a population of Hawks does poorly.
A population of all Doves is unstable to invasion by Hawks (because Hawk playing against Dove does better than Dove playing against Dove), and similarly a population of all Hawks is unstable to invasion by Doves. These two possible equilibria are therefore unstable, and the stable equilibrium consists of a mixed population of Hawks and Doves. In game theory, this mixed equilibrium is called a mixed Nash equilibrium, and is determined by assuming that the expected payoff to a Hawk in a mixed population of Hawks and Doves is the same as the expected payoff to a Dove.
With \(p\) the frequency of Hawks and \(q\) the frequency of Doves, the expected payoff to a Hawk is \(p E_{H H}+q E_{H D}\), and the expected payoff to a Dove is \(p E_{D H}+\) \(q E_{D D}\), so that the mixed Nash equilibrium satisfies
\[p E_{H H}+q E_{H D}=p E_{D H}+q E_{D D} \nonumber \]
Substituting in \(q=1-p\) and solving for \(p\), we obtain
\[p=\frac{E_{H D}-E_{D D}}{\left(E_{H D}-E_{D D}\right)+\left(E_{D H}-E_{H H}\right)} \nonumber \]
and with the numerical values in Table 5.12,
\[\begin{aligned} p_{*} &=\frac{2-1}{(2-1)+(0+2)} \\[4pt] &=1 / 3 . \end{aligned} \nonumber \]
Thus the stable polymorphic population maintained by frequency-dependent selection consists of \(1 / 3\) Hawks and \(2 / 3\) Doves.