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Cooperation in well-mixed populations: Difference between revisions
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Essentially the same results hold for well-mixed populations adopting mixed strategies, i.e. if every individual cooperates with a certain probability <math>p</math>. This propensity to cooperate <math>p</math> evolves towards one of the fixed points derived for pure strategies. To illustrate this, consider a homogenous population where all individuals cooperate with probability <math>p</math>. The fate of a rare mutant <math>q</math> is then given by | Essentially the same results hold for well-mixed populations adopting mixed strategies, i.e. if every individual cooperates with a certain probability <math>p</math>. This propensity to cooperate <math>p</math> evolves towards one of the fixed points derived for pure strategies. To illustrate this, consider a homogenous population where all individuals cooperate with probability <math>p</math>. The fate of a rare mutant <math>q</math> is then given by | ||
\begin{align} | |||
\qquad \frac{dm}{dt} = m(P_q-\bar P), | |||
\end{align} | |||
where <math>m</math> denotes the frequency of the mutant and <math>\bar P</math> the average population payoff. | where <math>m</math> denotes the frequency of the mutant and <math>\bar P</math> the average population payoff. As long as the mutant strategy is rare this can be approximated by | ||
\begin{align} | |||
\qquad \frac{dm}{dt} = m(P_q-P_p), | |||
\end{align} | |||
which means the mutant can invade whenever <math>P_q-P_p>0</math>. Again for small <math>m</math> we obtain | which means the mutant can invade whenever <math>P_q-P_p>0</math>. Again for small <math>m</math> we obtain | ||
\begin{align} | |||
\qquad P_q =&\ p q R + p(1-q)S+(1-p)q\, T+(1-p)(1-q) P\\ | |||
P_p =&\ p q R + p(1-q)T+(1-p)q\, S + (1-p)(1-q) P, | |||
\end{align} | |||
respectively. Thus, <math>q</math> successfully invades whenever | |||
\begin{align} | |||
\qquad (q - p)[p(R-T)+(1-p)(S-P)] > 0 | |||
\end{align} | |||
holds. This result can now be applied to the Prisoner's Dilemma and Snowdrift or Hawk-Dove games. The Prisoner's Dilemma is characterized by <math>T>R>P>S</math> and therefore the expression in square brackets is always negative. Consequentially, any mutant with <math>q<p</math> can invade and take over the population (the latter can be derived from the first equation which is not restricted to rare mutants). Thus, in the long run, the propensity to cooperate converges to <math>x_1 = 0</math> - just as in the pure strategy case. | holds. This result can now be applied to the Prisoner's Dilemma and Snowdrift or Hawk-Dove games. The Prisoner's Dilemma is characterized by <math>T>R>P>S</math> and therefore the expression in square brackets is always negative. Consequentially, any mutant with <math>q<p</math> can invade and take over the population (the latter can be derived from the first equation which is not restricted to rare mutants). Thus, in the long run, the propensity to cooperate converges to <math>x_1 = 0</math> - just as in the pure strategy case. | ||
The argument for the Snowdrift or Hawk-Dove game is slighlty more complicated. Because of the payoff ranking <math>T>R>S>P</math> the expression in square brackets can be both positive or negative, depending on the value of <math>p</math>. It switches sign for | The argument for the Snowdrift or Hawk-Dove game is slighlty more complicated. Because of the payoff ranking <math>T>R>S>P</math> the expression in square brackets can be both positive or negative, depending on the value of <math>p</math>. It switches sign for | ||
\begin{align} | |||
\qquad p=(S-P)/(T-R+S-P)=x_3. | |||
\end{align} | |||
For simplicity let us consider only mutants with arbitrarily small deviations from the resident strategy <math>p</math>. It follows that for <math>p<x_3</math>, a mutant with a slightly higher <math>q</math> can invade but if <math>p>x_3</math> only mutants with slightly lower <math>q</math> can invade. Thus, eventually the propensity to cooperate in the population converges to <math>x_3</math> - again as in the case of pure strategies. | For simplicity let us consider only mutants with arbitrarily small deviations from the resident strategy <math>p</math>. It follows that for <math>p<x_3</math>, a mutant with a slightly higher <math>q</math> can invade but if <math>p>x_3</math> only mutants with slightly lower <math>q</math> can invade. Thus, eventually the propensity to cooperate in the population converges to <math>x_3</math> - again as in the case of pure strategies. | ||
