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Evolutionary Games and Population Dynamics: Difference between revisions
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The emergence and abundance of cooperation in nature poses a tenacious and challenging puzzle to evolutionary biology. Cooperative behavior seems to contradict Darwinian evolution because altruistic individuals increase the fitness of other members of the population at a cost to themselves. Thus, in the absence of supporting mechanisms, cooperation should decrease and vanish, as predicted by classical models for cooperation in evolutionary game theory, such as the [[ | The emergence and abundance of cooperation in nature poses a tenacious and challenging puzzle to evolutionary biology. Cooperative behavior seems to contradict Darwinian evolution because altruistic individuals increase the fitness of other members of the population at a cost to themselves. Thus, in the absence of supporting mechanisms, cooperation should decrease and vanish, as predicted by classical models for cooperation in evolutionary game theory, such as the [[Prisoner's Dilemma]] and [[Public Goods Games]]. | ||
In typical public goods games individuals interact in groups of size \(N\). Cooperators contribute to a common pool at some cost \(c\) while defectors shirk their contributions. The total amount in the common pool is multiplied by a factor \(r\) and evenly distributed among all participants irrespective of their contributions. Defectors attempt to free ride on the contributions of others and because they avoid the costly contributions they are always better off than cooperators. However, if everyone reasons that way the public good is forfeit, yet everyone would prefer the outcome where everyone contributes and earns a payoff of \((r-1)c\). This marks the conflict of interest between the individual and the group that characterizes all [[ | In typical public goods games individuals interact in groups of size \(N\). Cooperators contribute to a common pool at some cost \(c\) while defectors shirk their contributions. The total amount in the common pool is multiplied by a factor \(r\) and evenly distributed among all participants irrespective of their contributions. Defectors attempt to free ride on the contributions of others and because they avoid the costly contributions they are always better off than cooperators. However, if everyone reasons that way the public good is forfeit, yet everyone would prefer the outcome where everyone contributes and earns a payoff of \((r-1)c\). This marks the conflict of interest between the individual and the group that characterizes all [[Social Dilemmas]]. | ||
Traditional approaches to the problem of cooperation based on the replicator dynamics assume constant (infinite) population sizes and thus neglect the ecology of the interacting individuals. Here we incorporate ecological dynamics into evolutionary games and reveal a new mechanism for maintaining cooperation whenever the population density depends on the average population payoff. Defection decreases the population density, due to small payoffs, resulting in smaller interaction group sizes in which cooperation may be favoured. This feedback between ecological dynamics and game dynamics generates fascinating and rich dynamical behavior. Such ''Ecological Public Goods Games'' represent natural extension of replicator dynamics to populations of varying densities. | Traditional approaches to the problem of cooperation based on the replicator dynamics assume constant (infinite) population sizes and thus neglect the ecology of the interacting individuals. Here we incorporate ecological dynamics into evolutionary games and reveal a new mechanism for maintaining cooperation whenever the population density depends on the average population payoff. Defection decreases the population density, due to small payoffs, resulting in smaller interaction group sizes in which cooperation may be favoured. This feedback between ecological dynamics and game dynamics generates fascinating and rich dynamical behavior. Such ''Ecological Public Goods Games'' represent natural extension of replicator dynamics to populations of varying densities. | ||
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===Payoffs of cooperators and defectors=== | ===Payoffs of cooperators and defectors=== | ||
In order to determine the average payoff for cooperators \(f_C\) and defectors \(f_D\) in a population with a normalized density of cooperators, \(u\), and of defectors, \(v\), with \(u+v\leq1\), it is convenient to introduce a new quantity \(w=1-u-v\) which represents reproductive opportunities. If \(w\) approaches one, competition for limited resources, such as space or food, becomes fierce and offspring may not reach adulthood but if \(w\) is small, population densities are low and reproductive opportunities abound. However, at low densities, individuals may no longer be able to find interaction partners. If the public goods group consist of only a single individual no payoffs are awarded as no social interaction occurs. In ecology such density dependence is known as the [ | In order to determine the average payoff for cooperators \(f_C\) and defectors \(f_D\) in a population with a normalized density of cooperators, \(u\), and of defectors, \(v\), with \(u+v\leq1\), it is convenient to introduce a new quantity \(w=1-u-v\) which represents reproductive opportunities. If \(w\) approaches one, competition for limited resources, such as space or food, becomes fierce and offspring may not reach adulthood but if \(w\) is small, population densities are low and reproductive opportunities abound. However, at low densities, individuals may no longer be able to find interaction partners. If the public goods group consist of only a single individual no payoffs are awarded as no social interaction occurs. In ecology such density dependence is known as the [https://en.wikipedia.org/wiki/Allee_effect Allee effect]. | ||
For a given player willing to join the public goods interaction, the probability to find itself in a group of \(S>1\) players is given by | For a given player willing to join the public goods interaction, the probability to find itself in a group of \(S>1\) players is given by | ||
\begin{align} | \begin{align} | ||
{N-1 \choose S-1}(1-w)^{S-1}w^{N-S}. | \qquad {N-1 \choose S-1}(1-w)^{S-1}w^{N-S}. | ||
\end{align} | \end{align} | ||
The probability that there are \(m\) cooperators among the \(S-1\) co-players is | The probability that there are \(m\) cooperators among the \(S-1\) co-players is | ||
\begin{align} | \begin{align} | ||
\left({\frac u{u+v}}\right)^m \left({\frac v{u+v}}\right)^{S-1-m}{S-1 \choose m}. | \qquad \left({\frac u{u+v}}\right)^m \left({\frac v{u+v}}\right)^{S-1-m}{S-1 \choose m}. | ||
\end{align} | \end{align} | ||
In that case the payoff for a defector is \(r m c/S\) where \(r\) is the multiplication factor of the total contributions to the public good and \(c\) the costs of contributing. Hence the expected payoff for a | In that case the payoff for a defector is \(r m c/S\) where \(r\) is the multiplication factor of the total contributions to the public good and \(c\) the costs of contributing. Hence the expected payoff for a | ||
defector in a group of \(S\) players (\(S=2,...,N\)) is | defector in a group of \(S\) players (\(S=2,...,N\)) is | ||
\begin{align*} | \begin{align*} | ||
{\frac{r c}S \sum _{m=0}^{S-1} m }\left({\frac u{u+v}}\right)^m \left({\frac v{u+v}}\right)^{S-1-m}{S-1 \choose m} | \qquad {\frac{r c}S \sum _{m=0}^{S-1} m }\left({\frac u{u+v}}\right)^m \left({\frac v{u+v}}\right)^{S-1-m}{S-1 \choose m} | ||
= \frac{r c}S(S-1)\frac u{u+v}. | = \frac{r c}S(S-1)\frac u{u+v}. | ||
\end{align*} | \end{align*} | ||
Thus, | Thus, | ||
\begin{align*} | \begin{align*} | ||
f_D =& r c\frac u{1-w}\sum _{S=1}^{N}{N-1 \choose S-1}(1-w)^{S-1}w^{N-S}\left(1-\frac{1}{S}\right)\\ | \qquad f_D =&\ r c\frac u{1-w}\sum _{S=1}^{N}{N-1 \choose S-1}(1-w)^{S-1}w^{N-S}\left(1-\frac{1}{S}\right)\\ | ||
=& r c\frac u{1-w}\left[1-\sum _{S=1}^{N}{N-1 \choose S-1}(1-w)^{S-1}w^{N-S}\frac1S\right] | =&\ r c\frac u{1-w}\left[1-\sum _{S=1}^{N}{N-1 \choose S-1}(1-w)^{S-1}w^{N-S}\frac1S\right] | ||
\end{align*} | \end{align*} | ||
and using | and using | ||
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leads to | leads to | ||
\begin{align*} | \begin{align*} | ||
f_D=& r\frac u{1-w}\left(1-\frac{1-w^N}{N(1-w)}\right)c. | \qquad f_D=&\ r\frac u{1-w}\left(1-\frac{1-w^N}{N(1-w)}\right)c. | ||
\end{align*} | \end{align*} | ||
In a group with \(S-1\) co-players playing the public goods game, switching from cooperation | In a group with \(S-1\) co-players playing the public goods game, switching from cooperation | ||
to defection yields \((1-r/S)c\). Hence, | to defection yields \((1-r/S)c\). Hence, | ||
\begin{align*} | \begin{align*} | ||
f_D-f_C=& \sum _{S=2}^N\left(1-{\frac rS}\right){N-1 \choose S-1}(1-w)^{S-1}w^{N-S}c. | \qquad f_D-f_C=& \sum _{S=2}^N\left(1-{\frac rS}\right){N-1 \choose S-1}(1-w)^{S-1}w^{N-S}c. | ||
\end{align*} | \end{align*} | ||
Using the same arguments as before, we obtain | Using the same arguments as before, we obtain | ||
\begin{align*} | \begin{align*} | ||
f_D-f_C=& \left(\!1+(r-1)w^{N-1}-\frac rN{\frac{1-w^N}{1-w}}\right)c =: F(w). | \qquad f_D-f_C=& \left(\!1+(r-1)w^{N-1}-\frac rN{\frac{1-w^N}{1-w}}\right)c =: F(w). | ||
\end{align*} | \end{align*} | ||
The payoff derivation follows <ref>Hauert, Ch., De Monte, S., Hofbauer, J. & Sigmund, K. (2002) Replicator Dynamics in Optional Public Goods Games, ''J. theor. Biol.'' '''218''', 187-194 [http://dx.doi.org/10.1006/jtbi.2002.3067 doi: 10.1006/jtbi.2002.3067].</ref>. Note that the payoff difference between cooperators and defectors only depends on the \(w\), that is on the population density. The sign of \(F(w)\) determines whether it pays to switch from cooperation to defection or vice versa. It turns out that for \(r<2\) it never pays to switch to cooperation but for \(r>2\), \(F(w)\) has a unique root for \(w\in(0,1)\) and hence a critical population density exists below which it pays to switch to cooperation. This occurs whenever the average interaction group size \(\bar S\) drops below the multiplication factor \(r\). Defectors still outperform cooperators in any mixed group but on average, cooperators are better off (such situations are known as [ | The payoff derivation follows <ref>Hauert, Ch., De Monte, S., Hofbauer, J. & Sigmund, K. (2002) Replicator Dynamics in Optional Public Goods Games, ''J. theor. Biol.'' '''218''', 187-194 [http://dx.doi.org/10.1006/jtbi.2002.3067 doi: 10.1006/jtbi.2002.3067].</ref>. Note that the payoff difference between cooperators and defectors only depends on the \(w\), that is on the population density. The sign of \(F(w)\) determines whether it pays to switch from cooperation to defection or vice versa. It turns out that for \(r<2\) it never pays to switch to cooperation but for \(r>2\), \(F(w)\) has a unique root for \(w\in(0,1)\) and hence a critical population density exists below which it pays to switch to cooperation. This occurs whenever the average interaction group size \(\bar S\) drops below the multiplication factor \(r\). Defectors still outperform cooperators in any mixed group but on average, cooperators are better off (such situations are known as [https://en.wikipedia.org/wiki/Simpson%27s_paradox Simpson's paradox]). However, in that case, defectors would be even better off by switching to cooperation because each dollar invested in the common pool has a positive return for the investor. | ||
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