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Evolutionary Games and Population Dynamics: Difference between revisions
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Evolutionary Games and Population Dynamics (view source)
Revision as of 11:18, 12 October 2023
, 12 October 2023→Payoffs of cooperators and defectors
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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} | ||
{N-1 \choose S-1}(1-w)^{S-1}w^{N-S}. | |||
\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} | ||
\left({\frac u{u+v}}\right)^m \left({\frac v{u+v}}\right)^{S-1-m}{S-1 \choose m}. | |||
\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 | ||
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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 [[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. | 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 [[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. | ||
{{-}} | {{-}} | ||
==References== | ==References== | ||
