https://wiki.evoludo.org/index.php?title=Evolutionary_Games_and_Population_Dynamics&feed=atom&action=history Evolutionary Games and Population Dynamics - Revision history 2024-03-28T08:57:31Z Revision history for this page on the wiki MediaWiki 1.40.0 https://wiki.evoludo.org/index.php?title=Evolutionary_Games_and_Population_Dynamics&diff=2684&oldid=prev Hauert at 15:47, 13 October 2023 2023-10-13T15:47:31Z <p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 08:47, 13 October 2023</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l2">Line 2:</td> <td colspan="2" class="diff-lineno">Line 2:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{TOCright}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{TOCright}}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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 [[<del style="font-weight: bold; text-decoration: none;">prisoner</del>'s <del style="font-weight: bold; text-decoration: none;">dilemma</del>]] and [[<del style="font-weight: bold; text-decoration: none;">public goods games</del>]].</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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 [[<ins style="font-weight: bold; text-decoration: none;">Prisoner</ins>'s <ins style="font-weight: bold; text-decoration: none;">Dilemma</ins>]] and [[<ins style="font-weight: bold; text-decoration: none;">Public Goods Games</ins>]].</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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 [[<del style="font-weight: bold; text-decoration: none;">social dilemmas</del>]].</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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 [[<ins style="font-weight: bold; text-decoration: none;">Social Dilemmas</ins>]].</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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 &#039;&#039;Ecological Public Goods Games&#039;&#039; represent natural extension of replicator dynamics to populations of varying densities.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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 &#039;&#039;Ecological Public Goods Games&#039;&#039; represent natural extension of replicator dynamics to populations of varying densities.</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l30">Line 30:</td> <td colspan="2" class="diff-lineno">Line 30:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Payoffs of cooperators and defectors===</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Payoffs of cooperators and defectors===</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>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 [<del style="font-weight: bold; text-decoration: none;">[</del>Allee effect<del style="font-weight: bold; text-decoration: none;">]</del>].</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>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 [<ins style="font-weight: bold; text-decoration: none;">https://en.wikipedia.org/wiki/Allee_effect </ins>Allee effect].</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For a given player willing to join the public goods interaction, the probability to find itself in a group of \(S&gt;1\) players is given by</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For a given player willing to join the public goods interaction, the probability to find itself in a group of \(S&gt;1\) players is given by</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{align}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{align}</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{N-1 \choose S-1}(1-w)^{S-1}w^{N-S}.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\qquad </ins>{N-1 \choose S-1}(1-w)^{S-1}w^{N-S}.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{align}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{align}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The probability that there are \(m\) cooperators among the \(S-1\) co-players is</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The probability that there are \(m\) cooperators among the \(S-1\) co-players is</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{align}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{align}</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\left({\frac u{u+v}}\right)^m \left({\frac v{u+v}}\right)^{S-1-m}{S-1 \choose m}.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\qquad </ins>\left({\frac u{u+v}}\right)^m \left({\frac v{u+v}}\right)^{S-1-m}{S-1 \choose m}.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{align}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{align}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>defector in a group of \(S\) players (\(S=2,...,N\)) is</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>defector in a group of \(S\) players (\(S=2,...,N\)) is</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{align*}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{align*}</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{\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}</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\qquad </ins>{\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}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>= \frac{r c}S(S-1)\frac u{u+v}.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>= \frac{r c}S(S-1)\frac u{u+v}.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{align*}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{align*}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Thus,</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Thus,</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{align*}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{align*}</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>f_D =&amp; 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)\\</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\qquad </ins>f_D =&amp;<ins style="font-weight: bold; text-decoration: none;">\ </ins>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)\\</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>=&amp; 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]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>=&amp;<ins style="font-weight: bold; text-decoration: none;">\ </ins>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]</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{align*}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{align*}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>and using</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>and using</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l55">Line 55:</td> <td colspan="2" class="diff-lineno">Line 55:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>leads to</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>leads to</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{align*}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{align*}</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>f_D=&amp; r\frac u{1-w}\left(1-\frac{1-w^N}{N(1-w)}\right)c.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\qquad </ins>f_D=&amp;<ins style="font-weight: bold; text-decoration: none;">\ </ins>r\frac u{1-w}\left(1-\frac{1-w^N}{N(1-w)}\right)c.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{align*}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{align*}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In a group with \(S-1\) co-players playing the public goods game, switching from cooperation</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In a group with \(S-1\) co-players playing the public goods game, switching from cooperation</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>to defection yields \((1-r/S)c\). Hence,</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>to defection yields \((1-r/S)c\). Hence,</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{align*}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{align*}</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>f_D-f_C=&amp; \sum _{S=2}^N\left(1-{\frac rS}\right){N-1 \choose S-1}(1-w)^{S-1}w^{N-S}c.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\qquad </ins>f_D-f_C=&amp; \sum _{S=2}^N\left(1-{\frac rS}\right){N-1 \choose S-1}(1-w)^{S-1}w^{N-S}c.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{align*}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{align*}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Using the same arguments as before, we obtain</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Using the same arguments as before, we obtain</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{align*}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{align*}</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>f_D-f_C=&amp; \left(\!1+(r-1)w^{N-1}-\frac rN{\frac{1-w^N}{1-w}}\right)c =: F(w).</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\qquad </ins>f_D-f_C=&amp; \left(\!1+(r-1)w^{N-1}-\frac rN{\frac{1-w^N}{1-w}}\right)c =: F(w).</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{align*}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{align*}</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The payoff derivation follows &lt;ref&gt;Hauert, Ch., De Monte, S., Hofbauer, J. &amp; 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].&lt;/ref&gt;. 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 \(r&lt;2\) it never pays to switch to cooperation but for \(r&gt;2\), \(F(w)\) 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 [<del style="font-weight: bold; text-decoration: none;">[</del>Simpson's paradox<del style="font-weight: bold; text-decoration: none;">]</del>]). 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.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The payoff derivation follows &lt;ref&gt;Hauert, Ch., De Monte, S., Hofbauer, J. &amp; 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].&lt;/ref&gt;. 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 \(r&lt;2\) it never pays to switch to cooperation but for \(r&gt;2\), \(F(w)\) 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 [<ins style="font-weight: bold; text-decoration: none;">https://en.wikipedia.org/wiki/Simpson%27s_paradox </ins>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.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{-}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{-}}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> </table> Hauert https://wiki.evoludo.org/index.php?title=Evolutionary_Games_and_Population_Dynamics&diff=2664&oldid=prev Hauert: /* Payoffs of cooperators and defectors */ 2023-10-12T18:18:52Z <p><span dir="auto"><span class="autocomment">Payoffs of cooperators and defectors</span></span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 11:18, 12 October 2023</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l33">Line 33:</td> <td colspan="2" class="diff-lineno">Line 33:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For a given player willing to join the public goods interaction, the probability to find itself in a group of \(S&gt;1\) players is given by</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For a given player willing to join the public goods interaction, the probability to find itself in a group of \(S&gt;1\) players is given by</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\<del style="font-weight: bold; text-decoration: none;">[</del>{N-1 \choose S-1}(1-w)^{S-1}w^{N-S}.\<del style="font-weight: bold; text-decoration: none;">]</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\<ins style="font-weight: bold; text-decoration: none;">begin{align}</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{N-1 \choose S-1}(1-w)^{S-1}w^{N-S}.</div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\<ins style="font-weight: bold; text-decoration: none;">end{align}</ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The probability that there are \(m\) cooperators among the \(S-1\) co-players is</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The probability that there are \(m\) cooperators among the \(S-1\) co-players is</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\<del style="font-weight: bold; text-decoration: none;">[</del>\left({\frac u{u+v}}\right)^m \left({\frac v{u+v}}\right)^{S-1-m}{S-1 \choose m}.\<del style="font-weight: bold; text-decoration: none;">]</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\<ins style="font-weight: bold; text-decoration: none;">begin{align}</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\left({\frac u{u+v}}\right)^m \left({\frac v{u+v}}\right)^{S-1-m}{S-1 \choose m}.</div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\<ins style="font-weight: bold; text-decoration: none;">end{align}</ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>defector in a group of \(S\) players (\(S=2,...,N\)) is</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>defector in a group of \(S\) players (\(S=2,...,N\)) is</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l64">Line 64:</td> <td colspan="2" class="diff-lineno">Line 68:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The payoff derivation follows &lt;ref&gt;Hauert, Ch., De Monte, S., Hofbauer, J. &amp; Sigmund, K. (2002) Replicator Dynamics in Optional Public Goods Games, &#039;&#039;J. theor. Biol.&#039;&#039; &#039;&#039;&#039;218&#039;&#039;&#039;, 187-194 [http://dx.doi.org/10.1006/jtbi.2002.3067 doi: 10.1006/jtbi.2002.3067].&lt;/ref&gt;. 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 \(r&lt;2\) it never pays to switch to cooperation but for \(r&gt;2\), \(F(w)\) 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&#039;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.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The payoff derivation follows &lt;ref&gt;Hauert, Ch., De Monte, S., Hofbauer, J. &amp; Sigmund, K. (2002) Replicator Dynamics in Optional Public Goods Games, &#039;&#039;J. theor. Biol.&#039;&#039; &#039;&#039;&#039;218&#039;&#039;&#039;, 187-194 [http://dx.doi.org/10.1006/jtbi.2002.3067 doi: 10.1006/jtbi.2002.3067].&lt;/ref&gt;. 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 \(r&lt;2\) it never pays to switch to cooperation but for \(r&gt;2\), \(F(w)\) 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&#039;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.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{-}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{-}}</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2" class="diff-side-added"></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr> </table> Hauert https://wiki.evoludo.org/index.php?title=Evolutionary_Games_and_Population_Dynamics&diff=1167&oldid=prev Hauert at 19:04, 31 March 2012 2012-03-31T19:04:58Z <p></p> <a href="//wiki.evoludo.org/index.php?title=Evolutionary_Games_and_Population_Dynamics&amp;diff=1167&amp;oldid=1042">Show changes</a> Hauert https://wiki.evoludo.org/index.php?title=Evolutionary_Games_and_Population_Dynamics&diff=1042&oldid=prev Hauert at 23:41, 21 March 2012 2012-03-21T23:41:23Z <p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:41, 21 March 2012</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{InCharge|author1=Christoph Hauert}}</ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{TOCright}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{TOCright}}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l25">Line 25:</td> <td colspan="2" class="diff-lineno">Line 26:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* </del>Wakano, J. Y., <del style="font-weight: bold; text-decoration: none;">Martin </del>A. <del style="font-weight: bold; text-decoration: none;">Nowak </del>&amp; Hauert, Ch. (2009) Spatial Dynamics of Ecological Public Goods, ''Proc. Natl. Acad. Sci. USA'' '''106'''<del style="font-weight: bold; text-decoration: none;">, </del>7910-7914.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"># </ins>Wakano, J. Y. <ins style="font-weight: bold; text-decoration: none;">&amp; Hauert</ins>, <ins style="font-weight: bold; text-decoration: none;">Ch. (2011) Pattern formation and chaos in spatial ecological public goods games, ''J. theor. Biol.'' '''268''' 30-38 [http://dx.doi.org/10.1016/j.jtbi.2010.09.036 doi: 10.1016/j.jtbi.2010.09.036].</ins></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* </del>Hauert, Ch., Wakano, J. Y. &amp; Doebeli, M. (2008) Ecological Public Goods Games: cooperation and bifurcation, ''Theor. Pop. Biol.'' '''73''', 257-263.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"># Wakano, J. Y., Nowak, M. </ins>A. &amp; Hauert, Ch. (2009) Spatial Dynamics of Ecological Public Goods, ''Proc. Natl. Acad. Sci. USA'' '''106''' 7910-7914 <ins style="font-weight: bold; text-decoration: none;">[http://dx.doi.org/10.1073/pnas.0812644106 doi: 10.1073/pnas.0812644106]</ins>.</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">* </del>Hauert, <del style="font-weight: bold; text-decoration: none;">Ch</del>., Holmes, M. &amp; Doebeli, M. (2006) Evolutionary games and population dynamics: maintenance of cooperation in public goods games, Proc. R. Soc. Lond B 273, 2565-2570. <del style="font-weight: bold; text-decoration: none;">Corrigendum</del>: ''Proc. R. Soc. Lond B'' '''273''', 3131-3132.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"># </ins>Hauert, Ch., Wakano, J. Y. &amp;<ins style="font-weight: bold; text-decoration: none;">amp; </ins>Doebeli, M. (2008) Ecological Public Goods Games: cooperation and bifurcation, ''Theor. Pop. Biol.'' '''73''', 257-263 <ins style="font-weight: bold; text-decoration: none;">[http://dx.doi.org/10.1016/j.tpb.2007.11.007 doi:10.1016/j.tpb.2007.11.007]</ins>.</div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"># </ins>Hauert, <ins style="font-weight: bold; text-decoration: none;">C</ins>., Holmes, M. &amp;<ins style="font-weight: bold; text-decoration: none;">amp; </ins>Doebeli, M. (2006) Evolutionary games and population dynamics: maintenance of cooperation in public goods games, <ins style="font-weight: bold; text-decoration: none;">''</ins>Proc. R. Soc. Lond<ins style="font-weight: bold; text-decoration: none;">. </ins>B<ins style="font-weight: bold; text-decoration: none;">'' '''</ins>273<ins style="font-weight: bold; text-decoration: none;">'''</ins>, 2565-2570 <ins style="font-weight: bold; text-decoration: none;">[http://dx.doi.org/10.1098/rspb.2006.3600 doi: 10.1098/rspb.2006</ins>.<ins style="font-weight: bold; text-decoration: none;">3600]; Addendum</ins>: ''Proc. R. Soc. Lond B'' '''273''', 3131-3132 <ins style="font-weight: bold; text-decoration: none;">[http://dx.doi.org/10.1098/rspb.2006.3717 doi: 10.1098/rspb.2006.3717]</ins>.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Tutorial]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Tutorial]]</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Christoph Hauert]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr> </table> Hauert https://wiki.evoludo.org/index.php?title=Evolutionary_Games_and_Population_Dynamics&diff=683&oldid=prev Hauert at 20:34, 19 July 2009 2009-07-19T20:34:50Z <p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:34, 19 July 2009</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l25">Line 25:</td> <td colspan="2" class="diff-lineno">Line 25:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==References==</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* Wakano, J. Y., Martin A. Nowak &amp; Hauert, Ch. (2009) Spatial Dynamics of Ecological Public Goods, ''Proc. Natl. Acad. Sci. USA'' '''<del style="font-weight: bold; text-decoration: none;">X</del>''', <del style="font-weight: bold; text-decoration: none;">X</del>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* Wakano, J. Y., Martin A. Nowak &amp; Hauert, Ch. (2009) Spatial Dynamics of Ecological Public Goods, ''Proc. Natl. Acad. Sci. USA'' '''<ins style="font-weight: bold; text-decoration: none;">106</ins>''', <ins style="font-weight: bold; text-decoration: none;">7910-7914</ins>.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Hauert, Ch., Wakano, J. Y. &amp; Doebeli, M. (2008) Ecological Public Goods Games: cooperation and bifurcation, &#039;&#039;Theor. Pop. Biol.&#039;&#039; &#039;&#039;&#039;73&#039;&#039;&#039;, 257-263.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Hauert, Ch., Wakano, J. Y. &amp; Doebeli, M. (2008) Ecological Public Goods Games: cooperation and bifurcation, &#039;&#039;Theor. Pop. Biol.&#039;&#039; &#039;&#039;&#039;73&#039;&#039;&#039;, 257-263.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Hauert, Ch., Holmes, M. &amp; Doebeli, M. (2006) Evolutionary games and population dynamics: maintenance of cooperation in public goods games, Proc. R. Soc. Lond B 273, 2565-2570. Corrigendum: &#039;&#039;Proc. R. Soc. Lond B&#039;&#039; &#039;&#039;&#039;273&#039;&#039;&#039;, 3131-3132.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Hauert, Ch., Holmes, M. &amp; Doebeli, M. (2006) Evolutionary games and population dynamics: maintenance of cooperation in public goods games, Proc. R. Soc. Lond B 273, 2565-2570. Corrigendum: &#039;&#039;Proc. R. Soc. Lond B&#039;&#039; &#039;&#039;&#039;273&#039;&#039;&#039;, 3131-3132.</div></td></tr> </table> Hauert https://wiki.evoludo.org/index.php?title=Evolutionary_Games_and_Population_Dynamics&diff=669&oldid=prev Hauert: /* Pattern formation in spatial populations */ 2009-07-10T22:24:37Z <p><span dir="auto"><span class="autocomment">Pattern formation in spatial populations</span></span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:24, 10 July 2009</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l18">Line 18:</td> <td colspan="2" class="diff-lineno">Line 18:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{-}}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{-}}</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>== [[Evolutionary Games and Population Dynamics / <del style="font-weight: bold; text-decoration: none;">Reaction</del>-diffusion|Pattern formation in spatial populations]] ==</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>== [[Evolutionary Games and Population Dynamics / <ins style="font-weight: bold; text-decoration: none;">Selection</ins>-diffusion|Pattern formation in spatial populations]] ==</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Image:Spatial Ecological PGG - Chaos.jpg|thumb|200px|Spatio-temporal chaos emerging from the spatial dynamics of ecological public goods.]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Image:Spatial Ecological PGG - Chaos.jpg|thumb|200px|Spatio-temporal chaos emerging from the spatial dynamics of ecological public goods.]]</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Spatial &#039;reaction-diffusion&#039; dynamics promotes cooperation based on different types of pattern formation processes. Individuals can migrate (diffuse) in order to populate new territories. Slow diffusion of cooperators fosters aggregation in highly productive patches (activation), whereas fast diffusion enables defectors to readily locate and exploit these patches (inhibition). These antagonistic forces promote co-existence of cooperators and defectors in static or dynamic patterns, including spatial chaos of ever changing configurations.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Spatial &#039;reaction-diffusion&#039; dynamics promotes cooperation based on different types of pattern formation processes. Individuals can migrate (diffuse) in order to populate new territories. Slow diffusion of cooperators fosters aggregation in highly productive patches (activation), whereas fast diffusion enables defectors to readily locate and exploit these patches (inhibition). These antagonistic forces promote co-existence of cooperators and defectors in static or dynamic patterns, including spatial chaos of ever changing configurations.</div></td></tr> </table> Hauert https://wiki.evoludo.org/index.php?title=Evolutionary_Games_and_Population_Dynamics&diff=538&oldid=prev Hauert: /* Pattern formation in spatial populations */ 2009-03-27T04:43:02Z <p><span dir="auto"><span class="autocomment">Pattern formation in spatial populations</span></span></p> <p><b>New page</b></p><div>{{TOCright}}<br /> <br /> 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&#039;s dilemma and public goods games.<br /> <br /> 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 &#039;&#039;Ecological Public Goods Games&#039;&#039; represent natural extension of replicator dynamics to populations of varying densities.<br /> <br /> {{-}}<br /> <br /> <br /> ==Ecological Public Goods==<br /> <br /> {{-}}<br /> == [[Evolutionary Games and Population Dynamics / Well-mixed populations|Bifurcations in well-mixed populations]] ==<br /> [[Image:Well-mixed Ecological PGG - stable limit cycle.png|thumb|200px|Stable limit cycles are observed in the vicinity of a sub-critical Hopf-bifurcation, which gives rise to ever lasting periodic oscillations of cooperators and defectors.]]<br /> In infinite populations where individuals randomly interact in public goods games, cooperators are doomed and readily disappear. In contrast, varying population densities can lead to stable coexistence of cooperators and defectors in public goods games. When increasing the efficiency of the public good the system undergoes a series of bifurcations and the dynamics ranges from extinction, to periodic oscillations and finally stable co-existence.<br /> <br /> <br /> {{-}}<br /> <br /> == [[Evolutionary Games and Population Dynamics / Reaction-diffusion|Pattern formation in spatial populations]] ==<br /> [[Image:Spatial Ecological PGG - Chaos.jpg|thumb|200px|Spatio-temporal chaos emerging from the spatial dynamics of ecological public goods.]]<br /> Spatial &#039;reaction-diffusion&#039; dynamics promotes cooperation based on different types of pattern formation processes. Individuals can migrate (diffuse) in order to populate new territories. Slow diffusion of cooperators fosters aggregation in highly productive patches (activation), whereas fast diffusion enables defectors to readily locate and exploit these patches (inhibition). These antagonistic forces promote co-existence of cooperators and defectors in static or dynamic patterns, including spatial chaos of ever changing configurations.<br /> <br /> {{-}}<br /> <br /> ==References==<br /> * Wakano, J. Y., Martin A. Nowak &amp; Hauert, Ch. (2009) Spatial Dynamics of Ecological Public Goods, &#039;&#039;Proc. Natl. Acad. Sci. USA&#039;&#039; &#039;&#039;&#039;X&#039;&#039;&#039;, X.<br /> * Hauert, Ch., Wakano, J. Y. &amp; Doebeli, M. (2008) Ecological Public Goods Games: cooperation and bifurcation, &#039;&#039;Theor. Pop. Biol.&#039;&#039; &#039;&#039;&#039;73&#039;&#039;&#039;, 257-263.<br /> * Hauert, Ch., Holmes, M. &amp; Doebeli, M. (2006) Evolutionary games and population dynamics: maintenance of cooperation in public goods games, Proc. R. Soc. Lond B 273, 2565-2570. Corrigendum: &#039;&#039;Proc. R. Soc. Lond B&#039;&#039; &#039;&#039;&#039;273&#039;&#039;&#039;, 3131-3132.<br /> <br /> [[Category:Tutorial]]<br /> [[Category:Christoph Hauert]]</div> Hauert