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2×2 Games: Difference between revisions
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== [[2×2 Games/Well-mixed populations|Well-mixed populations]] == | == [[2×2 Games/Well-mixed populations|Well-mixed populations]] == | ||
[[Image:Well-mixed 2x2 Games.png|thumb| | [[Image:Well-mixed 2x2 Games.png|thumb|300px|Equilibrium levels of \(A\) and \(B\) types in well-mixed populations.]] | ||
In this simplest scenario encounters between players are completely random. Such a mean-field approximation is valuable because for the replicator equation the dynamics of \(2\times2\) games can be fully analysed. With \(R=1\) and \(P=0\), this results in four dynamical scenarios: | In this simplest scenario encounters between players are completely random. Such a mean-field approximation is valuable because for the replicator equation the dynamics of \(2\times2\) games can be fully analysed. With \(R=1\) and \(P=0\), this results in four dynamical scenarios: | ||
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== [[2×2 Games/Spatial populations|Spatial populations]] == | == [[2×2 Games/Spatial populations|Spatial populations]] == | ||
[[Image:Spatial 2x2 Games.png| | [[Image:Spatial 2x2 Games.png|300px|thumb|Equilibrium levels of \(A\) and \(B\) types in spatially extended populations.]] | ||
<!--[[Image:Spatial 2×2 Games (difference).png|200px|thumb|Differences in equilibrium levels of \(A\) and \(B\) types in spatially extended populations as compared to well-mixed populations.]]--> | <!--[[Image:Spatial 2×2 Games (difference).png|200px|thumb|Differences in equilibrium levels of \(A\) and \(B\) types in spatially extended populations as compared to well-mixed populations.]]--> | ||
In structured populations players are arranged on a lattice or network and interact only with their nearest neighbors. The individuals' ability to form clusters can substantially alter the evolutionary outcome. In particular, comparisons with results from well-mixed populations highlight the effects of spatial structure for the four different scenarios of evolutionary dynamics. | In structured populations players are arranged on a lattice or network and interact only with their nearest neighbors. The individuals' ability to form clusters can substantially alter the evolutionary outcome. In particular, comparisons with results from well-mixed populations highlight the effects of spatial structure for the four different scenarios of evolutionary dynamics. | ||
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== [[2×2 Games/Stochastic dynamics|Stochastic dynamics in finite populations]] == | == [[2×2 Games/Stochastic dynamics|Stochastic dynamics in finite populations]] == | ||
[[Image:Stochastic dynamics - neutral selection, high mutation.png| | [[Image:Stochastic dynamics - neutral selection, high mutation.png|300px|thumb|Stationary distribution of three strategies \(x, y, z\) in a finite population (\(N=60\)) under neutral selection (\(w=0\)) for mutation rates exceeding the critical mutation rate \(u_c=1/(3+N)\).]] | ||
In infinite, well-mixed population, the fraction of players can change continuously, as described by the replicator dynamics in [[2×2 Games / Well-mixed populations|well-mixed populations]]. But there are only \(N\) players, then the fraction must change at least by \(1/N\). | In infinite, well-mixed population, the fraction of players can change continuously, as described by the replicator dynamics in [[2×2 Games / Well-mixed populations|well-mixed populations]]. But there are only \(N\) players, then the fraction must change at least by \(1/N\). | ||
