Stochastic dynamics in finite populations

In infinite, well-mixed population, the fraction of players can change continuously, as described by the replicator dynamics in well-mixed populations. But in finite populations of size $N$, the fraction must change at least by $1/N$.

In this case microscopic probabilities have to defined that describe how a player switches strategy, as in spatial evolutionary games. There are many ways to define such microscopic evolutionary process. In each of them, strategies that lead to higher payoffs are more likely to spread in the population. For example, two players can be chosen at random to compare their payoffs. The probability that a player adopts the strategy of the other player can be a linear function of the payoff difference. If only better strategies are adopted, the direction of the dynamics becomes deterministic in 2×2 games. But if also worse strategies are sometimes adopted with a small probability, then even a dominant strategy will only take over the population with a certain probability. This approach provides a natural connection between evolutionary game theory and theoretical population genetics, where such probabilities are routinely studied.

Besides the game, two parameters describe the dynamics: The population size $N$ and the intensity of selection $w$, which measures how much the adoption of someone else’s strategy depends on the payoffs. If the product of $w$ and $N$ is small, one speaks of weak selection and the dynamics is a small correction to random drift. If the product is large, then a deterministic replicator equation is recovered from finite population dynamics.

For weak selection, several new features appear in the system: In a bistable situation, one strategy can displace the other. Thus, a new concept of evolutionary stability is necessary. If we consider a single mutant in a population of size $N$, it will take over the population with probability $1/N$ without selection, because each individual is equally likely to eventually become the ultimate ancestor. Adding a little amount of selection, a mutant is first disfavored in a bistable situation, but once it has reached a critical fraction, it is favored. The probability that a mutant will take over is a global measure for this process. Interestingly, this probability is larger than $1/N$ if the mutants become advantageous at a frequency larger than $1/3$ and smaller then $1/N$ otherwise, independent of the other details of the underlying game. This result holds for many evolutionary processes under weak selection. Using tools from population genetics, it can be proven that it holds for all processes within the domain of Kingman’s coalescence.

Rock-Paper-Scissors game

Blabla.

Payoff matrix: ${\begin{matrix}~&{\begin{matrix}\ \ R\quad &S\quad &P\quad \end{matrix}}\\{\begin{matrix}R\\S\\P\end{matrix}}&{\begin{pmatrix}0&{\frac {s}{2}}&-1\\-1&0&2+s\\{\frac {1+s}{3}}&-1&0\end{pmatrix}}\end{matrix}}$

Fixed point: ${\hat {\mathbf {x} }}=\left({\frac {1}{2}},{\frac {1}{3}},{\frac {1}{6}}\right)$

Deterministic Dynamics

Replicator dynamics

Replicator-Mutator dynamics

Stochastic Dynamics

Stochastic differential equations

Stochastic differential equations, with mutations

Individual Based Simulations

Individual based simulations

Individual based simulations, with mutations

From finite to infinite populations

File:SDE vs IBS Performance

Performance comparison of individual based simulations (IBS) versus stochastic differential equations (SDE). a ratio of the CPU times $CPU_{\text{SDE}}/CPU_{\text{IBS}}$ as a function of the population size, $N$, and the number of strategic types, $d$. The bold contour indicates equal performance. For small $N$ and large $d$ IBS are faster (red region), but for larger $N$ and smaller $d$ SDE are faster (blue region). Each contour indicates a performance difference of one order of magnitude. b computational time with $d=10$ as a function of $N$ for IBD (red) and SDE (blue). As a reference for the scaling Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N^2} (red) and a constant (blue) are shown. c computational time with $N=5000$ as a function of $d$ for IBD (red) and SDE (blue). As a reference for the scaling $d^{1/2}$ (red) and $d^{3}$ (blue) are shown. For a proper scaling argument much larger $d$ are required but already $d=100$ far exceeds typical evolutionary models and hence is only of limited relevance in the current context. All comparisons use a constant payoff matrix and the local update process (such that ${\mathcal {A}}_{k}(\mathbf {x} )=0$ and $\gamma (\pi _{j},\pi _{k})=1/2$), a mutation rate of $1/N$ and are based on at least $1000$ time steps as well as at least one minute running time. CPU time is measured in milliseconds required to calculate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1000} time steps. The time increment for the SDE is $dt=0.01$.

In unstructured, finite populations of constant size, $N$, consisting of $d$ distinct strategic types and with a mutation rate, $\mu$, evolutionary changes can be described by the following class of birth-death processes: In each time step, one individual of type Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} produces a single offspring and displaces another randomly selected individual of type $k$. With probability $1-\mu$, no mutation occurs and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} produces an offspring of the same type. But with probability $\mu$, the offspring of an individual of type Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\neq j} ) mutates into a type $j$ individual. This results in two distinct ways to increase the number of $j$ types by one at the expense of decreasing the number of $k$ types by one, hence keeping the population size constant. Biologically, keeping Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} constant implies that the population has reached a stable ecological equilibrium and assumes that this equilibrium remains unaffected by trait frequencies. The probability for the event of replacing a type Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} individual with a type Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} individual is denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{kj}} and is a function of the state of the population Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf X=(X_1, X_2, \ldots X_d)} , with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_n} indicating the number of individuals of type Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textstyle\sum_{n=1}^d X_n = N} .

For such processes we can easily derive a Master equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P^{\tau+1}({\mathbf X}) = P^{\tau}({\mathbf X})+\!\sum_{j,k=1}^d\! P^{\tau}({\mathbf X}_j^k) T_{kj}({\mathbf X}_j^k)-P^{\tau}({\mathbf X})\ T_{jk}({\mathbf X}))}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P^\tau({\mathbf X})} denotes the probability of being in state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf X} at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tau} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbf X}_j^k=(X_1, \ldots X_j-1, \ldots X_k+1, \ldots X_d)} represents a state adjacent to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf X} . For large but finite Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} the Kramers-Moyal expansion yields a convenient approximation in the form of a Fokker-Planck equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot \rho({\mathbf x}) = -\sum_{k=1}^{d-1}\frac{\partial}{\partial x_k}\rho({\mathbf x}){\mathcal A}_{k}({\mathbf x})+\frac12\sum_{j,k=1}^{d-1}\frac{\partial^2}{\partial x_k\partial x_j}\rho({\mathbf x}){\mathcal B}_{jk}({\mathbf x})}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbf x} = {\mathbf X}/N} represents the state of the population in terms of frequencies of the different strategic types and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho({\mathbf x})} is the probability density in state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf x} . The drift vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal A}_{k}({\mathbf x})} is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal A}_{k}({\mathbf x}) = \sum_{j=1}^{d} \Big(T_{j k}({\mathbf x}) - T_{k j}({\mathbf x}) \Big) = -1+\sum_{j=1}^{d} T_{j k}({\mathbf x})}

For the second equality we have used Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textstyle\sum_{j=1}^d T_{kj}({\boldsymbol x})=1} , which simply states that a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} -type individual transitions to some other type (including staying type Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} ) with probability one. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal A}_{k}({\mathbf x})} is bounded in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [-1, d-1]} because the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_{jk}} are probabilities.

The diffusion matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal B}_{jk}({\mathbf x})} is defined as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $$\begin{array}{}\text{(1)}& {\mathcal{B}}_{jk}(\mathbf{x})& =-\frac{1}{N}[T_{jk}(\mathbf{x})+T_{kj}(\mathbf{x})]\mathrm{f}\mathrm{o}\mathrm{r}j\ne k\text{(2)}& {\mathcal{B}}_{jj}(\mathbf{x})& =\frac{1}{N}\sum _{l=1,l\ne j}^d(T_{jl}(\mathbf{x})+T_{lj}(\mathbf{x}))\end{array}$$ }

Note that the diffusion matrix is symmetric, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal B}_{jk}({\mathbf x}) = {\mathcal B}_{kj}({\mathbf x})} and vanishes as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim 1/N} in the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N\to\infty} .

The noise arising through demographic changes and mutations is uncorrelated in time and hence the Itô calculus can be applied to derive a Langevin equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot x_k = {\mathcal A}_k({\mathbf x}) + \sum_{j=1}^{d-1} {{\mathcal C}_{kj}({\mathbf x})} \xi_j(t) }

where the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \xi_j(t)} represent uncorrelated Gaussian white noise with unit variance, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \xi_k(t) \xi_j(t^\prime) \rangle = \delta_{kj} \delta(t-t^\prime)} . The matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {{\mathcal C}({\mathbf x})}} is defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal C}^T({\mathbf x}) {\mathcal C}({\mathbf x}) = {\mathcal B}({\mathbf x})} and its off-diagonal elements are responsible for correlations in the noise of different strategic types. In the limit Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N\to\infty} the matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathcal C}({\mathbf x})} vanishes with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sim 1/\sqrt{N}} and we recover a deterministic replicator mutator equation.

References

• Traulsen, A., Claussen, J. C. & Hauert, C. (2012) Stochastic differential equations for evolutionary dynamics with demographic noise and mutations. Phys. Rev. E in print.
• Traulsen, A., Claussen, J. C. & Hauert, C. (2006) Coevolutionary dynamics in large, but finite populations. Phys. Rev. E 74 011901.
• Traulsen, A., Claussen, J. C. & Hauert, C. (2005) Coevolutionary Dynamics: From Finite to Infinite Populations. Phys. Rev. Lett. 95 238701.