<p>Mean field Game (MFG) Theory derived by combining stochastic optimal control with a statistical description of multiple populations of competing agents is increasingly used in many applications, especially those involving acquisition of a resource. Using an ergodic, 2-population MFG model it is demonstrated that nonlinear dynamics caused by the model’s underlying bifurcation structure can dominate predictions as the influence of the optimal control is enhanced by lowering the stochastic diffusivity <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\sigma$</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>. Multiple ergodic states are predicted, and rapid segregation of the distributions is observed over small ranges of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\sigma$</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> close to bifurcation points. Introducing bias toward agents with more expertise breaks the bifurcation structure, however the effects of the bifurcations persist in the resulting multiple disconnected states. Including a common resource that is self-regenerating models the Tragedy of the Commons (TOTC) where the resource is depleted by over-acquisition by the competing populations. The dependence of the TOTC limit on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\sigma$</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation> and the regeneration rate <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(R_{N} )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mi>N</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> exhibits the residual influence of the bifurcation structure and controls the sensitivity of the limiting value of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$R_{N} .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mi>N</mi> </msub> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Large differences in the maximum resource acquisition between the two populations are predicted.</p>

Nonlinear Dynamics in 2-Population Mean Field Games Competing for a Resource: Influence on the Tragedy of the Commons

期刊论文
出版模式：

开放获取
发表日期：

2026年7月20日

查看全文

Robert A. Brown

摘要

Mean field Game (MFG) Theory derived by combining stochastic optimal control with a statistical description of multiple populations of competing agents is increasingly used in many applications, especially those involving acquisition of a resource. Using an ergodic, 2-population MFG model it is demonstrated that nonlinear dynamics caused by the model’s underlying bifurcation structure can dominate predictions as the influence of the optimal control is enhanced by lowering the stochastic diffusivity $\sigma$ $σ$ . Multiple ergodic states are predicted, and rapid segregation of the distributions is observed over small ranges of $\sigma$ $σ$ close to bifurcation points. Introducing bias toward agents with more expertise breaks the bifurcation structure, however the effects of the bifurcations persist in the resulting multiple disconnected states. Including a common resource that is self-regenerating models the Tragedy of the Commons (TOTC) where the resource is depleted by over-acquisition by the competing populations. The dependence of the TOTC limit on $\sigma$ $σ$ and the regeneration rate $(R_{N} )$ $(R_{N})$ exhibits the residual influence of the bifurcation structure and controls the sensitivity of the limiting value of $R_{N} .$ $R_{N} .$ Large differences in the maximum resource acquisition between the two populations are predicted.