The paper discusses a class of discrete-time zero-sum stochastic games where the objective of players is to control a large population of N interacting objects (e.g., agents, particles, data, etc.). Objects can be classified according to their characteristics within a finite or countable set. At each stage, once the players select actions, the objects move randomly among the classes under a transition law that depends on an unknown parameter. The fact that N is too large makes it almost impossible to apply standard procedures. Hence, the game problem is studied following a mean-field approach. That is, a zero-sum game model <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal{G}\mathcal{M}_{N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">G</mi> <msub> <mi mathvariant="script">M</mi> <mi>N</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, where the states are the proportions of objects in each class, is introduced. Next we study the mean-field limit as <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$N\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> to obtain a new game model <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal{G}\mathcal{M}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">G</mi> <mi mathvariant="script">M</mi> </mrow> </math></EquationSource> </InlineEquation>, independent of N, which is easier to analyze than <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal{G}\mathcal{M}_{N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">G</mi> <msub> <mi mathvariant="script">M</mi> <mi>N</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. By applying suitable estimation and control processes on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal{G}\mathcal{M}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">G</mi> <mi mathvariant="script">M</mi> </mrow> </math></EquationSource> </InlineEquation>, we construct a pair of strategies, and our objective is to prove that it is nearly optimal in the original game model <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal{G}\mathcal{M}_{N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">G</mi> <msub> <mi mathvariant="script">M</mi> <mi>N</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>.

Large population discounted zero-sum games with unknown disturbance distribution: a mean-field approach

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发表日期：

2026年5月5日

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Carmen G. Higuera-Chan,
J. Adolfo Minjárez-Sosa

摘要

The paper discusses a class of discrete-time zero-sum stochastic games where the objective of players is to control a large population of N interacting objects (e.g., agents, particles, data, etc.). Objects can be classified according to their characteristics within a finite or countable set. At each stage, once the players select actions, the objects move randomly among the classes under a transition law that depends on an unknown parameter. The fact that N is too large makes it almost impossible to apply standard procedures. Hence, the game problem is studied following a mean-field approach. That is, a zero-sum game model $\mathcal{G}\mathcal{M}_{N}$ $G M_{N}$ , where the states are the proportions of objects in each class, is introduced. Next we study the mean-field limit as $N\rightarrow \infty $ $N \to \infty$ to obtain a new game model $\mathcal{G}\mathcal{M}$ $G M$ , independent of N, which is easier to analyze than $\mathcal{G}\mathcal{M}_{N}$ $G M_{N}$ . By applying suitable estimation and control processes on $\mathcal{G}\mathcal{M}$ $G M$ , we construct a pair of strategies, and our objective is to prove that it is nearly optimal in the original game model $\mathcal{G}\mathcal{M}_{N}$ $G M_{N}$ .