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Potential in population games
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DOI：10.1007/s00199-025-01691-z.png)
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Abstract
En 中文
A general, novel notion of potential in population games is presented. A population game is defined, very broadly, as any bivariate function gx,y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\left( {x,y} \right)$$\end{document} on a convex set in a linear topological space. This function may specify the payoff for an individual population member from choosing strategyx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x$$\end{document} (in a symmetric population game) or the mean payoff to individuals from playing according to strategy profilex\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x$$\end{document} (in an asymmetric population game), with the choices in the population as a whole expressed by the population strategyy\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y$$\end{document}. These notions of population game and potential include a number of earlier notions as special cases. Potential is closely linked with (a general notion of) equilibrium. It increases along every improvement curve: the population-game analog of an improvement path in an N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N$$\end{document}-player game.
Keywords:
Population games
Potential games
Congestion games
Evolutionary game theory
Finite improvement property
C72
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