Potential in population games

Abstract A general, novel notion of potential in population games is presented. A population game is defined, very broadly, as any bivariate function $$g\left( {x,y} \right)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mfenced> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:math> on a convex set in a linear topological space. This function may specify the payoff for an individual population member from choosing strategy $$x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>x</mml:mi> </mml:math> (in a symmetric population game) or the mean payoff to individuals from playing according to strategy profile $$x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>x</mml:mi> </mml:math> (in an asymmetric population game), with the choices in the population as a whole expressed by the population strategy $$y$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>y</mml:mi> </mml:math> . 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 $$N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> -player game.