Continuum Nash bargaining solutions

Abstract We compute a mean field limit of Nash bargaining solutions with agents chosen from a smooth probability density and with utility determined by the negative exponential of a cost function. When the cost function has the property that Kantorovich potentials are unique, we show that both at finite stages and in the continuum limit, Nash bargaining solutions can be described by a solution to an optimal transport problem. Nash’s classical bargaining solution suggests that N players in a non-cooperative bargaining situation should find a solution that maximizes the product of each player’s utility functions. In our special case, the maximization problem becomes an optimal transport type problem, where the target density is the minimizer to the functional $$ F(\beta )=H_{\nu }(\beta )+C(\mu ,\beta ) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>β</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>ν</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>β</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>C</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>β</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> where $$H_{\nu }(\beta )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>ν</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>β</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is the relative entropy and $$C(\mu ,\nu )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo>(</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ν</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is the optimal transport distance. This minimization problem is also solved in the Jordan-Kinderlehrer-Otto scheme. We also show that for compact metric spaces, when c is a Lipschitz cost function, even in the presence of non-unique Kantorovich potentials, the solution the minimization problem has Lipschitz Radon-Nikodym derivative.