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Van Eenam Lecture #3 From Nash Equilibrium to Social Optimum and Back: A Mean Field Perspective

Rene Carmona (Princeton University)
Thursday, February 13, 2025
4:00-5:00 PM
4448 East Hall Map
Mean field games (MFG) and mean field control (MFC) problems have been introduced to study large populations of strategic players. They correspond respectively to non-cooperative or cooperative scenarios, and the goals of their analyses are to find the Nash equilibriums and social optimums. These frameworks provide approximate solutions to situations with a finite number of players and have found a wide range of applications, from economics to biology and machine learning. In this paper, we study how the players can transition from a non-cooperative to a cooperative regime, and back. The first direction is reminiscent of mechanism design, in which the game's definition is modified so that non-cooperative players reach an outcome similar to a cooperative scenario. To better understand the second direction we introduce the "price of instability" and study how players that are initially cooperative gradually deviate from a social optimum to reach a Nash equilibrium when they decide to optimize their individual cost very much in the spirit of the "free rider" phenomenon. To formalize these connections, we introduce two new classes of games which lie between MFG and MFC: $\lambda$-interpolated mean field games, in which the cost of an individual player is a $\lambda$-interpolation of the MFG and the MFC costs, and $p$-partial mean field games, in which a proportion $p$ of the population deviates from the social optimum by playing the game non-cooperatively. We shallconclude with algorithm for myopic players to learn a $p$-partial mean field equilibrium, and we illustrate it on a stylized model.

Joint work with G. Dayanikli, M. Lauriere and F. Delarue
Building: East Hall
Event Type: Workshop / Seminar
Tags: Mathematics
Source: Happening @ Michigan from Financial/Actuarial Mathematics Seminar - Department of Mathematics, Department of Mathematics