## Colloquium-Ronnie Sircar (Princeton University)-Oligopolies & Mean Field Games

Mathematics - Colloquium

Friday, April 4, 2014

11:00 AM-12:00 PM

Stratton Hall

203

ABSTRACT: One way to view energy markets is as competition between producers from different fuels and technologies with markedly varied characteristics. Motivated by dynamic oligopoly models of competition between heterogeneous energy producers, we discuss how continuous time Bertrand and Cournot competitions, in which firms producing similar goods compete with one another by setting prices or quantities respectively, can be analyzed as continuum dynamic mean field games under the constraint of finite supplies (or exhaustible resources). The continuum game is characterized by a coupled system of partial differential equations: a backward HJB PDE for the value function, and a forward Kolmogorov PDE for the density of players. Asymptotic approximation enables us to deduce certain qualitative features of the game in the limit of small competition. The equilibrium of the game is further studied using numerical solutions, which become very tractable by considering the tail distribution function instead of the density itself. This also allows us to consider Dirac delta distributions to use the continuum game to mimic finite N-player nonzero-sum differential games, the advantage being having to deal with two coupled PDEs instead of N. We find that, in accordance with the two-player game, a large degree of competitive interaction causes firms to slow down production. The continuum system can therefore be used as an effective approximation to even small player dynamic games.

Suggested Audiences:
Adult, College

E-mail:
ma-chair@wpi.edu

Last Modified: March 17, 2014 at 9:42 AM

## Colloquium-Ronnie Sircar (Princeton University)-Oligopolies & Mean Field Games

Mathematics - Colloquium

Friday, April 4, 2014

11:00 AM-12:00 PM

Stratton Hall

203

ABSTRACT: One way to view energy markets is as competition between producers from different fuels and technologies with markedly varied characteristics. Motivated by dynamic oligopoly models of competition between heterogeneous energy producers, we discuss how continuous time Bertrand and Cournot competitions, in which firms producing similar goods compete with one another by setting prices or quantities respectively, can be analyzed as continuum dynamic mean field games under the constraint of finite supplies (or exhaustible resources). The continuum game is characterized by a coupled system of partial differential equations: a backward HJB PDE for the value function, and a forward Kolmogorov PDE for the density of players. Asymptotic approximation enables us to deduce certain qualitative features of the game in the limit of small competition. The equilibrium of the game is further studied using numerical solutions, which become very tractable by considering the tail distribution function instead of the density itself. This also allows us to consider Dirac delta distributions to use the continuum game to mimic finite N-player nonzero-sum differential games, the advantage being having to deal with two coupled PDEs instead of N. We find that, in accordance with the two-player game, a large degree of competitive interaction causes firms to slow down production. The continuum system can therefore be used as an effective approximation to even small player dynamic games.

Suggested Audiences: Adult, College

E-mail: ma-chair@wpi.edu

Last Modified: March 17, 2014 at 9:42 AM