Bertrand and Cournot Mean Field Games

Abstract

We study 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. Interactions are of mean field type in the sense that the demand faced by a producer is affected by the others through their average price or quantity. Motivated by energy or consumer goods markets, we consider the setting of a dynamic game with uncertain market demand, and under the constraint of finite supplies (or exhaustible resources). The continuum game is characterized by a coupled system of partial differential equations: a backward Hamilton–Jacobi–Bellman partial differential equation (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 qualitative as an approximation to even small player dynamic games.

Appendices

Appendix 1: Asymptotic correction of arbitrary order

In this section we give expressions for successive terms in the small $\epsilon$ expansions (49) for $u$ and $m$ in the deterministic MFG where $\sigma =0$. It is straightforward to compute the expansions for the coefficients $\alpha (t) = a(\eta (t))$ and $\gamma (t) = c(\eta (t))$ using the expansion for $\eta (t)$ in (50). We also expand $\bar{p}(t)$. The terms in the three series are labeled $\alpha _k(t)$, $\gamma _k(t)$ and $\bar{p}_k(t)$ respectively, but we do not give their cumbersome formulas here.

1.1 Value function

Inserting the expansion of $u$ into the PDE (30) with $\sigma = 0$, we find that the $k$th-order value function correction $u_k$ satisfies the equation

$$\begin{aligned} \partial _t u_k - r u_k - \frac{1}{2} \Bigl ( 1 + \mathbb {W}\bigl ( \theta (x) \bigr ) \Bigr ) \partial _x u_k = f_k(t,x), \qquad u_k(t,0) = 0, \end{aligned}$$

(59)

where the inhomogeneous term $f_k$ is given by

$$\begin{aligned} f_k(t,x) = - \frac{1}{4} \left\{ \sum _{i=1}^{k-1} \xi _i \xi _{k-i} + 2 (1 - u_0') \left( \alpha _k + \sum _{j=0}^k \gamma _j \bar{p}_{k-j}\right) \right\} , \\ \qquad \xi _n = \alpha _n - \partial _x u_n + \sum _{i=0}^n \gamma _i \bar{p}_{n-i}. \end{aligned}$$

Observe that $f_k$ depends only on $u_j$ and $m_j$ for $0 \le j < k$. Having solved for these $u_j$ and $m_j$, the $u_k$ equations can be solved in closed-form:

$$\begin{aligned} u_k(t,x) = - \int _0^{\tau (x)} e^{-rs} f_k \left( t+s, X(s;x) \right) ds, \end{aligned}$$

(60)

where $X(t;x)$ is the capacity trajectory starting from $x$, given by (38).

1.2 Density

The $k$th-order density correction $m_k$ satisfies the equation

$$\begin{aligned} \partial _t m_k - \frac{1}{2} \partial _x \bigg ( (1 - u_0') m_k \bigg ) = g_k(t,x), \qquad m_k(0,x) = 0, \end{aligned}$$

(61)

where the inhomogeneous term $g_k$ is given by

$$\begin{aligned} g_k(t,x) = \frac{1}{2} \sum _{i=1}^k \partial _x \left( \xi _i m_{k-i} \right) . \end{aligned}$$

Again we see that $g_k$ depends only on $u_j$ and $m_j$ for $0 \le j < k$. Then $m_k$ can be written as

$$\begin{aligned} m_k(t,x) = \int _0^t \frac{1 + \mathbb {W}\bigl (\theta (x)\bigr ) e^{-r(t-s)}}{1 + \mathbb {W}\bigl (\theta (x)\bigr )} g_k \bigl (s, X(s-t;x) \bigr ) \ ds. \end{aligned}$$

(62)

Appendix 2: Cournot–Bertrand Equivalence in the Stochastic Dynamic CMFG

In this section we show that in the continuum mean field setting, the dynamic Cournot game and Bertrand games are identical. We first derive the Cournot MFG PDEs.

1.1 Dynamic Cournot Mean Field Game

As in the Bertrand game described in Section 3, there is an infinity of players on $x>0$ with initial density $M(x)$. Here they choose quantities of production $q_t=q(t,X_t)$ which deplete the remaining capacity of the producers $(X_t)$ following the dynamics

$$\begin{aligned}{}[dX_t = - q(t,X_t)\,dt + \sigma \,dW_t,] \end{aligned}$$

as long as $X_t > 0$, and $X_t$ is absorbed at zero. Here $W$ is a Brownian motion, and $\sigma > 0$ is a constant. The Cournot market model is specified by the inverse demand function $p_t = 1 - \left( q_t + \epsilon \bar{q}(t) \right)$, where % $\epsilon$ measures the degree of interaction, and $\bar{q}$ is the mean production. We will denote by $m^c(t,x)$ the “density” of firms with positive capacity at time $t>0$, and by $\eta ^c(t)=\int _{\mathbb {R}_+}m^c(t,x)\,dx$ the fraction of active firms remaining. Then the average quantity is % defined in (22). Then we have

$$\begin{aligned}{}[ \bar{q(t)} = \int _{\mathbb {R}_+} q(t,x) m^c(t,x) \, dx. ] \end{aligned}$$

The value function $u^c$ of the producers is

$$\begin{aligned} u^c(t,x) = \sup _q \mathbb {E} \left\{ \left. \int _t^{\infty } e^{-r(s-t)} p_s q_s 1\!\!1_{\{X_s > 0\}} \, ds \right| X_t = x \right\} , \qquad x > 0. \end{aligned}$$

(63)

In analogy to the Bertrand game, we define the Cournot exhaustion time $T^c$ to be the first time $\eta ^c$ reaches zero. The following quantities are defined for $t < T^c$. The associated HJB equation is

$$\begin{aligned} \partial _t u^c + \frac{1}{2} \sigma ^2 \partial ^2_{xx} u^c - r u^c + \max _{q \ge 0} \Bigl ( 1 - \bigl ( q + \epsilon \bar{q}(t) \bigr ) - \partial _x u^c \Bigr ) q = 0, \qquad x > 0.\nonumber \\ \end{aligned}$$

(64)

The internal optimization is the static continuum mean field Cournot game (Section 2.3.2) with cost function $s(x)\mapsto \partial _x u^c$. The first-order condition gives

$$\begin{aligned} q^*(t,x) = \frac{1}{2}\left( 1 - \epsilon \bar{q}(t) - \partial _x u^c(t,x) \right) , \end{aligned}$$

(65)

with the optimal (equilibrium) price given by $p^*(t,x) = \frac{1}{2} \left( 1 - \epsilon \bar{q}(t) + \partial _x u^c(t,x) \right)$. Therefore, the HJB equation becomes

$$\begin{aligned} \partial _t u^c + \frac{1}{2} \sigma ^2 \partial ^2_{xx} u^c - r u^c + \frac{1}{4} \bigg ( 1 - \epsilon \bar{q}(t) - \partial _x u^c \bigg )^2 = 0, \qquad x > 0. \end{aligned}$$

(66)

When all the reserves are exhausted, the game is over and $u^c(t,0) = 0$.

The density $m^c(t,x)$ of the distribution of reserves is the solution of the forward Kolmogorov equation

$$\begin{aligned} \partial _t m^c - \frac{1}{2} \sigma ^2 \partial ^2_{xx} m^c - \frac{1}{2} \partial _x \Bigl ( \bigl ( 1 - \epsilon \bar{q}(t) - \partial _x u^c \bigr ) m^c \Bigr ) = 0, \end{aligned}$$

(67)

with $m^c(0,x) = M(x)$. The average demand is computed by averaging (65) with respect to $m^c$, which leads to

$$\begin{aligned} \bar{q}(t) = \frac{1}{2 + \epsilon \eta ^c(t)} \left( \eta ^c(t) - \int _{\mathbb {R}_+} \partial _x u^c(t,x) m^c(t,x) \ dx \right) . \end{aligned}$$

(68)

1.2 Equivalence of Bertrand and Cournot Problems

We start by recalling the Bertrand MFG equations:

$$\begin{aligned} 0&= \partial _t u + \frac{1}{2} \sigma ^2 \partial _{xx} u - r u + \frac{1}{4} \left( a(\eta (t)) - \partial _x u + c(\eta (t)) \bar{p}(t) \right) ^2 \nonumber \\ 0&= \partial _t m - \frac{1}{2} \sigma ^2 \partial _{xx} m - \frac{1}{2} \partial _x \left( \left( a(\eta (t)) - \partial _x u + c(\eta (t)) \bar{p}(t) \right) m \right) \nonumber \\ \bar{p}(t)&= \frac{1}{2 - c(\eta (t))} \left( a(\eta (t)) + \frac{1}{\eta (t)} \int \partial _x u(t,x) m(t,x)\,dx \right) , \end{aligned}$$

(69)

where $\eta (t) = \int _{\mathbb {R}_+} m(t,x)\,dx$.

If we define

$$\begin{aligned} \bar{q}_b(t) = \frac{1}{2 + \epsilon \eta (t)} \left( \eta (t) + \int \partial _x u(t,x) m(t,x)\,dx \right) , \end{aligned}$$

then it follows that $(1 + \epsilon \eta ) \bar{q}_b(t) = \eta (1 - \bar{p}(t))$ and hence

$$\begin{aligned} a(\eta (t)) + c(\eta (t)) \bar{p}(t) = 1 - \epsilon \bar{q}_b(t). \end{aligned}$$

Then Eqs. (69) can be written

$$\begin{aligned} 0&= \partial _t u + \frac{1}{2} \sigma ^2 \partial _{xx} u - r u + \frac{1}{4} \left( 1 - \epsilon \bar{q}_b(t) - \partial _x u \right) ^2 \\ 0&= \partial _t m - \frac{1}{2} \sigma ^2 \partial _{xx} m - \frac{1}{2} \partial _x \left( \left( 1 - \epsilon \bar{q}_b(t) - \partial _x u \right) m \right) , \end{aligned}$$

and these are exactly the Cournot CMFG equations (66), (67) and (68). As the boundary conditions are the same, we have $u\equiv u^c$, $m\equiv m^c$ and $\bar{q}\equiv \bar{q}_b$, and the Bertrand and Cournot dynamic MFG problems are equivalent.