Multiple equilibria in mean-field game models for large oligopolies with strategic complementarities

Abstract

We consider continuous-time mean-field stochastic games with strategic complementarities. The interaction between the representative productive firm and the population of rivals comes through the price at which the produced good is sold and the intensity of interaction is measured by a so-called "strenght parameter" $ξ$. Via lattice-theoretic arguments we first prove existence of equilibria and provide comparative statics results when varying $ξ$. A careful numerical study based on iterative schemes converging to suitable maximal and minimal equilibria allows then to study in relevant financial examples how the emergence of multiple equilibria is related to the strenght of the strategic interaction.

Figures (8)

• Figure 1: The two equilibria $\underline{m}, \overline{m}$ in Example \ref{['Example Mean reverting log-dynamics and isoelastic inverse demand']}.
• Figure 2: Evolution of the distribution of firms $g_t$ for $t \in [0.05,T]$ in the high (red) and low (blue) equilibria in Example \ref{['Example Mean reverting log-dynamics and isoelastic inverse demand']}. Color darkness is increasing with $t$.
• Figure 3: The optimal investment $\hat{\alpha}(t,x)$ for $t \in [0,T)$ in the high (red) and low (blue) equilibria in Example \ref{['Example Mean reverting log-dynamics and isoelastic inverse demand']}. Color darkness is increasing with $t$.
• Figure 4: Equilibrium multiplicity as a function of the parameter $\xi$ in Example \ref{['Example Mean reverting log-dynamics and isoelastic inverse demand']}. The y-axis shows the level of $m^1$ (in logs) above which the model converges to the "high" equilibrium.
• Figure 5: The two equilibria $\underline{m}, \overline{m}$ in Example \ref{['Example Geometric dynamics and isoelastic inverse demand']}.

Theorems & Definitions (18)

• Definition 2.3
• Example 2.4: Mean reverting dynamics and linear demand function
• Example 2.5
• Example 2.6: Mean reverting log-dynamics and isoelastic inverse demand
• Example 2.7
• Example 2.8: Geometric dynamics and isoelastic inverse demand
• Remark 2.9
• Remark 2.10
• Lemma 2.11
• Lemma 2.12