Cooperation, Correlation and Competition in Ergodic N-player Games and Mean-field Games of Singular Controls: A Case Study

Abstract

We consider a class of $N$-player games and mean-field games of singular controls with ergodic performance criterion, providing a benchmark case for irreversible investment games featuring mean-field interaction and strategic complementarities. The state of each player follows a geometric Brownian motion, controlled additively through a nondecreasing process, while agents seek to maximize a long-term average reward functional with a power-type instantaneous profit, under strategic complementarity. We explore three different notions of optimality, which, in the mean-field limit, correspond to the mean-field control solution, mean-field coarse correlated equilibria, and mean-field Nash equilibria. We explicitly compute equilibria in the three cases and compare them numerically, in terms of yielded payoffs and existence conditions. Finally, we show that the mean-field control and mean-field equilibria can approximate the cooperative and competitive equilibria, respectively, in the corresponding $N$-player game when $N$ is sufficiently large. Our analysis of the mean-field control problem features a novel Lagrange multiplier approach, which proves crucial in establishing the approximation result, while the treatment of mean-field coarse correlated equilibria necessitates a new, specifically tailored definition for the stationary setting.

Figures (5)

• Figure 1: Values of the parameters $(u,v) \in \mathbb{R}_+^2$ so that the $(Z,\lambda^s,\theta_\infty) \in \mathcal{G}^s$ (on the left) and $(Z,\lambda^r,\theta_\infty) \in \mathcal{G}^r$ (on the right), $\theta_\infty \sim \Gamma(u,v)$ is a mean-field CCE outperforming the NE. Here, $\delta = 0.1$, $\sigma = 0.2$, $q = 2$, $\alpha = 0.3$ and $\beta = 0.5$. Notice that Assumption \ref{['assumption:dissipativity']} is satisfied.
• Figure 2: Reward associated to mean-field CCEs $(Z,\lambda^s,\theta_\infty) \in \mathcal{G}^s$ (on the left) and $(Z,\lambda^r,\theta_\infty) \in \mathcal{G}^r$ (on the right) which outperform the reward of the mean-field NE, when $\theta_\infty \sim \Gamma(u,v)$, $(u,v) \in \mathbb{R}_+^2$. Here, $\delta = 0.1$, $\sigma = 0.2$, $q = 2$, $\alpha = 0.3$ and $\beta = 0.5$.
• Figure 3: Reward associated to the best performing CCE $(Z,\lambda^s,\theta_\infty) \in \mathcal{G}^s$ (in blue) and $(Z,\lambda^r,\theta_\infty) \in \mathcal{G}^r$ (in cyan), when $\theta_\infty \sim \Gamma(u,v)$, $(u,v) \in \mathbb{R}_+^2$, compared with MFC solution (dashed orange line) and mean-field NE (dashed green line). Here, $\delta = 1$, $q=0.5$, $\alpha = 0.3$ and $\beta = 0.4$. $\sigma$ varies from $0$ to $\sqrt{2\delta}$.
• Figure 4: Reward associated to the best performing CCE $(Z,\lambda^s,\theta_\infty) \in \mathcal{G}^s$ (in blue) and $(Z,\lambda^r,\theta_\infty) \in \mathcal{G}^r$ (in cyan), for $\theta_\infty \sim \Gamma(u,v)$, $(u,v) \in \mathbb{R}_+^2$, compared with MFC solution (dashed orange line) and mean-field NE (dashed green line). Here, $\alpha=0.3$ and $0 \leq \beta \leq 1-\alpha$. On the left, $\delta=1$, $q_u=1$, $\sigma = 1$, so that condition \ref{['eq:mfc:condition_zero_infinity']} is satisfied. On the right, $\delta=0.1$, $q_u=2$, $\sigma = 0.2$, so that condition \ref{['eq:mfc:condition_zero_infinity']} is not satisfied.
• Figure 5: Value of $u^*$ as $\alpha$ varies in $[0,1]$, both for regular and singular recommendations. The blue dashed line is located at the value of $\alpha=\bar{\alpha}$ which satisfies \ref{['mfg:condition:infinitely_many']}. Here, $\delta = 0.1$, $\sigma = 0.2$, $q = 2$.

Theorems & Definitions (62)

• Definition 1: Open-loop strategies for the $N$-player game
• Definition 2
• Definition 3: Correlating device
• Definition 4: Correlated strategy profile
• Definition 5: $\varepsilon$-coarse correlated equilibrium within the set of strategies $\mathcal{B}$
• Definition 6: $\varepsilon$-Nash equilibrium within the set of strategies $\mathcal{B}$
• Definition 7: Open-loop strategy for the ergodic MFG
• Definition 8
• Definition 9
• Definition 10: Correlated stationary strategy