Optimal Stopping in Mean-Field Games

• Optimal stopping mean-field games are frameworks where agents select exit times, forming equilibria via a fixed point between individual optimal stopping and the population law.
• The approach employs obstacle-problem PDEs, variational inequalities, and relaxed occupation measures to manage equilibrium multiplicity and discontinuities.
• Applications span market exit strategies and energy models, with numerical methods like linear programming and fictitious play ensuring convergence and stability.

Optimal stopping mean-field games are mean field games in which each agent chooses an optimal stopping or exit time, rather than only a regular control, while the payoff and sometimes the dynamics depend on the evolving population distribution. Published formulations include continuum stopping games in which interaction occurs through the proportion of players who have already stopped (Nutz, 2016), obstacle-problem PDE systems in which the density of still-active agents is endogenously absorbed (Bertucci, 2017), relaxed occupation-measure formulations for control-and-stopping problems (Bouveret et al., 2018, Dumitrescu et al., 2020), and weak or randomized formulations that accommodate common noise, partial observation, and non-pure equilibrium selection (Possamaï et al., 2023, Ferrari et al., 25 Jul 2025). Across these formulations, the defining object is a fixed point between an individual optimal-stopping problem and a population law generated by the induced stopping behavior.

1. Canonical formulations

A standard continuous-time formulation considers a representative agent with state process $X$ and stopping time $\tau$. In the generic diffusion model, the state evolves as

$dX_t = b(X_t,\mu_t)\,dt + \sigma(X_t,\mu_t)\,dW_t,$

and the agent solves

$$J(x,\mu,\tau) = \mathbb E^x\!\left[\int_0^\tau f(X_s,\mu_s)\,ds + g(X_\tau,\mu_\tau)\right].$$

Here $\mu_t$ is the population law, $f$ is a running reward or cost, and $g$ is a terminal exit reward (Bassière et al., 2023). Closely related models allow a regular control $a_t$ in addition to stopping, so that the representative agent chooses both a control process and an exit time; in that setting the mean field is often described by a flow of sub-probability measures of still-active agents together with a terminal exit distribution (Dumitrescu et al., 2020).

The same basic structure appears in bounded-domain Brownian models, where agents stop before a finite horizon or upon hitting the boundary, and the density of remaining players enters the running cost and terminal obstacle (Bertucci, 2017). In discrete time, the state may instead be a Markov chain observed together with common noise, and the mean-field interaction is conditioned on the common-noise filtration; the control remains a stopping time, but the equilibrium object becomes a flow of conditional sub-probability measures (Dumitrescu et al., 2022).

Several extensions preserve the stopping core while changing the economic or informational environment. In electricity-market models, conventional producers choose an exit time and renewable producers choose an entry time, with interaction through an endogenous merit-order price process (Aïd et al., 2020, Dumitrescu et al., 2022). In the information-acquisition model, the agent controls a position process while facing a hidden state $S$, and the stopping decision is the time at which the agent pays to observe that hidden state and then continues under full information (D'Auria et al., 8 Jun 2026).

Formulation	Key ingredients	Representative papers
Diffusion stopping game	$X_t$, stopping time $\tau$0, running reward $\tau$1, exit reward $\tau$2	(Bassière et al., 2023, Bertucci, 2017)
Control-and-stopping MFG	Regular control $\tau$3, sub-probability flow $\tau$4, exit law $\tau$5	(Dumitrescu et al., 2020)
Discrete-time stopping MFG	Markov chain state, common noise, partial observation	(Dumitrescu et al., 2022)
Information-purchase stopping MFG	Hidden state $\tau$6, information cost $\tau$7, post-stopping full information	(D'Auria et al., 8 Jun 2026)

2. Obstacle problems, variational inequalities, and forward equations

For a fixed mean-field flow, the representative stopping problem is characterized by an obstacle-type Hamilton–Jacobi–Bellman equation. In the generic stopping formulation, the value function $\tau$8 satisfies the variational inequality

$\tau$9

where $dX_t = b(X_t,\mu_t)\,dt + \sigma(X_t,\mu_t)\,dW_t,$0 is the infinitesimal generator at frozen law $dX_t = b(X_t,\mu_t)\,dt + \sigma(X_t,\mu_t)\,dW_t,$1 (Bassière et al., 2023). Equivalent max-formulations are used in obstacle-problem treatments, for example

$dX_t = b(X_t,\mu_t)\,dt + \sigma(X_t,\mu_t)\,dW_t,$2

with $dX_t = b(X_t,\mu_t)\,dt + \sigma(X_t,\mu_t)\,dW_t,$3 the terminal or stopping payoff (Shen et al., 2023).

The forward equation describes how active mass evolves until absorption at the stopping set. In a pure-strategy equilibrium, if agents stop exactly on the contact set, the active-mass density $dX_t = b(X_t,\mu_t)\,dt + \sigma(X_t,\mu_t)\,dW_t,$4 satisfies

$dX_t = b(X_t,\mu_t)\,dt + \sigma(X_t,\mu_t)\,dW_t,$5

together with an initial condition (Shen et al., 2023). In measure form, one often writes

$dX_t = b(X_t,\mu_t)\,dt + \sigma(X_t,\mu_t)\,dW_t,$6

where $dX_t = b(X_t,\mu_t)\,dt + \sigma(X_t,\mu_t)\,dW_t,$7 is the absorbed mass at time $dX_t = b(X_t,\mu_t)\,dt + \sigma(X_t,\mu_t)\,dW_t,$8 (Bassière et al., 2023).

When pure stopping fails, the forward equation becomes a complementarity system. A mixed-strategy equilibrium allows the mass to split on the free boundary and is characterized by

$dX_t = b(X_t,\mu_t)\,dt + \sigma(X_t,\mu_t)\,dW_t,$9

together with the complementary condition

$$J(x,\mu,\tau) = \mathbb E^x\!\left[\int_0^\tau f(X_s,\mu_s)\,ds + g(X_\tau,\mu_\tau)\right].$$0

which expresses that randomization occurs precisely so that the net incentive on the stopping set is balanced (Shen et al., 2023). In control-and-stopping models the obstacle term may be combined with a control supremum, and in information-purchase models the stopping operator becomes a gain operator $$J(x,\mu,\tau) = \mathbb E^x\!\left[\int_0^\tau f(X_s,\mu_s)\,ds + g(X_\tau,\mu_\tau)\right].$$1 corresponding to paying the information cost and continuing under full information (Dumitrescu et al., 2020, D'Auria et al., 8 Jun 2026).

3. Equilibrium notions: pure, mixed, relaxed, weak, and randomized

The most direct equilibrium notion is the pure stopping equilibrium. Given a population law, the agent stops at the first time the state enters the stopping region, typically

$$J(x,\mu,\tau) = \mathbb E^x\!\left[\int_0^\tau f(X_s,\mu_s)\,ds + g(X_\tau,\mu_\tau)\right].$$2

and consistency requires that the induced law of the optimally stopped process reproduces the conjectured mean field (Bassière et al., 2023). This notion is natural when the free boundary is regular and the stopping rule is well defined as a first-hitting time.

A central development in the area is the recognition that pure solutions may fail to exist or may be too restrictive. Bertucci introduced the notion of a mixed solution for optimal-stopping MFGs, emphasizing that Nash equilibria of the game are in mixed strategies (Bertucci, 2017). In the stationary version, one keeps the obstacle equation for the value function but replaces the forward equation by inequalities and a complementary-slackness condition on the set $$J(x,\mu,\tau) = \mathbb E^x\!\left[\int_0^\tau f(X_s,\mu_s)\,ds + g(X_\tau,\mu_\tau)\right].$$3. The relaxed solution approach goes further by replacing stopping times with occupation measures of the stopped process. The admissible object is then a flow $$J(x,\mu,\tau) = \mathbb E^x\!\left[\int_0^\tau f(X_s,\mu_s)\,ds + g(X_\tau,\mu_\tau)\right].$$4 of positive finite measures satisfying a weak forward inequality, and a relaxed Nash equilibrium is a fixed point of the associated best-response correspondence (Bouveret et al., 2018).

The linear-programming formulation recasts the relaxed problem as an infinite-dimensional LP over a pair $$J(x,\mu,\tau) = \mathbb E^x\!\left[\int_0^\tau f(X_s,\mu_s)\,ds + g(X_\tau,\mu_\tau)\right].$$5, where $$J(x,\mu,\tau) = \mathbb E^x\!\left[\int_0^\tau f(X_s,\mu_s)\,ds + g(X_\tau,\mu_\tau)\right].$$6 is the occupation measure of still-active agents and $$J(x,\mu,\tau) = \mathbb E^x\!\left[\int_0^\tau f(X_s,\mu_s)\,ds + g(X_\tau,\mu_\tau)\right].$$7 is the exit law of $$J(x,\mu,\tau) = \mathbb E^x\!\left[\int_0^\tau f(X_s,\mu_s)\,ds + g(X_\tau,\mu_\tau)\right].$$8 (Dumitrescu et al., 2020). This relaxation is paired with an exact equivalence result: LP equilibria coincide with equilibria obtained via a controlled/stopped martingale formulation. A related but distinct weak formulation places the game on the canonical space $$J(x,\mu,\tau) = \mathbb E^x\!\left[\int_0^\tau f(X_s,\mu_s)\,ds + g(X_\tau,\mu_\tau)\right].$$9 with coordinates $\mu_t$0, encodes randomized stopping through a survival process $\mu_t$1, and defines a weak mean-field equilibrium as a fixed point $\mu_t$2 on the space of laws (Possamaï et al., 2023).

Randomization becomes particularly important with common noise. In that setting, strong randomized mean-field equilibria are defined on an enlarged space $\mu_t$3, with randomized stopping times compact in the Baxter–Chacon topology and mean-field laws conditioned on the common noise (Ferrari et al., 25 Jul 2025). In discrete-time major–minor games, the major player may use a relaxed control or stopping policy while minor players are represented through occupation measures; the resulting equilibrium is a relaxed fixed point, and entropy regularization is used to restore uniqueness in the major’s control problem (Yu et al., 15 Jan 2025).

4. Existence, uniqueness, and finite-player limits

Existence results are now available in several regimes, but the assumptions and proof technologies differ markedly. In relaxed continuous-time models with diffusion dynamics and exit, existence of a relaxed Nash equilibrium follows from compactness of the admissible occupation-measure set and a Fan–Glicksberg fixed-point argument (Bouveret et al., 2018). In the LP framework for control and optimal stopping, existence of an LP–MFG Nash equilibrium follows from nonemptiness, convexity, and compactness of the feasible set of occupation and exit measures, combined with Kakutani–Fan–Glicksberg (Dumitrescu et al., 2020). Discrete-time optimal-stopping MFGs with common noise and partial observation admit existence results through linear programming on conditional occupation measures (Dumitrescu et al., 2022), and weak equilibria in the canonical-law formulation exist under continuity of the payoff in the mean field and concavity in the law variable (Possamaï et al., 2023).

Uniqueness is substantially more delicate. In the relaxed solution approach, anti-monotonicity of the running reward yields uniqueness of the associated Nash value, even when equilibrium measures need not be unique (Bouveret et al., 2018). In generic obstacle–Fokker–Planck formulations, a prototypical existence–uniqueness theorem is stated under Lipschitz conditions and a Lasry–Lions monotonicity condition (Bassière et al., 2023). In entropy-regularized formulations, Lasry–Lions monotonicity yields uniqueness of the regularized equilibrium, and supermodularity can be used to characterize extremal equilibria (Dianetti et al., 23 Sep 2025).

A separate line of work studies the relation between mean-field equilibria and large finite games. Under suitable assumptions, mean-field equilibria provide approximate Nash equilibria for the $\mu_t$4-player game: this is shown for the weak formulation (Possamaï et al., 2023), for contest models with continuous reward schedules (Nutz et al., 2021), and for the information-purchase model, where approximate Nash equilibria can be constructed for a class of $\mu_t$5-player games with compatible information structure (D'Auria et al., 8 Jun 2026). However, this approximation theory has sharp limitations. In the case study of stopping games with interaction through the stopping proportion, increasing-transversal mean-field equilibria are limit points of $\mu_t$6-player equilibria, while strongly decreasing-transversal equilibria are not (Nutz et al., 2018). The contest model yields a related warning: when the prize function has a jump, the mean-field strategy ceases to be approximately Nash for large finite $\mu_t$7 (Nutz et al., 2021).

5. Structural regimes and explicitly solvable models

A notable feature of the field is that several nontrivial stopping MFGs admit closed-form or nearly closed-form analysis.

Model class	Equilibrium structure	Representative papers
Proportion-based stopping game	Scalar fixed-point equation for stopping proportion	(Nutz, 2016)
Hybrid LQG stopping MFG	Deterministic state-invariant stopping times	(Firoozi et al., 2018)
Rank-based contest	Unique closed-form MFE and explicit prize design	(Nutz et al., 2021)
Relative-performance stopping	Threshold $\mu_t$8	(Huang et al., 2022)
Information-purchase stopping MFG	Two equilibrium regimes: “No-buy” and “Buy-at-zero”	(D'Auria et al., 8 Jun 2026)

In the monotone continuum model of stopping through a Cox default intensity, the single-agent problem collapses to a first-crossing rule

$\mu_t$9

and equilibrium is characterized by a scalar fixed-point equation involving the distribution function of the idiosyncratic noise,

$f$0

This formulation makes multiplicity transparent: weak interaction can yield uniqueness, strong interaction can generate several roots, and at a critical parameter one may obtain infinitely many equilibria (Nutz, 2016).

In hybrid LQG mean field games with switching and stopping, the stopping times become state-invariant. The optimal stopping time is almost surely deterministic and is identified by the matrix-valued Hamiltonian-continuity equation $f$1, so that stopping depends only on the time-varying system matrices and not on the realized state (Firoozi et al., 2018). In the rank-based contest, the equilibrium stopping distribution and the value function are obtained in closed form, and the principal’s optimal contract is also explicit; the same model exhibits an asymptotic singularity in the finite-$f$2 distributions (Nutz et al., 2021).

Threshold phenomena also appear in relative-performance models. For geometric Brownian motion dynamics with “Joneses preference,” the free-boundary problem yields a threshold stopping rule

$f$3

together with a scalar consistency equation for the effective mean-field parameter (Huang et al., 2022). In the information-acquisition model, the linear–quadratic example produces exactly two equilibrium regimes. In the “No-buy” regime, $f$4 almost surely; in the “Buy-at-zero” regime, $f$5, after which the control switches to the full-information feedback law (D'Auria et al., 8 Jun 2026). These solvable cases show that the stopping component can encode threshold rules, deterministic intervention times, singular laws, or regime selection, depending on how the mean field enters the payoff.

6. Numerical methods and application domains

The two main numerical paradigms are linear programming and fictitious play. In the LP approach, one discretizes time and state, replaces the forward equation by finite-dimensional linear constraints, and solves a finite LP whose decision variables are occupation and exit measures. Convergence follows from weak-convergence and tightness arguments for the discrete occupation measures (Dumitrescu et al., 2020). This approach is particularly effective when measure flows are discontinuous, because it works directly in the space of sub-probability flows rather than through a value-function discretization.

Fictitious play has been developed in both LP and PDE settings. In the LP framework, the iteration alternates between a best-response LP for a frozen mean field and an averaging update of the empirical measure flow. The convergence analysis uses the topology of convergence in measure for flows of sub-probability measures, precisely to accommodate jumps caused by simultaneous exit; under the stated assumptions, exploitability decays as $f$6 (Dumitrescu et al., 2022). In the PDE framework, generalized fictitious play computes mixed equilibria by iteratively solving pure strategy systems: at each iteration one solves the obstacle problem against the averaged density, then solves the forward Fokker–Planck equation, and finally updates the average with learning rates $f$7 satisfying

$f$8

For potential games, any regular cluster point is a mixed equilibrium; finite-difference discretizations are given in implicit and semi-implicit form, and the numerical experiments show robust convergence, including cases in which no pure equilibrium exists (Shen et al., 2023).

Recent work introduces entropy regularization and randomized stopping to make the problem more amenable to learning-based methods. Randomized stopping times are reinterpreted as singular controls, existence of equilibria is proved for the entropy-regularized game, stability is established as the entropy parameter vanishes, and fictitious-play convergence is obtained under Lasry–Lions monotonicity and under supermodular assumptions (Dianetti et al., 23 Sep 2025). In major–minor stopping games, entropy regularization yields a Gibbs-form optimal policy for the major player and supports a fixed-point existence proof for relaxed equilibria (Yu et al., 15 Jan 2025).

Applications are concentrated in energy and market models, but the range is broader. Electricity-market models treat conventional producers as stoppers who choose exit times and renewable producers as entrants who choose investment times, with interaction through an endogenous price process determined by aggregate supply and demand (Aïd et al., 2020, Bassière et al., 2023). Under scenario uncertainty with common noise and partial observation, the discrete-time stopping MFG predicts that upward uncertainty accelerates conventional exit and renewable investment, while downward uncertainty delays them (Dumitrescu et al., 2022). Other papers explicitly identify bank-run stopping games, technology switching, capacity expansion under macro-shocks, and R&D entry/exit as natural application classes (Nutz, 2016, Dumitrescu et al., 2022).

7. Recurring issues and conceptual cautions

Several recurring issues distinguish optimal stopping MFGs from more standard control-only mean field games. First, pure equilibria are not generic. Obstacle systems can fail to admit pure-strategy solutions, and the mixed or relaxed formulations are not merely technical substitutes; they encode genuine equilibrium randomization at the free boundary (Bertucci, 2017, Shen et al., 2023, Bouveret et al., 2018). Second, multiplicity is common. In the monotone proportion-based model, the fixed-point equation may have three or more roots, and at a critical interaction strength there may be infinitely many equilibria (Nutz, 2016).

Third, the mean-field limit need not be an unproblematic proxy for large finite games. The transversality analysis shows that some mean-field equilibria are not limits of $f$9-player equilibria (Nutz et al., 2018). The contest model sharpens this point: the mean-field optimal prize design yields a singular two-point law, while the finite-$g$0 equilibrium remains atomless and the principal’s realized objective in the finite-$g$1 game converges to only about half of the mean-field optimum (Nutz et al., 2021). A plausible implication is that equilibrium selection and regularity of the reward schedule are not secondary modeling choices; they materially affect whether the mean-field solution has a faithful finite-player interpretation.

Finally, extensions involving common noise, partial observation, hidden states, or endogenous information acquisition change the equilibrium object itself. In these settings the mean field must be adapted to the common noise, the stopping time may need to be randomized, and consistency is formulated through conditional laws rather than deterministic flows (Dumitrescu et al., 2022, Ferrari et al., 25 Jul 2025). In the information-purchase model, the stopping decision is no longer just exit from the game; it is the stopping time at which the player purchases information and switches to a new control problem under a richer filtration (D'Auria et al., 8 Jun 2026). This suggests that “optimal stopping” in mean-field games should be understood broadly: not only as irreversible exit, but as a singular decision that changes the admissible strategy class, the information structure, or both.