Quantilized Mean-Field Game Models

• Quantilized mean-field games are models where agents’ rewards depend on their ranking relative to a specific population quantile rather than the average outcome.
• They utilize target-based and threshold-based formulations to derive equilibrium strategies by coupling individual optimal controls with a quantile consistency condition.
• Numerical and analytic analyses confirm that these models yield ε-Nash equilibria and efficient approximations for competitive scenarios, such as venture investment selections.

Quantilized mean-field game (Q-MFG) models are a class of mean-field games in which the equilibrium and agent interactions are determined not by the population mean or aggregate, but by population quantiles—specifically, the α-quantile of state distributions. These models provide a rigorous framework for rank-based competition in large populations, where payoffs and strategies hinge on whether agents attain or surpass a specific performance threshold defined by a quantile. Q-MFGs generalize classical mean-field approaches to contexts where selection, ranking, or rare-event performance is central, such as tournaments, financial rankings, selective investment, and prize allocation.

1. Quantilized Equilibrium and Population Quantiles

Q-MFGs are characterized by payoff functionals or selection rules that explicitly depend on an endogenous quantile of the population’s terminal state distribution. For a given α ∈ (0,1), the α-quantile $q^\alpha_T$ at time T divides the population such that a fraction α have terminal states below $q^\alpha_T$, and the rest above. The equilibrium structure necessitates that this threshold emerges from the joint strategic behavior of all agents, leading to a self-consistency requirement: $q^\alpha_T$ is both a function of, and a determinant of, the agents’ optimal controls.

Agents’ terminal rewards or penalties are explicitly tied to their rank relative to $q^\alpha_T$. This setting models competitions in which only the top (1–α)% are selected or rewarded, introducing a nontrivial dependency between aggregate dynamics and rank ordering in the sense of the induced population law.

2. Mathematical Formulations: Target- and Threshold-Based Models

Two primary formulations structure quantilized MFGs for ranking games:

a. Target-Based Formulation

Agents are penalized for deviation—either above or below—from the target quantile: $$J_i^{[N]}(u^i, u^{-i}, \alpha) = \mathbb{E} \left[ \int_0^T \frac{r}{2} (u^i_t)^2 dt + \frac{\lambda}{2} (x_T^i - q_T^{\alpha,[N]})^2 \right]$$ with $q_T^{\alpha,[N]}$ being the empirical α-quantile. In the large-population limit, the cost is replaced by its continuous counterpart, and the equilibrium condition becomes a coupled forward-backward ODE system for $\bar{q}^\alpha_t$ and auxiliary variables. The best-response strategy is linear feedback: $$u_t^* = -\frac{b}{r}\left( \eta_t x_t^* + \pi_t \bar{q}_t^\alpha + \phi_t^\alpha \right)$$ where all coefficients are determined by the equilibrium ODEs.

b. Threshold-Based Formulation

Only deviations below the quantile incur a penalty: $$J_i^{[N]}(u^i, u^{-i}, \alpha) = \mathbb{E} \left[ \int_0^T \frac{r}{2} (u^i_t)^2 dt + \frac{\lambda}{2} (x_T^i - q_T^{\alpha,[N]})^2 \mathbf{1}_{\{ x_T^i < q_T^{\alpha,[N]} \}} \right]$$ The mean-field solution employs the stochastic maximum principle, yielding a semi-explicit feedback law depending on conditional probabilities and means relative to the quantile, coupled with a fixed-point quantile-consistency condition: $q_T^\alpha = Q(\alpha, \mathcal{L}(x_T^*))$ This system lacks a closed analytic form but is solvable iteratively via numerical fixed-point schemes.

Both formulations hinge on nonlocal consistency: the distribution of agents, propagating under optimal controls, must realize the candidate quantile at equilibrium.

3. Existence, Analytic Solutions, and ε-Nash Equilibria

The target-based formulation admits an explicit analytic solution for both the best-response strategies and the equilibrium quantile in the linear-Gaussian case. The forward-backward system determining $\bar{q}^\alpha_t$ and its associated controls is fully decoupled and solvable for general parameters, ensuring both tractability and transparency in determining the impact of model coefficients.

Crucially, the target-based Q-MFG exhibits the ε-Nash property: for any finite but large N, the equilibrium profile achieves Nash error

$$\epsilon_N^\alpha = \mathcal{O} \left( \sqrt{ \frac{1}{N} \frac{ \sqrt{ \alpha(1-\alpha) } }{ p(T, \bar{q}_T^\alpha ) } } \right)$$

where $p(T, \bar{q}_T^\alpha)$ is the equilibrium terminal density at the quantile. Thus, Q-MFG strategies yield asymptotically optimal outcomes in large but finite games, justifying the mean-field approximation for large populations.

The threshold-based model, while lacking a closed-form solution due to the indicator nonlinearity, is amenable to a numerical fixed-point iterative procedure, which converges reliably in simulation. The resulting equilibrium and strategies closely approximate those of the target-based case, particularly as N increases.

4. Numerical Analysis and Population Effects

Computational experiments confirm several central features of Q-MFGs:

• Equilibrium quantile accuracy: The calculated mean-field quantile matches the empirical quantile in large simulated populations.
• Strategy concentration: Under both formulations, individual agent trajectories cluster more tightly around the equilibrium quantile as the population size grows, indicating the controlling effect of the quantile-based incentive.
• Selection dynamics: The estimator for the probability of exceeding the quantile threshold increases over time under optimal control, and the population variance diminishes, leading to sharp phase transitions at selection thresholds.
• Approximations: The difference between target-based and threshold-based equilibrium outcomes is small in practical settings, with the target-quadratic penalty slightly regularizing the distribution of successful agents.

The following table summarizes key comparative aspects:

Aspect	Target-Based Formulation	Threshold-Based Formulation
Terminal Cost	Quadratic (penalizes all deviations)	Quadratic below quantile, none above
Analytic Solution	Yes (via ODE system)	No (semi-explicit, numerical fixed point)
ε-Nash Guarantee	Explicit, order $\mathcal{O}(N^{-1/2})$	Not established
Equilibrium Quantile	Explicit ODE-based	Numerical fixed point
Realism (VC selection)	Direct as competitive target	More realistic, but well approximated

5. Application: Early-Stage Venture Investment

The framework is applied to the modeling of competitive selection processes in venture capital (VC) investment, wherein a VC firm seeks to allocate further funding only to the top (1–α)% performers (e.g., startups with the highest valuation at a fixed date). Here, each startup’s strategic control (effort and investment over time) is optimized for selection under diffusion-driven market uncertainty, and the global quantile outcome ($q^\alpha_T$) dictates the selection cutoff.

• Determinants of the selection threshold: The competitive equilibrium quantile is explicitly computed, allowing prediction of the cutoff value as a function of market volatility, cost of effort, and reward structure.
• Effort dynamics: Higher selection stringency (smaller α) induces greater initial effort and more compressed final outcomes, mirroring real-world dynamics in high-stakes tournaments or investment rounds.
• Efficient approximation: The analytic target-based Q-MFG provides reliable and computationally efficient estimates of the quantile and strategic outcomes for VC-style multistage selection.

6. Broader Significance and Theoretical Implications

Quantilized mean-field games extend classical mean-field approaches to situations where ranks, percentiles, or rare events are the main drivers of competition and selection. The analysis provides:

• Rigorous existence and uniqueness results for equilibrium quantiles and strategies in linear-quadratic diffusion models.
• Explicit expressions for the mean-field error in large populations, validating these models for empirical and computational applications.
• Demonstration that rank-based nonlinearities (as in threshold selection) do not break the mean-field analysis, with target-quadratic formulations serving as effective surrogates.

Applications extend beyond venture investment to any competitive scenario with rank-based incentives—prize tournaments, elite admissions, competitive procurement, and sports—where equilibrium is set not by mean or average but by quantiles of the evolving population performance.

7. Summary Table: Formulation and Solution Properties

Feature	Target-Based	Threshold-Based
Penalization	All deviations	Only below quantile
Analytic Solution	Yes	No, but semi-explicit
Equilibrium Quantile	ODE-based, explicit	Numerically fixed-point
ε-Nash Error	Explicit, order $N^{-1/2}$	Not established
Application Suitability	Direct, computationally efficient	More general, nearly identical numerically

Quantilized MFGs thus furnish a mathematically and computationally tractable way to model, understand, and simulate large-scale competitive selection processes where ranking and quantiles—rather than averages—govern the incentives and outcomes.