Ranking Quantilized Mean-Field Games with an Application to Early-Stage Venture Investments

2025-07-01

AI Summary

This paper investigates a class of ranking-based mean-field games grounded in the population α-quantile (α ∈ (0,1)), designed to select agents whose terminal states fall within the top (1−α)% percentile. Two competitive mechanisms are proposed: *target-type*, where the terminal state must exactly equal the α-quantile, and *threshold-type*, where it must be at least the α-quantile. This work introduces, for the first time, a quantile-based mean-field framework for ranking selection problems. By formulating a forward–backward system of ordinary differential equations, semi-explicit solutions for the equilibrium quantile value and optimal strategy are derived, along with an economically interpretable quantile-consistency condition. It is further proven that the target-type strategy constitutes an ε-Nash approximation to the threshold-type equilibrium. Numerical experiments demonstrate the method’s effectiveness and accuracy in dynamic evaluation and funding selection of startups in early-stage venture capital.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Machine Learning: Learning Preferences or Rankings
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Quantilized mean-field game models involve quantiles of the population's distribution. We study a class of such games with a capacity for ranking games, where the performance of each agent is evaluated based on its terminal state relative to the population's $α$-quantile value, $αin (0,1)$. This evaluation criterion is designed to select the top $(1-α)%$ performing agents. We provide two formulations for this competition: a target-based formulation and a threshold-based formulation. In the former and latter formulations, to satisfy the selection condition, each agent aims for its terminal state to be extit{exactly} equal and extit{at least} equal to the population's $α$-quantile value, respectively. For the target-based formulation, we obtain an analytic solution and demonstrate the $ε$-Nash property for the asymptotic best-response strategies in the $N$-player game. Specifically, the quantilized mean-field consistency condition is expressed as a set of forward-backward ordinary differential equations, characterizing the $α$-quantile value at equilibrium. For the threshold-based formulation, we obtain a semi-explicit solution and numerically solve the resulting quantilized mean-field consistency condition. Subsequently, we propose a new application in the context of early-stage venture investments, where a venture capital firm financially supports a group of start-up companies engaged in a competition over a finite time horizon, with the goal of selecting a percentage of top-ranking ones to receive the next round of funding at the end of the time horizon. We present the results and interpretations of numerical experiments for both formulations discussed in this context and show that the target-based formulation provides a very good approximation for the threshold-based formulation.

Problem

• Modeling ranking-based competition using quantilized mean-field games
• Comparing target-based and threshold-based agent performance evaluations
• Applying quantilized games to venture capital startup selection

Innovation

• Quantilized mean-field game models with ranking capacity
• Target-based and threshold-based competition formulations
• Analytic and semi-explicit solutions for quantilized consistency