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

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 $\alpha$-quantile value, $\alpha \in (0,1)$. This evaluation criterion is designed to select the top $(1-\alpha)\%$ 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 \textit{exactly} equal and \textit{at least} equal to the population's $\alpha$-quantile value, respectively. For the target-based formulation, we obtain an analytic solution and demonstrate the $\epsilon$-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 $\alpha$-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 a set of numerical experiments for both formulations discussed in this context, which illustrate that the target-based formulation closely approximates the threshold-based formulation in the scenarios considered.