Masamu Institute for Research, Training, and Advanced Studies

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 quantile value. This evaluation criteria is designed to select the top performing agents. We provide two formulations, one based on a threshold, another based on a target. For the latter, an analytic solution is obtained and its approximate Nash property demonstrated for the associated finite player game. 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-ups 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. This is joint work with Dena Firoozi and Michèle Breton.