标题： 基于排名的量化平均场博弈及其在早期阶段风险投资中的应用 显示英文标题

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

摘要： 量化均场博弈模型涉及种群分布的分位数。 我们研究了一类具有排名博弈能力的此类博弈，其中每个代理的性能是根据其终端状态相对于种群的$\alpha$分位数值的$\alpha \in (0,1)$进行评估的。 此评估标准旨在选择表现最好的$(1-\alpha)\%$个代理。 我们为此竞争提供了两种表述方式：基于目标的表述和基于阈值的表述。 在前一种和后一种表述中，为了满足选择条件，每个代理旨在使其终端状态分别等于种群的$\alpha$分位数值，且与之\textit{精确地}相等和\textit{至少}相等。 针对基于目标的公式，我们得到了一个解析解，并展示了在$N$人博弈中渐近最优反应策略的$\epsilon$-Nash 性质。 具体而言，量化平均场一致性条件被表示为一组前向-后向常微分方程，描述了平衡状态下的$\alpha$-分位数值。 对于基于阈值的公式，我们得到了一个半显式解，并数值求解了由此产生的量化平均场一致性条件。 随后，我们在早期阶段的风险投资背景下提出了一种新的应用，其中风险投资公司资助一组在有限时间范围内进行竞争的初创公司，目标是在时间范围结束时选择一定比例的排名靠前的公司以获得下一轮资金支持。 我们展示了在此背景下讨论的两种公式的数值实验结果和解释，并表明基于目标的公式对基于阈值的公式提供了非常好的近似。 显示英文摘要

摘要： 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 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.