We study a portfolio optimization problem for competitive agents with CRRA utilities and a common finite time horizon. The utility of an agent depends not only on her absolute wealth and consumption but also on her relative wealth and consumption when compared to the averages among the other agents. We derive a closed form solution for the n-player game and the corresponding mean field game. This solution is unique in the class of equilibria with constant investment and continuous time-dependent consumption, both independent of the wealth of the agent. Compared to the classical Merton problem with one agent, the competitive model exhibits a wide range of highly nonlinear and non-monotone dependence on the agents’ risk tolerance and competitiveness parameters. Counter-intuitively, competitive agents with high risk tolerance may behave like non-competitive agents with low risk tolerance.

中文翻译：

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在相对性能标准下具有最佳消费和投资的多玩家游戏

我们研究具有CRRA实用程序和共同的有限时间范围的竞争性代理商的投资组合优化问题。与其他代理人的平均值相比，代理人的效用不仅取决于她的绝对财富和消费，还取决于她的相对财富和消费。我们为n导出一个封闭形式的解决方案玩家游戏和相应的平均场游戏。该解决方案在均衡类中是唯一的，具有不变的投资和连续的时间依赖型消耗，而这与代理的财富无关。与带有一个代理的经典默顿问题相比，竞争模型对代理的风险承受能力和竞争参数表现出广泛的高度非线性和非单调依赖性。与直觉相反，具有高风险承受能力的竞争行为者的行为可能类似于具有低风险承受能力的非竞争行为者。