Stochastic Controls in Competitions and Mean Field Games

Ye, Jiaxuan

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Stochastic Controls in Competitions and Mean Field Games

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Problems combining games and controls for multiple players become widely studied due to the complexity of the world and the interactions among populations. In this thesis, we propose two models for fund managers who compete with their relative performance and one Mean Field Game model with common noise for cost minimization. For the first model, we consider a group of managers competing for the cash flows based on their relative performance by choosing between an idiosyncratic and a common risky investment opportunity. Since investors may choose to invest or withdraw continuously conditional on the real-time performance of funds, the model is of continuous competition. The unique constant equilibrium is derived in closed form, which implies that funds generally decrease the investments in their idiosyncratic risky assets under competition, in order to lower the risk of the relative performance. It pushes all funds to herd and hurts their after-fee performance. However, sufficiently disadvantaged funds with poor idiosyncratic investment opportunities or highly risk-averse managers may take the excessive risk for a better chance of attracting new investments, and their performance may improve compared to the case without competition, which benefits the investors. For the second model, we propose a principle agent model where the principle is a policy maker who decides the optimal capital gain tax rate and agents are fund managers who choose optimal portfolios in their investment opportunities. The optimal tax rate and unique portfolios are derived for one policy maker and one representative fund. Moreover, with one policy maker but N funds competing with each other based on the terminal relative performance, there exist multiple Nash equilibria and a unique Pareto optimal equilibrium can be found. Our findings also suggest that managers may take more risks with the higher tax rate, which is different from the existing literature. For the third model, we study Mean Field Games with a common noise given by a continuous time Markov chain or an independent Brownian motion under a quadratic cost structure. The theory implies that the optimal path under the equilibrium is a Gaussian process conditional on the common noise. Interestingly, it reveals the Markovian structure of the random equilibrium measure flow, which can be characterized via a deterministic finite dimensional system. Moreover, the counterpart N-player game can be embedded in the probability space generated by two Brownian motions, which concludes the convergence of the N-player game to Mean Field Games, both in the sense of the processes and the empirical measure.

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