## Mean-Field Game Strategies for a Major-Minor Agent Optimal Execution Problem

#### Sebastian JaimungalUniversity of Toronto

We introduce, for the first time, a mean-field game framework for a multiple agent optimal execution problem with continuous trading. This modeling generalizes the classical single agent optimal liquidation problem to a setting with (i) a major agent who is liquidating a large portion of shares, and (ii) a number of minor agents (high-frequency traders (HFTs)) who detect and trade along with the liquidator. As in the classical framework, all agents are exposed to temporary price impact and attempt to balance their impact against price uncertainty. Unlike most other works, we account for the permanent price impact that order-flow from all agents have on the midprice and this induces a distinct cross interaction between major and minor agents. This formulation falls into the realm of stochastic dynamic game problems with mean-field couplings in the dynamics, and we analyze the problem using a mean-field game approach. We obtain a set of decentralized feedback trading strategies for the major and minor agents, and express the solution explicitly in terms of a deterministic fixed point problem. For a finite population of $N$ HFTs, the set of major-minor agent mean-field game strategies is shown to have an epsilon-Nash equilibrium property where $\epsilon \to 0$ as $N \to \infty$.

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