Mean-Field Games and Ambiguity Aversion

Abstract

This thesis focuses on incorporating the idea of ambiguity aversion into mean-field games.Intuitively, mean-field games describes the dynamics of a system with an infinite population. It is useful in approximating systems with a large population, for which an exact result is computationally intractable. By adding in ambiguity aversion, the mean-field game reflects how players in the population should act if they wish to protect themselves from model misspecification.Two applications of mean-field games are considered through two distinct approaches.The broker execution problem is investigated in a multi-agent framework containing (i) a major agent who is liquidating a large number of shares, (ii) a number of minor agents (high-frequency traders (HFTs)) who search for statistical arbitrage strategies, and (iii) noise traders who buy and sell for exogenous reasons. All optimizing agents (the broker and HFTs) trade against noise traders as well as one another. We use a mean-field game approach to solve the problem and obtain a set of decentralized feedback trading strategies for the major and minor agents. Furthermore, the mean-field game strategies have an N -Nash equilibrium property where N → 0 as N → ∞.The second application focuses on interbank borrowing and lending, which may induce systemic risk into financial markets. A simple model of this is to assume that log-monetary reserves are coupled, and that banks can also borrow/lend from/to a central bank. When all banks optimize their cost of borrowing and lending, this leads to a stochastic game. We account for model uncertainty by recasting the problem as a robust stochastic game and succeed in providing a strategy which leads to a Nash equilibria for ii both the finite game and the mean-field game limit. We prove that an -Nash equilibrium exists and show that when firms are ambiguity-averse, default probabilities can be reduced relative to their ambiguity-neutral counterparts.Finally, this thesis develops a modified stochastic maximum principle for min-max problems, and derives new existence and uniqueness results for a mean-field game with ambiguity averse players. An -Nash equilibrium is shown to exist for this class of mean-field games, and the mean-field game equations are explicitly derived in the linearquadratic framework.iii

Mean‐field games with differing beliefs for algorithmic trading

“…Another direction of research is to restrict agents to trade at stopping times, rather than continuously as done here. Lastly, accounting for model uncertainty/ambiguity aversion along the lines of [10] and [21] would be a very interesting direction to explore so that agents' strategies become more robust.…”