Mean Field Games for Modeling Liquidity and Market Interaction

# Game theoretic description of stock market


# Game theoretic description of stock market












I read the book "Theory of Games and Economic Behavior" by John von Neumann and Oskar Morgenstern. I think that stock market may be described by game theory. But here are the problems:

- number of players (traders) varies. In this book that number is constant.

- A player may act (e.g. place a limit order) whenever he wants. In this book, the moves are placed in a specific given order.

- This game has infinite time horizon, that is why we can't describe a payoff functions at the end of the game.

I am interested in such description of stock market, that does not involve any chance moves (i.e. moves that are random, that no player can understand its "nature"). Indeed, each move in stock market is a personal move (i.e. a move made by player's will). I focus mostly on short time horizon.

I would be grateful if someone gave me a general game theoretic description of a stock market that solves these problems (1-3). I would also ask for references in books, articles about this topic. Are there any books about game theoretic view on market microstructure?

Regards.

## Answer by lehalle (score 3, accepted)

https://quant.stackexchange.com/a/69897

You should have a look at Mean Field Games (MFG). Mean Field Game theory studies optimal control problems with infinitely many interacting agents. The solution of the problem is an equilibrium configuration in which no agent has interest to deviate (a Nash equilibrium).

The terminology and many substantial ideas were introduced in the seminal papers by Lasry and Lions (remember that PL Lions obtained the Fields medal in 1994) [Lasry and Lions 2006a, Lasry and Lions 2006b, Lasry and Lions 2007]. Similar models were discussed at the same time by Caines, Huang and Malhamé (see, e.g., [Huang et al., 2006]), who computed explicit solutions for the linear- quadratic (LQ) case (see also [Bensoussan et al. 2016]).

Applications to economics and finance were first developed in [Guéant et al. 2011]. Since these pioneering works the literature on MFG has grown very fast: see for instance the monographs or the survey papers [Bensoussan et al. 2013, Caines 2015, Gomes et al. 2014].

This description comes from Mean Field Game of Controls and An Application To Trade Crowding, by Cardaliaguet et L. They are different papers using MFG for financial applications, my opinion is that the good (and easiest way) to do it is to consider liquidity as a mean field.

Remember that a simple example of mean field is the atmospheric pressure in a room: every particle of air is submitted to the mean field (atmospheric pressure) that influences them, and no particule in isolation can influence the mean field. But if all particles moves simultaneously in the same direction, it influences the mean field.

This is a good definition for liquidity: liquidity implicitly synchronises market participants, no participant can significantly change the state of the liquidity, but a decent fraction of participants can move liquidity a way that influences the operation of markets (essentially via trading costs and market impact, including what happens during fire sales).