# Efficiency of the Price Formation Process in Presence of High Frequency Participants: a Mean Field Game analysis

arXiv: https://arxiv.org/abs/1305.6323 · HTML: https://mlsanity.com/paper/1305.6323/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 May 2013.

## Built on by (most cited first, 6 of 6)

- [Bridging Mean-Field Games and Normalizing Flows with Trajectory Regularization](https://mlsanity.com/paper/2206.14990/) (2022)
- [Signatured Deep Fictitious Play for Mean Field Games with Common Noise](https://mlsanity.com/paper/2106.03272/) (2021)
- [Solving Continuous Mean Field Games: Deep Reinforcement Learning for Non-Stationary Dynamics](https://mlsanity.com/paper/2510.22158/) (2025)
- [Consensus Based Stochastic Control](https://mlsanity.com/paper/2501.17801/) (2025)
- [Reliability and Identifiability in Persona-Trained Monte Carlo: Variance Decomposition, Stability Bounds, and the Identifiability of Heterogeneous News Reaction](https://mlsanity.com/paper/2607.04627/) (2026)
- [Inverse Learning of the Altruism and Cost Level in Mixed-Individual Mean Field Games](https://mlsanity.com/paper/2609.13469/) (2026)

## Abstract (arXiv)

This paper deals with a stochastic order-driven market model with waiting costs, for order books with heterogenous traders. Offer and demand of liquidity drives price formation and traders anticipate future evolutions of the order book. The natural framework we use is mean field game theory, a class of stochastic differential games with a continuum of anonymous players. Several sources of heterogeneity are considered including the mean size of orders. Thus we are able to consider the coexistence of Institutional Investors and High Frequency Traders (HFT). We provide both analytical solutions and numerical experiments. Implications on classical quantities are explored: order book size, prices, and effective bid/ask spread. According to the model, in markets with Institutional Investors only we show the existence of inefficient liquidity imbalances in equilibrium, with two symmetrical situations corresponding to what we call liquidity calls for liquidity. During these situations the transaction price significantly moves away from the fair price. However this macro phenomenon disappears in markets with both Institutional Investors and HFT, although a more precise study shows that the benefits of the new situation go to HFT only, leaving Institutional Investors even with higher trading costs.