# 2024 Optimization Days, (algorithmic) collusions in games

Tomorrow, I will attend the 2024 Optimization Days, in Montréal. I will present some work we did last Fall with Philipp Ratz and Suzie Grondin, on (algorithmic) collusions in games, “Market Pricing with Reinforcement Learning” (the paper will be available soon)

Several recent articles have attempted to gain a better understanding of algorithmic collusion (Calvano et al. (2020), Klein (2021), Banchio & Mantegazza (2022) Rocher et al. (2023)). For example, in Calvano et al. (2020), a simulation study showed that for a simplified market environment, basic Q-Learning Agents can learn to collude tacitly, in order to propose higher prices and increase their combined profit. Inspired by some Iterated Prisoners Dilemma, we derive some reinforcement learning algorithm to investigate and discuss several recent results and their robustness, and explain how reinforcement learning differs from simpler strategies and which conditions lead to unfavorable outcomes from a consumer perspective. In particular, we first describe the reinforcement learning problem in a more general manner and investigate the influence of the hyper-parameters. We then consider two situations separately. One, similar in spirit to Rocher et al. (2023), assumes that the market is in equilibrium and that a general agent tries to exploit a pricing strategy of an incumbent agent. The second, more general, approach consists of an agent continuously updating their own policy.

The starting point was Calvano et al. (2020),

For classical games, the mathematical framework is the following

for example, with the prisoner’s dilemma

Then, consider repeated games, and possible collusion

The next step is to include randomness, with (dynamic) stochastic games

and standard equations

(I describe quickly the different concepts). Finally, we can move from here to reinforcement learning, and Q-learning

The idea will be to play (or to interact) to learn that matrix

with the following interpretations, for the different parameters

Then, we will play a little bit, on the framework introduced to present the prisoner’s dilemma, for instance to understand the importance of $\beta$, using in the $\epsilon$-greedy approach, with $\epsilon_t=\exp(-\beta t)$

That is our first approach to the concept of collusion : agents don’t need to “cooperate” to have collusion

Then, we will use the experiment of Calvano et al. (2020) to get more complex discussions…

# Apprentissage par renforcement, jeux et collision

Ce matin, Suzie Grondin nous expliquait les applications du RL (apprentissage par renforcement) en théorie des jeux, pour comprendre la collusion, pour la fin de son stage (6 mois de césure à l’ENSAE Paris) 😪 . Avec un lien avec le papier de Luc Rocher sur les approches adversariales, et une rapide introduction au “Offline-to-Online Reinforcement Learning”…

Merci à toute l’équipe (Philipp Ratz, François HU, Agathe Fernandes Machado, Dante Mata López) qui est venu l’écouter, ainsi que Louis Abraham… Superbe travail ! on veut un papier maintenant 😉 !

# Government Intervention in Catastrophe Insurance Markets

Our paper, Government Intervention in Catastrophe Insurance Markets: A Reinforcement Learning Approach, jointly written with Menna Hassan and Nourhan Sakr is now available on ArXiv.

This paper designs a sequential repeated game of a micro-founded society with three types of agents: individuals, insurers, and a government. Nascent to economics literature, we use Reinforcement Learning (RL), closely related to multi-armed bandit problems, to learn the welfare impact of a set of proposed policy interventions per \$1 spent on them. The paper rigorously discusses the desirability of the proposed interventions by comparing them against each other on a case-by-case basis. The paper provides a framework for algorithmic policy evaluation using calibrated theoretical models which can assist in feasibility studies.

# Dynamic Programming in Distributional Reinforcement Learning

Last summer, I supervised a summer intern, Cédric Odin, student at Ecole Normale in Ker Lann, France, on Dynamic Programming in Distributional Reinforcement Learning. A state-of-the-art is now available online https://hal.archives-ouvertes.fr/hal-03168889

The classic approach to reinforcement learning is limited in that it only predicts the expected return. The specialized literature has long tried to remedy this problem by studying risk-sensitive models, but the distributional approach will not emerge until 2017. Since the seminal article Bellemare, Dabney, and Munos 2017 and the state-of-the-art performance of the C51 algorithm in the ATARI 2600 suite of benchmark tasks (Bellemare, Naddaf, et al. 2013), research has focused on understanding the behaviour of distributional algorithms. In this paper we place Bellemare’s original results in distributional dynamic programming in parallel with the classic results.

# Reinforcement Learning in Economics and Finance

With Romuald Elie and Carl Remlinger we recently uploaded on ArXiv a paper on Reinforcement Learning in Economics and Finance

Reinforcement learning algorithms describe how an agent can learn an optimal action policy in a sequential decision process, through repeated experience. In a given environment, the agent policy provides him some running and terminal rewards. As in online learning, the agent learns sequentially. As in multi-armed bandit problems, when an agent picks an action, he can not infer ex-post the rewards induced by other action choices. In reinforcement learning, his actions have consequences: they influence not only rewards, but also future states of the world. The goal of reinforcement learning is to find an optimal policy — a mapping from the states of the world to the set of actions, in order to maximize cumulative reward, which is a long term strategy. Exploring might be sub-optimal on a short-term horizon but could lead to optimal long-term ones. Many problems of optimal control, popular in economics for more than forty years, can be expressed in the reinforcement learning framework, and recent advances in computational science, provided in particular by deep learning algorithms, can be used by economists in order to solve complex behavioral problems. In this article, we propose a state-of-the-art of reinforcement learning techniques, and present applications in economics, game theory, operation research and finance.