Algorithmic collusion: theory & practice

Thesis

We develop a framework to analyze the evolution of bounded memory strategies in a repeated game. In this framework, we introduce the algorithmic learning equations, a set of ordinary differential equations which characterizes the finite-time and asymptotic behavior of the stochastic interaction between learning algorithms that learn a bounded memory strategy in a repeated game. Our framework allows us to study repeated games under a variety of monitoring structures, including perfect, public, private, and any of the combinations.

Using this framework, we use a dynamic generalization of smooth fictitious play with bounded m-memory strategies to model learning with bounded rationality that is consistent with learning by algorithms. With this learning model, we prove a Folk theorem when players with bounded rationality learn as they play a repeated potential game. In a repeated potential game with perfect monitoring, we use this learning model to show that for any feasible and individually rational payoff profile, if players have sufficient memory, are sufficiently patient, and best respond with sufficiently few mistakes, then the players have a non-zero probability of learning an m-memory strategy profile that achieves an average payoff close to the specified payoff profile for an appropriate continuation game. Moreover, the strategy profile learned is an m-memory ε-subgame perfect equilibrium of the repeated game.

Finally, we examine a case study where high-frequency traders (HFTs) in the European ETF market break the pre-trade anonymity of limit orders by signaling their type in an otherwise anonymous market. We explain the behavior of HFTs with a model that considers competitive and collusive equilibria. The model shows that the behavior of the HFTs is consistent with that in a collusive equilibrium where HFTs signal themselves to avoid sniping each other's limit orders. Signaling enables the HFTs to share the benign flow from retail limit orders, and to share the additional benign flow from impatient investors who otherwise would have traded with a retail investor’s limit order.

Authors: Chang, P

• DOI: 10.5287/ora-2zzb1a9gw
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: learning; collusion; games; Folk theorem
• Subjects: Games; Collusion; Learning