Algorithmic Collusion in Electronic Markets : The Impact of Tick Size

We characterise the stochastic interaction of independent learning algorithms as a deterministic system of ordinary differential equations and use it to understand the long-term behaviour of the algorithms in a repeated game. In a symmetric bimatrix repeated game, we prove that the dynamics of many learning algorithms converge to the outcomes of pure strategy Nash equilibria of the stage game. In contrast, we prove that competition between Q-learning algorithms does not always lead to outcomes of Nash equilibria of the stage game. We apply these results to study how the size of the tick in a limit order book facilitates or obstructs tacit collusion among algorithms that compete to provide liquidity. We characterise the set of pure strategy Nash equilibria of the market making stage game with a discrete action space (e.g., the price grid of a limit order book) and its relation to the Bertrand–Nash equilibrium of the game with a continuous action space (i.e., an idealised limit order book with a zero tick size). We derive the bounds that define the set of Nash equilibria of the stage game when the action space is discrete and show that the bounds converge to the Bertrand–Nash equilibrium as the tick size of the limit order book tends to zero. In the repeated game played by all the algorithms we consider, our findings show that a large tick size obstructs competition; a smaller tick size lowers trading costs for liquidity takers, but slows down the speed of convergence to a rest point. For the algorithms with theoretical guarantees to reach a Nash equilibrium outcome, there is no assurance that the outcome reached is the outcome of the most competitive equilibrium. Indeed, tacit collusion can and often occurs when the algorithms tacitly coordinate on a more profitable Nash equilibrium outcome. However, the excess profits are bounded by the range of those of the possible Nash equilibria of the stage game, which shrinks as the tick size decreases. Finally, for Q-learning, many of the outcomes are sub-optimal for both the market makers and the liquidity takers