Janusz M Meylahn

Janusz M. Meylahn, Maximilian Schäfer

Algorithmic collusion among pricing algorithms has raised concerns about sustained supra-competitive prices and their implications for social welfare. Existing work has largely focused on the probability that reinforcement-learning algorithms converge to cooperative strategies, typically under the assumption that exploration vanishes over time. Motivated by the observation that algorithms deployed in practice are likely to continue exploring in order to remain adaptive to changing environments, we study learning dynamics under constant exploration. In this setting, the relevant question is no longer whether an algorithm converges to a particular strategy profile, but rather what fraction of time the algorithms spend playing cooperative strategies. Even in the benchmark case of the repeated Prisoner's Dilemma with one-period memory, this yields high-dimensional stochastic learning dynamics, for which a complete analytic treatment is intractable. We show that cooperative strategies can be dominant in this time-averaged sense and derive a boundary predicting when such dominance arises, based on the expected dynamics of the Q-learning process. Extensive simulations show that this boundary is a strong predictor for non-defection-dominated behaviour under epsilon-greedy Q-learning.

Wolfram Barfuss, Janusz Meylahn

In this work, we ask for and answer what makes classical temporal-difference reinforcement learning with epsilon-greedy strategies cooperative. Cooperating in social dilemma situations is vital for animals, humans, and machines. While evolutionary theory revealed a range of mechanisms promoting cooperation, the conditions under which agents learn to cooperate are contested. Here, we demonstrate which and how individual elements of the multi-agent learning setting lead to cooperation. We use the iterated Prisoner's dilemma with one-period memory as a testbed. Each of the two learning agents learns a strategy that conditions the following action choices on both agents' action choices of the last round. We find that next to a high caring for future rewards, a low exploration rate, and a small learning rate, it is primarily intrinsic stochastic fluctuations of the reinforcement learning process which double the final rate of cooperation to up to 80%. Thus, inherent noise is not a necessary evil of the iterative learning process. It is a critical asset for the learning of cooperation. However, we also point out the trade-off between a high likelihood of cooperative behavior and achieving this in a reasonable amount of time. Our findings are relevant for purposefully designing cooperative algorithms and regulating undesired collusive effects.

Leon Halgryn, Sophie Langer, Janusz M. Meylahn et al.

Finite-sample analyses of deep Q-learning typically treat replayed data as independent, even though it is sampled from temporally dependent state-action trajectories. We study the Deep Q-networks (DQN) algorithm under explicit dependence by modelling the minibatches used for updating the network as $τ$-mixing. We show that this assumption holds under certain dependence conditions on the underlying trajectories and the mechanism used to sample minibatches. Building on this observation, we extend statistical analyses of DQN with fully connected ReLU architectures to dependent data. We formulate each update as a nonparametric regression problem with $τ$-mixing observations and derive finite-sample risk bounds under this dependence structure. Our results show that temporal dependence leads to a degradation in the statistical rate by inducing an additional dimensionality penalty in the rate exponent, reflecting the reduced effective sample size of $τ$-mixing data. Moreover, we derive the sample complexity of DQN under $tau$-mixing from these risk bounds. Finally, we empirically demonstrate on standard Gymnasium environments that the independence assumption is systematically violated and that replay sampling yields approximately exponentially decaying correlations, supporting our theoretical framework.