Robust predictions under finite depth of reasoning

This paper studies a robustness of solutions in finite depth of reasoning models. The level-k and cognitive hierarchy models often assume that each player has a commonly known anchor behavior (so called a level-0 action) to obtain a sharp prediction. We provide a general framework to examine whether the obtained prediction is robust to small uncertainty about other players’ anchors. Our main result shows that any $$\mathbf p$$p-dominant equilibrium is robust if players put sufficiently small probability (decreasing in $$\mathbf p$$p) on other players’ having high reasoning levels. This result highlights a distinction between the two prominent models: a risk-dominated equilibrium is robust in the cognitive hierarchy model, but not in the level-k model.