Closed-form Stackelberg leader strategy in the Bertrand pricing stage game

Derive a closed-form analytical expression for the Stackelberg leader commitment strategy in the Bertrand pricing stage game G^B(k), in which two sellers choose prices from the discretized set P = {1/k, 2/k, ..., 1} under the Bertrand allocation rule C^B (the lower price captures all demand and ties split demand). Specify the resulting leader’s mixed strategy explicitly as a function of k and, as a motivation, enable rigorous analysis of the corresponding payoffs and induced average prices across all discretizations k.

Background

The paper studies algorithmic collusion in repeated pricing games and analyzes both Nash and Stackelberg equilibria in algorithm space and in the underlying stage games. In numerical experiments for the Bertrand stage game with discretized prices, the authors compute the Stackelberg leader strategy and observe that the leader and follower payoffs are approximately (k−1)/(e k), implying an average buyer price of about 2(k−1)/(e k) ≥ 2/3 for k up to 200.

However, there is no analytical characterization of the Stackelberg leader’s optimal mixed strategy in the stage game, and obtaining a closed-form expression would provide a foundation for proving the observed behavior for all k and rigorously characterizing the induced supra-competitive prices without threats.