Is Simple Rounding of Quantity Estimates Suboptimal for the First-Mover in a Duopoly Game?

The classic Stackelberg model (CSM) generally yields fractional units given its continuous divisibility assumption. We consider whether the first-mover in a duopoly game is hurt by rounding budgetary estimates of quantity as calculated from the CSM. We derive our benchmark for not rounding by considering a dynamic “game within a game” in which we relax the continuous divisibility assumption, and each player optimizes the deviation between their CSM quantity with a fractional remainder and an integer solution. Using a game-theoretical framework, we obtain a closed-form subgame perfect Nash equilibrium given integer constraints. We find that the leader is generally better off than under the CSM, in contrast to the follower who is often worse off. We show that a universal rounding strategy by the leader is suboptimal. As a verification test, we use a brute-force numerical search to confirm our theoretical solutions.