When a Local Pricing Guard Is Not a Welfare Promise

A local pricing rule can improve observed customer benefit without guaranteeing its target. We study a bounded logit duopoly where a guard permits near-best immediate profit against the rival’s previous price. A one-time cap appears protective in a coarse game, but the continuous worst long-run benefit is 66.272, below the unchanged 95% target of 68.513. An attainable price envelope characterizes continuing use of the same cap. A funded calendar minimizes worst-path cap frequency within causal cap-or-guard controllers protecting every activation prefix; base effective-continuous frequency falls about 7.29% from its best fixed period. Separate finite and continuous calculations retain 120 target labels. An earlier 192-run learning study has unequal clock capacities. A prospectively frozen 24-run followup matches five phases and dense memory: recurring limits improve final frozen-cycle benefit by 0.956 units, with all twelve paired differences positive. A further twelve-run reference removes the post-startup guard over the original grid at the same budget. Customer benefit falls in all twelve pairs, by 12.733 units on average; not every shop gains against static Nash. Separately authored replay checks every round of both follow-ups. Static anchors and a 48-label frozen-partner audit keep observed behavior distinct from equilibrium and individual incentives. These finite-budget results combine learning and execution constraints and do not establish convergence, collusion prevention or intent. The contribution is a model-specific guarantee audit, controlled behavioral comparisons and a graphical game using established methods.