The Welfare Gap of Strategic Storage: Universal Bounds and Price Non-Linearity

This paper studies the efficiency of battery storage operations in electricity markets by comparing the social welfare gain achieved by a central planner to that of a decentralized profit-maximizing operator. The problem is formulated in a generalized continuous-time stochastic setting, where the battery follows an adaptive, non-anticipating policy subject to periodicity and other constraints. We quantify the efficiency loss by bounding the ratio of the optimal welfare gain to the gain under profit maximization. First, for linear price functions, we prove that this ratio is tightly bounded by $4\mathrm{/}3$. We show that this bound is a structural invariant: it is robust to arbitrary stochastic demand processes and accommodates general convex operational constraints. Second, we demonstrate that the efficiency loss can be unbounded for general convex price functions, implying that convexity alone is insufficient to guarantee market efficiency. Finally, to bridge these regimes, we analyze monomial price functions, where the degree controls the curvature. For specific discrete demand scenarios, we demonstrate that the ratio is bounded by $2$, independent of the degree.