No Coin Left Behind: Maximizing Strategic Surplus Against No-Regret Dynamics

We investigate the **strategic surplus** obtainable against a **Follow-the-Regularized-Leader (FTRL)** learner with constant step size $\eta$ in $n \times m$ two-player zero-sum games played over $T$ rounds against a clairvoyant optimizer. In contrast with prior analysis, we show that the extraction of such regret-scale surplus is an inherent feature of the FTRL family, rather than an artifact of specific instantiations. First, for a fixed max-min optimizer, we establish a sweeping law of order $\Omega(N/\eta)$, proving that utility surplus scales with the number of the learner's suboptimal actions $N$ and vanishes in their absence. Second, for an alternating optimizer, a surplus of $\Omega(\eta T/\text{poly}(n,m))$ can be guaranteed regardless of the equilibrium structure, with high probability, in random games. Our analysis uncovers a sharp geometric dichotomy: **non-steep** regularizers allow the optimizer to realize the maximal transient surplus via finite-time elimination of suboptimal actions, whereas **steep** regularizers introduce a vanishing tail correction that can delay surplus saturation. Finally, we discuss whether this leverage persists under bilateral payoff uncertainty and propose a susceptibility measure quantifying which regularizers are most vulnerable to learner-aware strategic steering.