When Best-of-Both-Worlds Is Impossible: Terminal-Ratio Bandits

We study finite-armed bandits whose performance is the ratio of cumulative expected reward to cumulative expected cost. This apparently mild change from additive regret creates a sharp causal obstruction. We prove that, for the standard best-fixed-arm terminal-ratio benchmark, every learner suffers constant adversarial regret, even with full information and even when a genie, at the midpoint, reveals the continuation before round $T/2+1$. At denominator floor $V_{\min}=1/4$, the minimax regret is at least $4/15$ for every even horizon. Consequently, a best-of-both-worlds theorem combining vanishing adversarial regret with logarithmic stochastic regret is impossible under this benchmark. We then isolate what remains achievable. In the i.i.d. regime, a clipped optimistic-ratio policy, Root-UCB, obtains a stabilization-free, gap-dependent $O(\log T/T)$ fractional pseudo-regret bound with an explicit burn-in scale. The result requires neither Robbins--Monro convergence nor a postulated probability-collapse event. We also explain why drifting-dual decompositions necessarily leave a comparator mismatch in the adversarial model. The findings separate a statistical optimization problem that admits fast rates from a non-additive adversarial objective that is not online learnable.