Computable Countermarkets and the Limits of Universal Trading

Abstract

We explain why no trading algorithm can guarantee profit in every market. For each deterministic program that always returns an exact rational position, we construct a fixed, algorithmically generated price path on which every active position loses and inactivity earns nothing. This holds with positive, continually changing prices, costless trading, and unlimited computation time. Conversely, a supplied total computable price program with nonzero moves admits a computable trader that profits at every step. Mutual countering depends on information, timing, and termination. Approximation-based real outputs preserve the cumulative obstruction but not the loss on every active position. Separate arguments limit learning market rules, certifying future events, and establishing randomness from finite data. Betting-martingale characterizations connect trading with computable and Martin-Löf randomness. Useful strategies may exploit market structure, information, or compensation for risk, while benchmark performance need not imply profit. Reversing and rearranging price histories within the assumed market class provide practical stress tests, distinguishing conditional success from universal guarantees.

arXiv comment15 pages, 1 table, added a section on adaptive mutual countering, invited contribution to ECONVN2026/2027