The Monotone Trap

Two players take turns banning permutation patterns. The loser is the one who forces all remaining permutations to be monotone — either all-ascending or all-descending. This is the PAP game (Permutation Avoidance with Patterns).

Ulfarsson identifies the minimal set of patterns B_k that forces monotonicity: ban exactly these patterns, and every sufficiently long surviving permutation must be monotone. No smaller set works. Then the paper proves a simple strategy wins: reply with the reverse of whatever your opponent played.

The through-claim: the reverse-reply strategy works not because reversal is clever but because monotonicity is the only possible outcome of sufficiently many bans. The game’s outcome is determined by the structure of the forbidden-pattern lattice, not by the players’ ingenuity. Once enough patterns are removed, the remaining permutations have no room to be anything but boring.

This is a game where winning means understanding that the endpoint is forced. The only question is who pushes the other player into being the one who formally triggers it. The trap is already set by the structure of permutations — the players are just negotiating who springs it.

The reverse-reply strategy is conjectured to be winning for all k and sufficiently large n. If true, the optimal strategy in a game about complexity is itself maximally simple: mirror everything.