On Opponent Ineptitude as a Path to Victory: A Structural Game-Theoretic Model of Sequential Attempts under Conjunctive Success

Winning a sequential turn-based game does not depend only on a player’s own skill: it also depends on how competently the opponent plays, since the opponent’s choices determine which of a player’s own targets remain blocked and which become open. Using a simplified model inspired by pool and billiards, we show that attacking one’s hardest remaining target first robustly beats the naive instinct to clear easy targets first, across three levels of opponent competence. This resembles a known result from stochastic scheduling theory, but a direct test shows the resemblance is only partial: the effect is substantially rooted in the board’s spatial structure, not the general scheduling principle it echoes. We then generalize the board to six structurally distinct shapes and find the advantage is not universal: it traces a clear gradient tied to how much alternative routing the board offers, strong on boards with little redundancy and negligible or reversed on richly connected ones. A standard graph-theoretic measure of overall connectivity predicts this gradient closely. We further show the advantage depends on the win condition itself, reversing once a player no longer needs to clear every target, and that “inept” opponent play is not a single category: an opponent who consistently inverts sound instinct, despite understanding nothing of the game, is harder to beat than one who plays with no pattern at all. Along the way, an internal check caught and corrected a subtle assignment bias that had inflated our earliest results on the most extreme board shape; we report this openly, since the correction is itself part of what this work offers. Together these results sharpen a single central claim: prioritizing one’s hardest target is not a universal law of sequential competition, but a context-dependent strategy whose value depends on the shape of the board, the demands of the objective, and precisely how an opponent falls short of playing well.