A unified convention for achievement positional games

We introduce achievement positional games, a convention for positional games which encompasses the Maker-Maker and Maker-Breaker conventions. We consider two hypergraphs, one red and one blue, on the same vertex set. Two players, Left and Right, take turns picking a previously unpicked vertex. Whoever first fills an edge of their color, blue for Left or red for Right, wins the game (draws are possible). We establish general properties of such games. In particular, we show that a lot of principles which hold for Maker-Maker games generalize to achievement positional games. We also study the algorithmic complexity of deciding whether Left has a winning strategy as the first player when blue edges and red edges have respective sizes at most $p$ and $q$. This problem is in P for $p,q \leq 2$, but it is NP-hard for $p \geq 3$ and $q=2$, coNP-complete for $p=2$ and $q \geq 3$, and PSPACE-complete for $p,q \geq 3$ even when the 3-edges are the same for both colors. That last result has an interesting consequence on the Maker-Maker convention: for 3-uniform hypergraphs, which is the only case whose complexity is currently open (for starting positions of the game), we show PSPACE-completeness for positions obtained after one round of play.

💡 Research Summary

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The paper introduces “achievement positional games,” a unified framework that simultaneously captures the classic Maker‑Maker and Maker‑Breaker conventions. A game is defined by a common vertex set V together with two hypergraphs (V, EL) and (V, ER), representing the winning sets of Left (blue) and Right (red). Players alternately claim previously unclaimed vertices; the first to completely occupy an edge of their own colour wins, and a draw occurs if all vertices are taken without a winner. After each move the relevant hyperedges are updated: the player’s own edges shrink by the chosen vertex, while the opponent’s edges containing that vertex become dead. This formulation subsumes Maker‑Maker (EL = ER) and Maker‑Breaker (ER empty or the transversal of EL).

The authors first establish that many fundamental properties of Maker‑Maker games—strategy stealing, the existence of non‑losing strategies for the second player, and the determinability of the outcome of a disjoint union—carry over unchanged to the achievement setting. They then focus on the decision problem AchievementPos(p,q): given a game where every blue edge has size at most p and every red edge at most q, does Left have a winning strategy as the first player? They prove that the problem lies in PSPACE for all p,q, and they completely classify its complexity for every pair (p,q).

For p,q ≤ 2 the problem is solvable in polynomial time (indeed in L). When p ≥ 3 and q = 2 the problem becomes NP‑hard, extending known hardness for Maker‑Maker games. When p = 2 and q ≥ 3 it is coNP‑complete, reflecting the dual difficulty of preventing the opponent’s larger edges. The most striking result is that for p,q ≥ 3 the problem is PSPACE‑complete; in particular, for (p,q) = (3,3) the authors obtain PSPACE‑hardness even when the 3‑edges are identical for both colours. This yields the first general complexity classification for Maker‑Maker games on 3‑uniform hypergraphs after a single round of (non‑optimal) play, a case that had remained open for starting positions.

A comprehensive table summarizes known results for Maker‑Maker and Maker‑Breaker games and highlights the new contributions of this work (the yellow cells). The paper also notes a recent refinement showing that the (3,2) case is PSPACE‑complete, strengthening the earlier NP‑hardness claim.

In conclusion, the unified achievement model provides a powerful lens for studying positional games with asymmetric winning sets, clarifies the algorithmic landscape across all edge‑size regimes, and opens avenues for further exploration of multi‑colour or more intricate goal structures.