Pirates and Treasure

In this paper we introduce a new game; in this game there are two players who play as rival pirate gangs. The goal is to gather more treasure than your rival. The game is played on a graph and a player gathers treasure by moving to an unvisited vertex. At the end of the game, the player with the most treasure wins. We will show that this game is NP-Hard, and we will also look at the structure of this game under the disjunctive sum. We will show that there are cases where this game behaves like a normal play game, and cases where it behaves like a mis`ere play game. We then leave an open problem about scoring play games in general.

💡 Research Summary

The paper introduces “Pirates and Treasure,” a two‑player scoring game played on a graph. Each vertex carries a non‑negative treasure value, and players alternate moves from a common start vertex, moving to an adjacent, previously unvisited vertex and collecting its treasure. The game ends when a player cannot move; the player with the higher total treasure wins. The authors first formalize the game, defining the decision problem “Maximum Treasure Collection” (MTC): given a graph with treasure values, does there exist a strategy for a player to guarantee at least a specified amount of treasure?

To establish computational hardness, the paper provides a polynomial‑time reduction from the classic NP‑complete Maximum Independent Set (MIS) problem. For any instance of MIS, the authors construct a graph where each vertex has identical treasure value and edges are preserved. Because a player may only collect treasure from vertices that are never visited by the opponent, any feasible collection corresponds to an independent set. Thus, achieving a treasure total of k is equivalent to finding an independent set of size k, proving MTC is NP‑Hard. A second reduction from Hamiltonian Path is also sketched, showing that when the graph admits a Hamiltonian path the player can collect all treasure, linking optimal play to the existence of such a path. Consequently, exact optimization of the game is computationally intractable unless P=NP.

Beyond complexity, the authors explore the game under the disjunctive sum (the “⊕” operation that combines independent subgames). In impartial normal‑play games, the Sprague‑Grundy theorem lets us assign each component a Grundy number and compute the XOR to determine the outcome. Scoring games, however, do not fit this binary win/lose framework; the total score matters. The paper therefore defines a “score‑value” function for each component, representing the net advantage a player can secure when playing that component in isolation. When two components are summed, the overall advantage is not simply the XOR of Grundy numbers but the sum of the individual score‑values, adjusted for turn order.

Two contrasting regimes are identified. In the first regime, all vertices have equal treasure and the underlying graph is highly connected, so each subgame behaves like a normal‑play impartial game: the player who makes the last move (i.e., cannot move) loses the point differential, mirroring the usual “last‑move‑loses” rule. In the second regime, treasure values are highly asymmetric or the graph imposes strict movement constraints (e.g., a tree). Here, a player may deliberately avoid taking the final move because doing so would hand the opponent a large treasure, effectively turning the game into a misère‑style scenario where “last move wins” in the traditional sense but “last move loses” in terms of score. The authors illustrate this with concrete examples where a single high‑value vertex dominates the outcome, forcing optimal play to deviate from the normal impartial strategy.

The paper concludes by outlining open problems for the broader class of scoring games. Key questions include: (i) characterizing graph topologies and treasure distributions that trigger a transition between normal‑play and misère‑play behavior; (ii) determining whether polynomial‑time approximation schemes (PTAS) exist for maximizing guaranteed treasure; (iii) extending the score‑value framework to arbitrary disjunctive sums and establishing algebraic properties analogous to the Sprague‑Grundy theorem. These challenges point toward a richer theory that unifies combinatorial game analysis with optimization and algorithmic complexity.

Overall, the work contributes a novel game model, proves its NP‑Hardness via classic reductions, and provides an initial theoretical treatment of its compositional behavior, highlighting both normal and misère characteristics and setting a research agenda for scoring‑play game theory.