Solving combinatorial games has been a classic research topic in artificial intelligence because solutions can offer essential information to improve gameplay. Several definitions exist for `solving the game,' but they are markedly different regarding computational cost and the detail of insights derived. In this study, we introduce a novel definition called `semi-strongly solved' and propose an algorithm to achieve this type of solution efficiently. This new definition addresses existing gaps because of its intermediate computational cost and the quality of the solution. To demonstrate the potential of our approach, we derive the theoretical computational complexity of our algorithm under a simple condition, and apply it to semi-strongly solve the game of 6x6 Othello. This study raises many new research goals in this research area.

The concept of nimbers--a.k.a. Grundy-values or nim-values--is fundamental to combinatorial game theory. Nimbers provide a complete characterization of strategic interactions among impartial games in their disjunctive sums as well as the winnability. In this paper, we initiate a study of nimber-preserving reductions among impartial games. These reductions enhance the winnability-preserving reductions in traditional computational characterizations of combinatorial games. We prove that Generalized Geography is complete for the natural class, $\cal{I}^P$ , of polynomially-short impartial rulesets under nimber-preserving reductions, a property we refer to as Sprague-Grundy-complete. In contrast, we also show that not every PSPACE-complete ruleset in $\cal{I}^P$ is Sprague-Grundy-complete for $\cal{I}^P$ . By considering every impartial game as an encoding of its nimber, our technical result establishes the following striking cryptography-inspired homomorphic theorem: Despite the PSPACE-completeness of nimber computation for $\cal{I}^P$ , there exists a polynomial-time algorithm to construct, for any pair of games $G_1$, $G_2$ of $\cal{I}^P$ , a prime game (i.e. a game that cannot be written as a sum) $H$ of $\cal{I}^P$ , satisfying: nimber($H$) = nimber($G_1$) $\oplus$ nimber($G_2$).

Recently, a standardized framework was proposed for introducing quantum-inspired moves in mathematical games with perfect information and no chance. The beauty of quantum games-succinct in representation, rich in structures, explosive in complexity, dazzling for visualization, and sophisticated for strategic reasoning-has drawn us to play concrete games full of subtleties and to characterize abstract properties pertinent to complexity consequence. Going beyond individual games, we explore the tractability of quantum combinatorial games as whole, and address fundamental questions including: Quantum Leap in Complexity: Are there polynomial-time solvable games whose quantum extensions are intractable? Quantum Collapses in Complexity: Are there PSPACE-complete games whose quantum extensions fall to the lower levels of the polynomial-time hierarchy? Quantumness Matters: How do outcome classes and strategies change under quantum moves? Under what conditions doesn't quantumness matter? PSPACE Barrier for Quantum Leap: Can quantum moves launch PSPACE games into outer polynomial space We show that quantum moves not only enrich the game structure, but also impact their computational complexity. In settling some of these basic questions, we characterize both the powers and limitations of quantum moves as well as the superposition of game configurations that they create. Our constructive proofs-both on the leap of complexity in concrete Quantum Nim and Quantum Undirected Geography and on the continuous collapses, in the quantum setting, of complexity in abstract PSPACE-complete games to each level of the polynomial-time hierarchy-illustrate the striking computational landscape over quantum games and highlight surprising turns with unexpected quantum impact. Our studies also enable us to identify several elegant open questions fundamental to quantum combinatorial game theory (QCGT).

Multiplayer Online Battle Arena (MOBA) games have received increasing popularity recently. In a match of such games, players compete in two teams of five, each controlling an in-game avatars, known as heroes, selected from a roster of more than 100. The selection of heroes, also known as pick or draft, takes place before the match starts and alternates between the two teams until each player has selected one hero. Heroes are designed with different strengths and weaknesses to promote team cooperation in a game. Intuitively, heroes in a strong team should complement each other's strengths and suppressing those of opponents. Hero drafting is therefore a challenging problem due to the complex hero-to-hero relationships to consider. In this paper, we propose a novel hero recommendation system that suggests heroes to add to an existing team while maximizing the team's prospect for victory. To that end, we model the drafting between two teams as a combinatorial game and use Monte Carlo Tree Search (MCTS) for estimating the values of hero combinations. Our empirical evaluation shows that hero teams drafted by our recommendation algorithm have significantly higher win rate against teams constructed by other baseline and state-of-the-art strategies.

Temperature Discovery Search (TDS) is a forward search method for computing or approximating the temperature of a combinatorial game. Temperature and mean are important concepts in combinatorial game theory, which can be used to develop efficient algorithms for playing well in a sum of subgames. A new algorithm TDS+ with five enhancements of TDS is developed, which greatly speeds up both exact and approximate versions of TDS. Means and temperatures can be computed faster, and fixed-time approximations which are important for practical play can be computed with higher accuracy than before.

Dots-And-Boxes is a well-known and widely-played combinatorial game. While the rules of play are very simple, the state space for even small games is extremely large, and finding the outcome under optimal play is correspondingly hard. In this paper we introduce a Dots-And-Boxes solver which is significantly faster than the current state-of-the-art: over an order-of-magnitude faster on several large problems. We describe our approach, which uses Alpha-Beta search and applies a number of techniques—both problem-specific and general—to reduce the number of duplicate states explored and reduce the search space to a manageable size. Using these techniques, we have determined for the first time that Dots- And-Boxes on a board of 4x5 boxes is a tie given optimal play. This is the largest game solved to date.