Motivated by the results for Magic: The Gathering presented in [CBH20] and [Bid20], we study a computability problem related to optimal play in Yu-Gi-Oh! Trading Card Game, a popular card game developed and published by Konami. We show that the problem of establishing whether, from a given game state, a computable strategy is winning is undecidable. In particular, not only do we prove that the Halting Problem can be reduced to this problem, but also that the same holds for the Kleene's $\mathcal{O}$, thereby demonstrating that this problem is actually $Π^1_1$-complete. We extend this last result to all strategies with a reduction on the set of countable well orders, a classic $\boldsymbolΠ^1_1$-complete set. For these reductions we present two legal decks (according to the current Forbidden & Limited List of Yu-Gi-Oh! Trading Card Game) that can be used by the player who goes first to perform them.

翻译：受[CBH20]与[Bid20]中《万智牌》相关研究结果的启发，我们针对科乐美开发并发行的热门卡牌游戏《游戏王》集换式卡牌游戏，研究了与最优策略求解相关的可计算性问题。我们证明了从给定游戏状态出发，判定可计算策略是否必胜的问题是不可判定的。特别地，我们不仅证明了停机问题可归约至该问题，还证明了克林$\mathcal{O}$同样可归约至该问题，从而表明该问题实际上是$Π^1_1$-完全的。我们通过将可数良序集（经典的$\boldsymbolΠ^1_1$-完全集）归约至该问题，将最终结论推广至所有策略。针对这些归约，我们提出了两个符合当前《游戏王》集换式卡牌游戏禁限卡表规定的合法卡组，可供先攻玩家用于执行归约过程。