e-Valuate：一个关于算术表达式的双人游戏--更新
e-Valuate: A Two-player Game on Arithmetic Expressions -- An Update
摘要：
e-Valuate是一款关于算术表达式的游戏。参与者扮演着最大化和最小化给定表达式的不同角色。最大化者提出价值，最小化者用价值代替他选择的变量。当表达式被完全实例化时，它的值将与某个极大极小值进行比较，如果参与者使用他们的最优策略，则会产生该值。获胜者将根据这一比较结果宣布。我们使用一个对策树来表示游戏的状态，并展示了如何使用反向归纳法和α-β剪枝来有效地计算极大极小值。alpha-beta剪枝的有效性取决于评估节点的顺序。进一步的改进可以通过使用换位表来防止对相同节点的重新评估来获得。我们提出了一个启发式的节点排序。我们展示了如何使用的启发式和换位表导致改进的性能，通过比较每个方法修剪的节点数。我们描述了这个游戏的一些特定领域的变体。第一个是图论公式，其中两个玩家通过着色相关集合来共享图的一组元素，每个玩家都希望最大化他的份额。被共享的集合可以是顶点、边或面的集合（对于可平面图）。这一点的一个应用是共享由可平面图包围的区域，其中每个玩家的目标是最大化他的份额的面积。另一种变体是平铺游戏，其中玩家交替地将多米诺骨牌放置在$8 \times 8$棋盘上以构建最大部分平铺。我们表明，瓷砖$x$的大小满足$22 \le x \le 32$证明，任何最大的部分瓷砖至少需要$22$多米诺骨牌。
英文摘要：
e-Valuate is a game on arithmetic expressions. The players have contrasting roles of maximizing and minimizing the given expression. The maximizer proposes values and the minimizer substitutes them for variables of his choice. When the expression is fully instantiated, its value is compared with a certain minimax value that would result if the players played to their optimal strategies. The winner is declared based on this comparison. We use a game tree to represent the state of the game and show how the minimax value can be computed efficiently using backward induction and alpha-beta pruning. The efficacy of alpha-beta pruning depends on the order in which the nodes are evaluated. Further improvements can be obtained by using transposition tables to prevent reevaluation of the same nodes. We propose a heuristic for node ordering. We show how the use of the heuristic and transposition tables lead to improved performance by comparing the number of nodes pruned by each method. We describe some domain-specific variants of this game. The first is a graph theoretic formulation wherein two players share a set of elements of a graph by coloring a related set with each player looking to maximize his share. The set being shared could be either the set of vertices, edges or faces (for a planar graph). An application of this is the sharing of regions enclosed by a planar graph where each player's aim is to maximize the area of his share. Another variant is a tiling game where the players alternately place dominoes on a $8 \times 8$ checkerboard to construct a maximal partial tiling. We show that the size of the tiling $x$ satisfies $22 \le x \le 32$ by proving that any maximal partial tiling requires at least $22$ dominoes.
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