e-Valuate: A Two-player Game on Arithmetic Expressions -- An Update
This work addresses optimization in game theory for researchers, but it is incremental as it builds on existing methods like alpha-beta pruning with minor enhancements.
The paper tackles the problem of efficiently computing minimax values in a two-player arithmetic expression game by proposing a heuristic for node ordering and using transposition tables, resulting in improved performance as measured by the number of nodes pruned. It also explores domain-specific variants, such as a graph-sharing game and a tiling game, proving bounds like at least 22 dominoes in maximal partial tilings.
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.