Quality Diversity (QD) is a branch of evolutionary computation that seeks high-quality and behaviorally diverse solutions to a problem. While adversarial problems are common, classical QD cannot be easily applied to them, as both fitness and behavior depend on the opposing solutions. Recently, Generational Adversarial MAP-Elites (GAME) has been proposed to coevolve both sides of an adversarial problem by alternating the execution of a multi-task QD algorithm against previous elites, called tasks. The original algorithm selects new tasks based on a behavioral criterion, which may lead to undesired dynamics due to inter-side dependencies. In addition, comparing sets of solutions cannot be done directly using classical QD metrics due to inter-side dependencies. In this paper, we propose (1) 6 metrics of adversarial quality and diversity based on an inter-variants tournament to compare the sets of solutions, ensuring a fair comparison, and (2) propose two tournament-informed task selection methods to promote higher quality and diversity at each generation. We evaluate the variants across three adversarial problems: Pong, a Cat-and-mouse game, and a Pursuers-and-evaders game. We show that the tournament-informed task selection method leads to higher adversarial quality and diversity. We hope that this work will help further advance adversarial QD.