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Artificial intelligence and robotic competitions are accompanied by a classof game paradigms in which each player privately commits a strategy to a gamesystem which simulates the game using the collected joint strategy and thenreturns payoffs to players. This paper considers the strategy commitment fortwo-player symmetric games in which the players' strategy spaces are identicaland their payoffs are symmetric. First, we introduce two digraph-based metricsat a meta-level for strategy evaluation in two-agent reinforcement learning,grounded on sink equilibrium. The metrics rank the strategies of a singleplayer and determine the set of strategies which are preferred for the privatecommitment. Then, in order to find the preferred strategies under the metrics,we propose two variants of the classical learning algorithm self-play, calledstrictly best-response and weakly better-response self-plays. By modelinglearning processes as walks over joint-strategy response digraphs, we provethat the learnt strategies by two variants are preferred under two metrics,respectively. The preferred strategies under both two metrics are identifiedand adjacency matrices induced by one metric and one variant are connected.Finally, simulations are provided to illustrate the results.

Evaluation and Learning in Two-Player Symmetric Games via Best and Better Responses