Towards a rigorous framework for studying 2-player continuous games

The use of 2-player strategic games is one of the most common frameworks for studying the evolution of economic and social behavior. Games are typically played between two players, each given two choices that lie at the extremes of possible behavior (e.g. completely cooperate or completely defect). Recently there has been much interest in studying the outcome of games in which players may choose a strategy from the continuous interval between extremes, requiring the set of two possible choices be replaced by a single continuous equation. This has led to confusion and even errors in the classification of the game being played. The issue is described here specifically in relation to the continuous prisoners dilemma and the continuous snowdrift game. A case study is then presented demonstrating the misclassification that can result from the extension of discrete games into continuous space. The paper ends with a call for a more rigorous and clear framework for working with continuous games.