A comment on "Towards a rigorous framework for studying 2-player continuous games" by Shade T. Shutters, Journal of Theoretical Biology 321, 40-43, 2013.

Continuous evolutionary games are generalisations of games with a finite or discrete set of strategies to games with infinitely many, continuously varying strategies (e.g., Mar and St. Denis, 1994; Doebeli and Knowlton, 1998; Day and Taylor, 1998; Killingback et al., 1999; Wahl and Nowak, 1999; Ahmed and Elgazaar, 2000; Killingback and Doebeli, 2002; Doebeli et al., 2004; Cressman and Hofbauer, 2005; André and Day, 2007; McGill and Brown, 2007; Killingback et al., 2010). A recent paper, Shutters (2013), claims that earlier treatments of continuous games were “confusing” and based on “misconceptions” that led to “misclassifications”. These claims seem to be based on a misunderstanding of earlier work and, in particular, of one of our papers (Doebeli et al., 2004, which was identified as DHK in Shutters (2013)). In continuous games, the strategy of an individual is given by the cooperative investment x, a quantitative trait that can vary continuously in an interval of positive real numbers. In DHK, we further assumed that continuous games are defined by two functions, the benefit function B and the cost function C. For example, in analogy with widely used parameterizations of the Prisoner's Dilemma game in terms of costs and benefits, one can assume that the payoff of an individuals with trait x playing against an individual with trait y is given by B(y)−C(x). We termed this continuous analogue the “Continuous Prisoner’s Dilemma game”. Similarly, benefits could accrue from investments of both interacting players, such that the payoff to the x-individual in an (x−y)-interaction is P(x, y)1⁄4B(x+y)−C(x). In analogy to parameterizations of the conventional Snowdrift game, we gave the resulting continuous game the name “Continuous Snowdrift game”. Clearly, for two distinct trait values x and y, the Continuous Snowdrift game turns into a standard 2 2-game with a payoff matrix whose entries are determined by the cost and benefit functions, evaluated at x, y, and x+y, respectively. The nature of this standard 2 2-game depends on the parameter values defining the cost and benefit functions, as well as on the traits x and y. Importantly, for the Continuous Snowdrift game the resulting standard 2 2-game can be of any type and, in particular, does not necessarily result in a standard Snowdrift game. In DHK this is explained in detail in Fig. 2, where the panels are explicitly designed to illustrate the standard games that can result from substituting given values for x and y into the Continuous Snowdrift game. More specifically, the figure legend explicitly states that “Local games between strategies in the vicinity of the singular point can be of any type.” In fact, this dependence of the standard 2 2-game resulting from substituting particular values of x and y into the Continuous