Strategic Games and Truly Playable Effectivity Functions

A well known (and often used) result by Marc Pauly states that for every playable effectivity function E there exists a strategic game that assigns to coalitions exactly the same power as E, and vice versa. While the latter direction of the correspondence is correct, we show that the former does not always hold in the case of infinite game models. We point out where the proof of correspondence goes wrong, and we present examples of playable effectivity functions in infinite models for which no equivalent strategic game exists. Then, we characterize the class of truly playable effectivity func- tions, that does correspond to strategic games. Moreover, we discuss a construction that transforms any playable effec- tivity function into a truly playable one while preserving the power of most (but not all) coalitions. We also show that Coalition Logic is not expressive enough to distinguish be- tween playable and truly playable effectivity functions, and we extend it to a logic that can make this distinction while enjoying finite axiomatization and finite model property.