Finite ß-Playable Effectivity Functions

The ß-effectivity function of a strategic game form G describes the decision power of coalitions under G as contingent on the ability of each coalition to predict the behaviour of the complementary coalition. An e¤ectivity function E is ß-playable if there exists a strategic game form G such that E is the ß-effectivity function of G. It is shown that whenever the player set and the outcome set are fi?nite an effectivity function E is ß-playable if and only if E is both outcome-monotonic and polar-superadditive. Moreover, the underlying strategic game form only needs ?small?strategy spaces, whose size is linear in the size of the monotonic co-basis of E. As a by-product of that result, a few new characterizations of tight finite e¤ectivity functions are also obtained.