Equivalent game forms and coalitional power
A game form is a strategic game from which the payoffs are suppressed. It consists of a finite player set N, a finite strategy space \(S_ i\) for each player i and a map g which assigns to every n-tuple of strategies a point of an outcome space Z (or a probability measure on Z if the game form contains a non-trivial `chance player'). Introducing for each player a utility function on the outcome space Z makes the game form to a strategic game. If we identify equivalent strategies (i.e. strategies which give the same utility for every strategy (n-1)-tuple of the opponents) and delete redundant strategies (strategies which have the same payoff as a convex combination of other strategies) we find a reduced strategic game. Two game forms are called equivalent if introduction of utilities generates the `same' reduced games. Also game forms can be reduced by identifying equivalent strategies and skipping redundant strategies. Game forms are called identical if the strategy sets can be identified under a bijective map and linear dependencies between the outcome probabilities of a set of strategy n-tuples in the first game form are transformed into linear dependencies between the outcome probabilities in the second game form and vice versa. In section 2 of this paper it is proved that game forms are equivalent if and only if the reduced game forms are identical. In section 3 \(\alpha\)- and \(\beta\)-effectivity functions are associated with reduced game forms and `identity' of effectivity functions is defined. If two game forms have identical \(\alpha\)- and \(\beta\)-effectivity functions the game forms need not be equivalent but if the \(\alpha\)-effecivity function is moreover agent-additive (i.e. if \(A_ i\) is effective for \(\{\) \(i\}\), (i\(\in N)\) then \(\cap_{i\in N}A_ i\) is effective for N) then the two game forms are equivalent. Further it is proved that agent-additivity follows from superadditivity or convexity of the \(\alpha\)-effectivity function or the property of the extensive game form to have unique outcomes at all points of the game tree.
scientific article; zbMATH DE number 4202066