Equivalent game forms and coalitional power
type: Item
instance of: scholarly article
title: Equivalent game forms and coalitional power
zbMATH Open document ID: 0727.90101
DOI: 10.1016/0165-4896(90)90004-Q
published in: Mathematical Social Sciences
publication date: 1990-01-01T00:00:00Z
review text: 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 pr…
Mathematics Subject Classification ID: 91A12
zbMATH DE Number: 4202066
zbMATH Keywords: game form
zbMATH Keywords: outcome space
zbMATH Keywords: reduced strategic game
zbMATH Keywords: effectivity functions
zbMATH Keywords: agent-additivity
zbMATH Keywords: superadditivity
zbMATH Keywords: convexity
author: V. W. Kolpin
reviewed by: Jos Potters
MaRDI profile type: Publication
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full work available at URL: 0165-4896(90)90004-q
OpenAlex ID: W1975250831
description: scientific article; zbMATH DE number 4202066