Constitutions, effectivity functions, and game forms

In this chapter we expound on Gärdenfors’s (1981) theory of rights-systems or constitutions. Gärdenfors formalizes rights-systems as follows. If S is a coalition (a group of individuals, members, players,...) and B is a set of social states (outcomes, alternatives,...), then B is a ‘right’ of S in the sense of Gärdenfors if S is legally entitled to the final social state being in B. The set of all pairs (S,B) where S is a coalition and B is a right of S, is a rights-system. Under very mild conditions a rights-system is a so-called effectivity function. (Effectivity functions are formally introduced in Definition 2.3.1.) Under some additional intuitive conditions, implying the requirements of monotonicity and consistency as postulated in Gärdenfors (1981), a rights-system is a monotonic and superadditive effectivity function (see Section 2.3). Gärdenfors’s definition of rights is somewhat indirect, as it is based on attainability of social states. Therefore, we first introduce Peleg’s (1998) model of a constitution (Section 2.2). This model distinguishes between rights and social states and describes explicitly how rights of groups result in attainable sets of social states. Nevertheless, for a given assignment of rights the model can be reduced to a rights-system in the sense of Gärdenfors (see Section 2.3). Section 2.2 also contains some important examples – such as the example underlying Gibbard’s Paradox (1974) – which are used throughout Part I of this book.