# Representation of constitutions under incomplete information

We model constitutions by effectivity functions. We assume that the constitution is common knowledge among the members of the society. However, the preferences of the citizen are private information. We investigate whether there exist decision schemes (i. e., functions that map profiles of (dichotomous) preferences on the set of outcomes to lotteries on the set of social states), with the following properties: i) The distribution of power induced by the decision scheme is identical to the effectivity function under consideration; and ii) the (incomplete information) game associated with the decision scheme has a Bayesian Nash equilibrium in pure strategies. If the effectivity function is monotonic and superadditive, then we find a class of decision schemes with the foregoing properties. When applied to n-person games in strategic form, a decision scheme d is a mapping from profiles of (dichotomous) preferences on the set of pure strategy vectors to probability distributions over outcomes (or equivalently, over pure strategy vectors). We prove that for any feasible and individually rational payoff vector of a strategic game, there exists a decision scheme that yields that payoff vector as a (pure) Nash equilibrium payoff in the game induced by the strategic game and the decision scheme. This can be viewed as a kind of purification result.