Welfare economy on belief in data mining - a rough set theoretical approach

We investigate a pure exchange economy under uncertainty with emphasis on the logical point of view from data base theory; the traders are assumed to have a multi-modal logic of belief and to make their decision under uncertainty represented by rough sets. We propose a generalized notion of expectations equilibrium for the economy, and we show the fundamental welfare theorem: An allocation in the economy is ex-ante Pareto optimal if and only if it is an expectations equilibrium allocation in belief for some initial endowment with respect to some price system.