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Some basic facts of public goods theory are presented in the primitive set-up of a collection of projects devoid of any linear structure. There is a single private good. Characterizations of Pareto optimal and core states in terms of valuation functions (i.e. supporting "prices") are obtained. Voluntary financing schemes are discussed.

[A price equilibrium existence theorem is proved for exchange economies whose consumption sets are the positive orthant of arbitrary topological vector lattices. The motivation comes from economic applications showing the need to bring within the scope of equilibrium theory commodity spaces whose positive orthant has empty interior, a typical situation in infinite dimensional linear spaces.]

Andreu Mas-Colell, Hugo Sonnenschein; General Possibility Theorems for Group Decisions12, The Review of Economic Studies, Volume 39, Issue 2, 1 April 1972,

[The fact that preference maximizing consumers generate aggregate excess demand is utilized to prove (i) a statement on the values of the excess demand correspondence and (ii) that the economies having an excess demand function are dense in the set of all economies. This is applied to get a straightforward proof for the existence of an equilibrium distribution.]

In this paper we provide a statement of the relationship between the weak axiom of revealed preference (WA) and the negative semidefiniteness of the matrix of substitution terms (NSD). As a corollary we determine the relation between WA and the strong axiom of revealed preference (SA). The latter is equivalent to NSD and the symmetry of the matrix of substitution terms. The former, WA, implies NSD but is not implied by NSD. Also, WA is implied by the condition that the matrix of substitution terms is negative definite (ND), but it does not imply ND. Application of these results yield an infinity of demand functions which satisfy WA but not SA.

We study a general implementation problem for exchange economies with a continuum of players and private information, and test the robustness of the results for sequences of approximating finite economies. Assuming that the designer knows the distribution of the characteristics in the economy we consider continuous and unique implementation in both its equilibrium and dominant strategies versions and obtain results for general self-selective (first- and second-best) allocations. An upper hemicontinuity property of Bayesian equilibria of approximating economies for continuous mechanisms is demonstrated. Using this, we can, for example, conclude that if a given continuous mechanism implements uniquely a Walrasian allocation in the continuum economy then all Bayesian equilibria of large approximation economies will (with probability close to one) yield an allocation which is almost ex-post Pareto optimal.

We propose a simple adaptive procedure for playing a game. In this procedure, players depart from their current play with probabilities that are proportional to measures of regret for not having used other strategies (these measures are updated every period). It is shown that our adaptive procedure guaranties that with probability one, the sample distributions of play converge to the set of correlated equilibria of the game. To compute these regret measures, a player needs to know his payoff function and the history of play. We also offer a variation where every player knows only his own realized payoff history (but not his payoff function).