We introduce the notion of an antichain-convex set to extend Debreu (1954)'s version of the second welfare theorem to economies where either the aggregate production set or preference relations are not convex. We show that-possibly after some redistribution of individuals' wealth-the Pareto optima of some economies which are marked by certain types of non-convexities can be spontaneously obtained as valuation quasi-equilibria and equilibria: both equilibrium notions are to be understood in Debreu (1954)'s sense. From a purely structural point of view, the mathematical contribution of this work is the study of the conditions that guarantee the convexity of the Minkowski sum of …nitely many possibly non-convex sets. Such a study allows us to obtain a version of the MinkowskinHahn-Banach separation theorem which dispenses with the convexity of the sets to be separated and which can be naturally applied in standard proofs of the second welfare theorem; in addition-and equally importantly-the study allows to get a deeper understanding of the conditions on the single production sets of an economy that guarantee the convexity of their aggregate.

Considered are imperfectly discriminating contests in which players may possess private information about the primitives of the game, such as the contest technology, valuations of the prize, cost functions, and budget constraints. We find general conditions under which a given contest of incomplete information admits a unique pure-strategy Nash equilibrium. In particular, provided that all players have positive budgets in all states of the world, existence requires only the usual concavity and convexity assumptions. Information structures that satisfy our conditions for uniqueness include independent private valuations, correlated private values, pure common values, and examples of interdependent valuations. The results allow dealing with inactive types, asymmetric equilibria, population uncertainty, and the possibility of resale. It is also shown that any player that is active with positive probability ends up with a positive net rent. Keywords Imperfectly discriminating contests • Private information • Existence and uniqueness of equilibrium • Budget constraints • Rent dissipation JEL Classification C72 • D23 • D72 • D82 This paper was drafted while the second-named author was visiting the University of Zurich in spring 2013. Two anonymous reviewers and a Co-Editor provided extremely valuable comments. For useful discussions, we would like to thank

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