This paper characterizes the equilibrium sets of an intrinsic common agencygame with discrete types and direct revelation mechanisms. After presentinga general algorithm to find the pure-strategy equilibria of this game, we use itto characterize these equilibria when the two principals control activitieswhich are complements in the agent's objective function. Some of thoseequilibria may entail allocative inefficiency. For the case of substitutes, wedemonstrate non-existence of such equilibria with direct mechanisms, butexistence may be obtained with indirect mechanisms. Finally, we relax theequilibrium concept and analyze quasi-equilibria. We show that existence isthen guaranteed and characterize the corresponding allocations.