A contract with multiple agents may be susceptible to collusion. We show that agents' collusion imposes no cost in a large class of circumstances with risk neutral agents, including both uncorrelated and correlated types. In those circumstances, any payoff the principal can attain in the absence of collusion, including the second best level, can be attained in the presence of collusion in a way robust to many aspects of collusion behavior. The collusion-proof implementation generalizes to a setting in which only a subset of agents may collude, provided that noncollusive agents' incentives can be protected via an ex post incentive compatible and ex post individually rational mechanism. Our collusion-proof implementation also sheds light on the extent to which hierarchical delegation of contracts can optimally respond to collusion.

Peter DeMarzo, Ilan Kremer, and Andrzej Skrzypacz (2005) analyzed auctions in which bidders compete in securities. They show that a steeper security leads to a higher expected revenue for the seller, and also use this to establish the revenue ranking between standard auctions. In this comment, we obtain the opposite results to DKS's by assuming that a higher return requires a higher investment cost. Given this latter assumption, steeper securities are more vulnerable to adverse selection, and may yield lower expected revenue, than flatter ones.

We develop a network-flow approach for characterizing interim-allocation rules that can be implemented by ex post allocations. Our method can be used to characterize feasible interim allocations in general multi-unit auctions where agents face capacity constraints, both ceilings and floors. Applications include a variety of settings of practical interest, ranging from individual and group-specific capacity constraints, set-aside sale, partnership dissolution, and government license reallocation.

We study stability of two-sided many-to-one matching in which firms' preferences for workers may exhibit complementarities. Although such preferences are known to jeopardize stability in a finite market, we show that a stable matching exists in a large market with a continuum of workers, provided that each firm's choice is convex and changes continuously as the set of available workers changes. We also study the existence and structure of stable matchings under preferences exhibiting substitutability and indifferences in a large market. Building on these results, we show that an approximately stable matching exists in large finite economies. We extend our framework to ensure a stable matching with desirable incentive and fairness properties in the presence of indifferences in firms' preferences.

We give two characterizations of Pareto optimality via “near” weighted utilitarian welfare maximization. One characterization sequentially maximizes utilitarian welfare functions using a finite sequence of nonnegative and eventually positive welfare weights. The other maximizes a utilitarian welfare function with a certain class of positive hyperreal weights. The social welfare ordering represented by these “near” weighted utilitarian welfare criteria is characterized by the standard axioms for weighted utilitarianism under a suitable weakening of the continuity axiom.