This paper proposes a new general class of strategic games and develops an associated new existence result for pure-strategy Nash equilibrium. For a two-player game with scalar and compact action sets, existence entails that one reaction curve be increasing and continuous and the other quasi-increasing (i.e., not have any downward jumps). The latter property amounts to strategic quasi-complementarities. The paper provides a number of ancillary results of independent interest, including sufficient conditions for a quasi-increasing argmax (or non-monotone comparative statics), and new sufficient conditions for uniqueness of fixed points. For maximal accessibility of the results, the main results are presented in a Euclidean setting. We argue that all these results have broad and elementary applicability by providing simple illustrations
with commonly used models in economic dynamics and industrial organization.

This paper reconsiders the well-known comparison of equilibrium entry levels into a Cournot industry under free entry, second best (control of entry but not production) and first best (control of entry and production). Allowing for the possibility of limited increasing returns to scale in production, this paper generalizes the conclusion of Mankiw and Whinston (1986) [10], that under business-stealing competition, free entry yields more firms than the second-best solution. We also show that under-entry always holds under business-enhancing competition. This confirms the general intuition given by Mankiw and Whinston, which does not rely on the convexity of the cost function. The same result is shown to extend (at a similar level of generality) to the comparison between free entry and the first best socially optimal solution, irrespective of business-stealing. Three illustrative examples are provided, one showing that the second-best and free entry solutions may actually coincide.

A preference is invariant with respect to a set of transformations if the ranking of acts is unaffected by reshuffling the states under these transformations. For example, transformations may correspond to the set of finite permutations, or the shift in a dynamic choice model. Our main result is that any invariant preference must be parametric: there is a unique sufficient set of parameters such that the preference ranks acts according to their expected utility given the parameters. Parameters are characterized in terms of objective frequencies, and can thus be interpreted as objective probabilities. By contrast, uncertainty about parameters is subjective. The preferences for which the above results hold are only required to be reflexive, transitive, monotone, continuous, and mixture linear.

We prove the existence of monotonic pure strategy equilibrium for many kinds of asymmetric auctions with n bidders and unitary demands, interdependent values and independent types. The assumptions require monotonicity only in the own bidder’s type. The payments can be a function of all bids. Thus, we provide a new equilibrium existence result for asymmetrical double auctions and a small number of bidders. The generality of our setting requires the use of special tie-breaking rules. We present an example of a double auction with interdependent values where all equilibria are trivial, that is, they have zero probability of trade. This is related to Akerlof’s “market for lemmons” example and to the “winner’s curse,” establishing a connection between them. However, we are able to provide sufficient conditions for non-trivial equilibrium existence.

We introduce the idea of implementation under ambiguity. In particular, we study maximin efficient notions for an ambiguous asymmetric information economy (i.e., economies where agents’ preferences are maximin à la Wald, 1950). The interest on the maximin preferences lies in the fact that maximin efficient allocations are always incentive compatible (de Castro and Yannelis, 2009), a result which is false with Bayesian preferences. A noncooperative notion called maximin equilibrium is introduced which provides a noncooperative foundation for individually rational and maximin efficient notions. Specifically, we show that given any arbitrary individually rational and ex-ante maximin efficient allocation, there is a direct revelation mechanism that yields the efficient allocation as its unique maximin equilibrium outcome.  Thus, an incentive compatible, individually rational and efficient outcome can be reached by means of noncooperative behavior under ambiguity.

What is the effect of ambiguity aversion on trade? Although in a Bewley’s model, ambiguity aversion always leads to less trade; in other models, this is not always true. However, we show that if the endowments are unambiguous, then more ambiguity aversion implies less trade for a very general class of preferences. The reduction in trade caused by ambiguity aversion can be as severe as to lead to no trade. In an economy with MEU decision makers, we show that if the aggregate endowment is unanimously unambiguous, then every Pareto optima allocation is also unambiguous. We also characterize the situation in which every unanimously unambiguous allocation is Pareto optimal. Finally, we show how our results can be used to explain the home-bias effect. As a useful result for our methods, we also obtain an additivity theorem for CEU and MEU decision makers that does not require comonotonicity.

Many conditions have been introduced to ensure equilibrium existence in games with discontinuous payoff functions. This paper introduces a new condition, called regularity, that is simple and easy to verify. Regularity requires that if there is a sequence of strategies converging to s∗ such that the players’ payoffs along the sequence converge to the best-reply payoffs at s∗, then s∗ is an equilibrium. We show that regularity is implied both by Reny’s better-reply security and Simon and Zame’s endogenous sharing rule approach. This allows us to explore a link  between these two distinct methods. Although regularity implies that the limits of epsilon-equilibria are equilibria, it is in general too weak for implying equilibrium existence. However, we are able to identify extra conditions that, together with regularity, are sufficient for equilibrium existence. In particular, we show how regularity allows the technique of approximating games both by payoff functions and space of strategies.

Discontinuous games, such as auctions, may require special tie-breaking rules to guarantee equilibrium existence. The best results available ensure equilibrium existence only in mixed strategy with endogenously defined tie-breaking rules and communication of private information. We show that an all-pay auction tie-breaking rule is sufficient for the existence of pure strategy equilibrium in a class of auctions. The rule is explicitly defined and does not require communication of private information. We also characterize when special tie-breaking rules are really  needed.

This note points out a mistake in the proof of one of the main results of Jackson and Swinkels (2005) and provides a correct proof.