Affiliation, equilibrium existence and the revenue ranking of auctions

We consider private value auctions where bidders’ types are dependent, a case usually treated by assuming affiliation. We show that affiliation is a restrictive assumption in three senses: topological, measure-theoretic and statistical (affiliation is a very restrictive characterization of positive dependence). We also show that affiliation’s main implications do not generalize for alternative definitions of positive dependence. From this, we propose new approaches to the problems of pure strategy equilibrium existence in first-price auctions (PSEE) and the characterization of the revenue ranking of auctions. For equilibrium existence, we slightly restrict the set of distributions considered, without loss of economic generality, and offer a complete characterization of PSEE. For revenue ranking, we obtain a characterization of the expected revenue differences between second and first price auctions with general dependence of types

Corrigendum to "existence of equilibrium in single and double private value actions"

This is a corrigendum to Theorem 15 of Jackson and Swinkels (2005) [Existence of Equilibrium in Single and Double Private Value Auctions, Econometrica, 73, 93-140], which proves the existence of equilibrium with positive probability of trade for private value auctions

CORRIGENDUM TO "EXISTENCE OF EQUILIBRIUM IN SINGLE AND DOUBLE PRIVATE VALUE ACTIONS"

This is a corrigendum to Theorem 15 of Jackson and Swinkels (2005) [Existence of Equilibrium in Single and Double Private Value Auctions, Econometrica, 73, 93-140], which proves the existence of equilibrium with positive probability of trade for private value auctions.

Affiliation, equilibrium existence and the revenue ranking of auctions

We consider private value auctions where bidders’ types are dependent, a case usually treated by assuming affiliation. We show that affiliation is a restrictive assumption in three senses: topological, measure-theoretic and statistical (affiliation is a very restrictive characterization of positive dependence). We also show that affiliation’s main implications do not generalize for alternative definitions of positive dependence. From this, we propose new approaches to the problems of pure strategy equilibrium existence in first-price auctions (PSEE) and the characterization of the revenue ranking of auctions. For equilibrium existence, we slightly restrict the set of distributions considered, without loss of economic generality, and offer a complete characterization of PSEE. For revenue ranking, we obtain a characterization of the expected revenue differences between second and first price auctions with general dependence of types.

Equilibria Existence in Regular Discontinuous Games

Many conditions have been introduced to weaken the continuity re- quirements for equilibrium existence in games. We introduce a new con- dition, called regularity, that is simple and easy to verify. It is implied both by Reny's better-reply security and Simon and Zame's endogenous sharing rule method. Regularity implies that the limits of epsilon-equilibria are equilibria. Since this condition is weak, it is yet not enough to ensure pure strategy equilibrium existence, but we are able to identify extra conditions that, together with regularity, are sufficient for equilibrium existence. One is the marginal continuity property introduced by Prokopovych (2008), while the second is the well behavior of a sequence of approximating con- tinuous functions. In this way, we provide new equilibrium existence re- sults for discontinuous games under conditions that are simpler and easier to check than most of the available alternatives.

Pure strategy equilibria of single and double auctions with interdependent values

We prove the existence of monotonic pure strategy equilibrium for many types 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 a reasonable counterexample for interdependent values auctions that shows that sometimes all equilibria are trivial, that is, they have zero probability of trade. Nevertheless, we give sufficient conditions for non-trivial equilibrium existence

PURE STRATEGY EQUILIBRIA OF SINGLE AND DOUBLE AUCTIONS WITH INTERDEPENDENT VALUES

We prove the existence of monotonic pure strategy equilibrium for many types 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 a reasonable counterexample for interdependent values auctions that shows that sometimes all equilibria are trivial, that is, they have zero probability of trade. Nevertheless, we give sufficient conditions for non-trivial equilibrium existence.

Ambiguity Aversion and Trade

What is the effect of ambiguity aversion on trade? Although in a Bewley's model ambiguity aversion always lead 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. JEL Classification Numbers: D51, D6, D8

Characterization of Bidding Behavior in Multi-Unit

This paper provides first order conditions for a very general class of single or multi-unit auctions. Under mild conditions we characterize bidding-behavior and generalize previous standard results in the literature. As an application we obtain sufficient conditions for truth-telling, monotonic best reply strategies and identification results for multi-unit auctions.Multi-unit auctions, first order conditions, truth-telling, identification, monotonic bestreply.

Non-monotoniticies and the all-pay auction tie-breaking rule

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