Antonio Penta

Interactive Epistemology and Solution Concepts for Games with Asymmetric Information

We use an interactive epistemology framework to provide a systematic analysis of some solution concepts for games with asymmetric information. We characterize solution concepts using expressible epistemic assumptions, represented as events in the canonical space generated by primitive uncertainty about the payoff relevant state, payoff irrelevant information, and actions. In most of the paper, we adopt an interim perspective, which is appropriate to analyze genuine incomplete information. We relate Delta-rationalizability (Battigalli and Siniscalchi, Advances in Theoretical Economics 3, 2003) to interim correlated rationalizability (Dekel, Fudenberg and Morris, Theoretical Economics 2, 2007) and to rationalizability in the interim strategic form. We also consider the ex ante perspective, which is appropriate to analyze asymmetric information about an initial chance move. We prove the equivalence between interim correlated rationalizability and an ex ante notion of correlated rationalizability.

On the structure of rationalizability for arbitrary spaces of uncertainty

Weinstein and Yildiz (Econometrica, 2007) have shown that only very weak predictions are robust to mispecifications of higher order beliefs. Whenever a type has multiple rationalizable actions, any of these actions is uniquely rationalizable for some arbitrarily close type. Hence, refinements of rationalizability are not robust. This negative result is obtained under a richness condition, which essentially means that all common knowledge assumptions on payoffs are relaxed. In many settings this condition entails an unnecessarily demanding robustness test. It is therefore natural to explore the structure of rationalizability when arbitrary common knowledge assumptions are relaxed (i.e., without assuming richness). For arbitrary spaces of uncertainty, and for every player i, I construct a set A_{i}^{∞} of actions that are uniquely rationalizable for some hierarchy of beliefs. The main result shows that for any type t_{i}, and any action a_{i} rationalizable for t_{i}, if a_{i} belongs to A_{i}^{∞} and is justified by conjectures concentrated on A_{-i}^{∞}, then there exists a sequence of types converging to t_{i} for which a_{i} is uniquely rationalizable. This result significantly generalizes Weinstein and Yildizs. Some of its implications are discussed in the context of auctions, equilibrium refinements and in connection with the literature on global games.

Robust Dynamic Implementation

This paper extends the belief-free approach to robust mechanism design to dynamic environments, in which agents obtain information over time. A social choice function (SCF) is robustly partially implemented if it is perfect Bayesian incentive compatible for all possible beliefs. It is shown that this is possible if and only if the SCF is ex-post incentive compatible. Robust full implementation imposes the stronger condition that, for all possible beliefs, all Perfect Bayesian Equilibria induce outcomes consistent with the SCF. Characterizing the set of such equilibria is a key difficulty in studying this problem. This paper shows that, for a weaker notion of equilibrium, the set of all such equilibria can be computed by means of a recursive procedure which combines the logic of rationalizability and backward induction reasoning. These results are then used to show that, in environments with single-crossing preferences and well-behaved intertemporal effects, strict ex-post incentive compatibility and a condition which limits the strength of preference interdependencies are sufficient to guarantee robust full implementation.

Common Agency and Coordinated Bids in Sponsored Search Auctions

As auctions are becoming the main mechanism for selling advertisement space on the web, marketing agencies specialized in bidding in online auctions are proliferating. We analyze theoretically how bidding delegation to a common marketing agency can undermine both revenues and efficiency of the generalized second price auction, the format used by Google and Microsoft-Yahoo!. Our characterization allows us to quantify the revenue losses relative to both the case of full competition and the case of agency bidding under an alternative auction format (specifically, the VCG mechanism). We propose a simple algorithm that a search engine can use to reduce efficiency and revenue losses.

On the Structure of Rationalizability for Arbitrary Spaces of Uncertainty, Second Version

This note characterizes the set Ai∞ of actions of player i that are uniquely rationalizable for some hierarchy of beliefs on an arbitrary space of uncertainty. It is proved that for any rationalizable action ai for the type ti, if ai belongs to Ai∞ and is justified by conjectures concentrated on A-i∞, then there exists a sequence of types converging to ti for which ai is uniquely rationalizable.

Robustness of the Uniqueness of Walrasian Equilibrium with Cobb-Douglas Utilities

The majority of results in the literature on general equilibrium are not for an economy (i.e. given an endowment and preferences), but rather, for a set of economies (i.e. a set of endowments given preferences). Therefore, we argue that the most appropriate robustness result requires perturbing economies uniformly over the space of endowments for which the result is obtained. In this paper, we examine the robustness of the uniqueness of Walrasian endowment economies with Cobb-Douglas utility functions under this interpretation of robustness. Namely, we prove that for economies described by Cobb-Douglas utilities and all endowments in a fixed set, uniqueness of equilibrium is robust to perturbations of the utility functions.