Timothy Van Zandt

- Information Overload in a Network of Targeted Communication
- Robustness of Adaptive Expectations as an Equilibrium Selection Device
- Hidden Information Acquisition and Static Choice

- Monotone Equilibria in Bayesian Games of Strategic Complementarities
- Interim Bayesian Nash Equilibrium on Universal Type Spaces for Supermodular Games

- A Theorem of the Maximin and Applications to Bayesian Zero-Sum Games
- Berge's Maximum Theorem with Two Topologies on the Space of Actions
- The Hausdorff Metric of Sigma-Fields and the Value of Information
- Information, Measurability and Continuous Behavior

**Published:**
2004, *Rand Journal of Economics*, **35**:542-560.

**Abstract:**
As the costs of generating and transmitting information fall, the
main bottlenecks in communication networks are becoming the human
receivers, who are overloaded with information. For networks of
targeted communication, this paper discusses the meaning of
information overload, provides a theoretical treatment of its
causes as the outcome of strategic interaction between senders, and
examines mechanisms for allocating the attention of receivers.
Mechanisms for allocating attention include surcharges on
communication and auctions. These mechanisms increase the cost of
sending messages and shift the task of screening messages from the
receivers to the senders. This shift may benefit both the receivers
and the senders because the senders know the contents of the
messages whereas the receivers do not. We show that, if the
communication cost is low, then an increase in the communication
cost benefits most (but not all) receivers. More surprisingly, the
increase benefits all the senders if either the extra cost is a tax
that is redistributed to them as lump-sum transfers or their
information about the receivers is sufficiently accurate.

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**Supplementary notes:**
These notes contain a few generalizations and extensions.

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**Coauthor:**
Martin Lettau, Federal Reserve Bank of New York

**Published:**
2003, *Macroeconomic Dynamics*, **7**:89-118.

**Abstract:**
Dynamic models in which agents' behavior depends on expectations of future prices or other endogenous variables can have steady states that are stationary equilibria for a wide variety of expectations rules, including rational expectations. When there are multiple steady states, stability is a criterion for selecting among them as predictions of long-run outcomes. The purpose of this paper is to study how sensitive stability is to certain details of the expectations rules, in a simple OLG model with constant government debt that is financed through seigniorage. We compare simple recursive learning rules, learning rules with vanishing gain, and OLS learning, and also relate these to expectational stability. One finding is that two adaptive expectation rules that differ only in whether they use current information can have opposite stability properties.

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**Supplementary notes:**
These are extensive supplementary notes that complement this paper:

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**Published:**
1996, *Theory and Decision*, **40**:235-247.

**Abstract:**
This note explores the consequence of hidden information
acquisition for static choice theory. We show that any choice
function in the observable problem can be consistent with some
well-behaved choice function in a metaproblem with unobservable
costly information acquisition. This illustrates how choices may
not satisfy consistency conditions because a decision maker's
decision process (in this case, information acquisition) depends
on her feasible set. It also illustrates the importance of modeling
the source of violations of consistency conditions, rather than
simply weakening axioms on preferences.

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**Coauthor:** Xavier Vives.

**Published:**
2007, *Journal of Economic Theory*, 134:339–360.

**Abstract:**
For Bayesian games of strategic complementarities, we provide a constructive
proof of the existence of a greatest and a least Bayesian Nash equilibrium, each one in strategies that are monotone in type. Our main assumptions, besides strategic complementarities, are that each player's payoff displays
increasing differences in own action and the profile of types and that each player's interim beliefs are increasing in type with respect to first-order stochastic dominance (e.g., types are affiliated). The result holds for general action and type
spaces (single-, multi-, or infinite-dimensional; continuous or discrete) and no prior is assumed. We also provide the following comparative statics result: the greatest and least equilibria are higher if there is a first-order stochastic dominant shift in the interim beliefs. We apply this result to strategic information revelation in games of voluntary
disclosure.

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**Supplementary notes:**
These are minor supplementary notes that complement the paper:

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**Published:**
2010, *Journal of Economic Theory*, 145:249–263.

**Abstract:**
We prove the existence of a greatest and a least interim Bayesian Nash equilibrium for supermodular games of incomplete information. There are two main differences from the earlier proofs in Vives (1990) and Milgrom and Roberts (1990): (a) we use the interim formulation of a Bayesian game, in which each player's beliefs are part of his or her type rather than being derived from a prior; (b) we use the interim formulation of a Bayesian Nash equilibrium, in which each player and every type (rather than almost every type) chooses a best response to the strategy profile of the other players. Given also the mild restrictions on the type spaces, we have a proof of interim Bayesian Nash equilibrium for universal type spaces (for the class of supermodular utilities), as constructed, for example, by Mertens and Zamir (1985). We also weaken restrictions on the set of actions.

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**Coauthor:**
Kaifu Zhang.

**Date:**
3 February 2010.

**Abstract:**
Consider a family of zero-sum games indexed by a parameter that determines each
player’s payoff function and feasible strategies. Our first main result characterizes
continuity assumptions on the payoffs and the constraint correspondence such that
the equilibrium value and strategies depend continuously and upper hemicontinuously
(respectively) on the parameter. This characterization uses two topologies in
order to overcome a topological tension that arises when players’ strategy sets are
infinite-dimensional. Our second main result is an application to Bayesian zero-sum
games in which each player’s information is viewed as a parameter. We model each
player’s information as a sub-field, so that it determines her feasible strategies:
those that are measurable with respect to the player’s information. We thereby characterize
conditions under which the equilibriumvalue and strategies depend continuously
and upper hemicontinuously (respectively) on each player’s information. This
clarifies and extends related results of Einy et al. (2008).

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**Coauthors:**
Anthony Horsley and A.J. Wrobel.

**Published:**
1998, *Economics Letters*, **61**:285-291.

**Abstract:**
A topological tension arises in optimization in general choice
spaces because the topology on the choice space should be weak enough
for the choice set to be compact and strong enough for the preference
relation to be continuous. This paper states and proves a version of
the Maximum Theorem that relaxes this tension by using two different
topologies on the choice set. Upper semicontinuity of the objective
function and upper hemicontinuity of the constraint
correspondence are assumed with respect to one topology, and lower
semicontinuity and lower hemicontinuity are assumed with respect
to the other topology. This theorem is particularly useful when the
choice set is in an infinite-dimensional vector space.

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**Published:**
1993, *The Annals of Probability*, **21**:161-167.

**Abstract:**
Using a result of Landers and Rogge, it is shown that the Hausdorff metric of sigma-fields is uniformly equivalent to the metric induced by the Hausdorff distance between sets of measurable functions. An application is given to the continuity of the value of information with respect to the Hausdorff distance of sigma-fields.

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**Published:**
2002, *Journal of Mathematical Economics*, **38**:293-309.

**Abstract:**
The stability of optimal plans with respect to information is studied
given the representation of information and sub- fields of a probability
space. A decision maker is constrained to choose a plan measurable with
respect to her information. Continuity is derived by characterizing the
continuity of the measurability constraint correspondence and then
applying a generalized maximum theorem. This approach can be simpler and
require fewer assumptions than an approach based on continuity of
conditional expectations.

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**Published:**
2010, *Quantitative Economics and Marketing*, **8**:35-60.

**Abstract:**
We consider a seller with uncertain demand for its product. If the
demand curve were certain, then setting price and setting quantity would be
equivalent ways to frame the seller’s problem of choosing a profit-maximizing
point on its demand curve. With uncertain demand, these become distinct sales
mechanisms. We distinguish between uncertainty about the market size and
uncertainty about the consumers’ valuations. Our main results are that (i) for a
given marginal cost, an increase in uncertainty about valuations favors setting
quantity whereas an increase in uncertainty about market size favors setting
price; (ii) keeping demand uncertainty fixed, there is a nonmonotonic relationship
between marginal costs and the optimal selling mechanism (setting price
or quantity); and (iii) in a bilateral monopoly channel setting, coordination
occurs except for a conflict zone in which the retailer’s choice of a selling
mechanism deviates from the coordinated channel selling mechanism.

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