A tight sufficient condition for Radner-Stiglitz nonconcavity in the value of information

This paper deals with the existence of a nonconcavity in the value of information, as was first explained by Radner and Stiglitz [A nonconcavity in the value of information, in: M. Boyer, R.E. Kihlstrom (Eds.), Bayesian Models in Economic Theory, Elsevier Science Publishers, Amsterdam, 1984, pp. 33-52 (Chapter 3)]. After defining infinitesimal information distance variationIIDV, we find that IIDV=0 is sufficient for a zero marginal value of information at the null. This is a condition only on the information structure and in particular is independent of the decision maker's preferences. This condition is tight: when IIDV>0, there exists a payoff function for which the marginal value of information at the null is positive under general assumptions.     

On the learnability of majority rule

We establish how large a sample of past decisions is required to predict future decisions of a committee with few members. The committee uses majority rule to choose between pairs of alternatives. Each member's vote is derived from a linear ordering over all the alternatives. We prove that there are cases in which an observer cannot predict precisely any decision of a committee based on its past decisions. Nonetheless, approximate prediction is possible after observing relatively few random past decisions.     

Private Information: Similarity as Compatibility

We introduce a new topology on information which evaluates the similarity between information fields taking into account their compatibility, that is, the events that are commonly observed. With this “topology of common information”, the Walrasian expectations equilibrium (Radner 1968) and the private core (Yannelis 1991) are upper semicontinuous     

Imperfect decisions in organizations: Toward a theory of internal structure

I argue that an organization's internal structure systematically depends on how its members use information imperfectly, as distinct from their information also being imperfect. Certain reliability principles are developed to analyze the effects of decision errors: involving the probability of failing to select actions when they are superior to others based on observed information, and the probability of still selecting actions when they are inferior to others based on observed information. A two-stage reliability model is also developed in order to explicitly distinguish between imperfect information and imperfect decisions. The above results imply the need to use rules and procedures to constrain individual decision and information spaces within an organization, and the dynamic flow of information between them, thereby explaining why organizations evolve an internal decision structure in the first place. The analysis is also briefly compared with organization models that incorporate only imperfect information; such as ‘architecture’ theory by Sah and Stiglitz and ‘team’ theory models by Marschak, Radner and Arrow.     

Information Processing Hierarchies

This paper characterizes optimal stationary hierarchies based on an information processing model introduced by Radner. We show how the delay, size (number of processors), capacity (the size of each cohort processed), and throughput (frequency of cohort arrival) are related and determine a feasibility frontier in the space of these variables. The structure of efficient hierarchies implementing the points on the feasibility frontier is also specified. These structures are always nonregular; i.e., every agent has subordinates from several different levels. The nature of the long accepted maxim of decreasing returns to scale in management hierarchies is demonstrated. When the production function of the hierarchy is taken to be the number of information items it can process, we find that returns to size and delay are always decreasing in efficient hierarchies, but these variables are complementary in the design problem. Finally, we discuss how the underlying abilities of the processors affect the marginal returns to size and delay. Journal of Economic Literature Classification Numbers: L22, D83.     

Robust estimation and control under commitment

In a Markov decision problem with hidden state variables, a decision maker expresses fear that his model is misspecified by surrounding it with a set of alternatives that are nearby as measured by their expected log likelihood ratios (entropies). Sets of martingales represent alternative models. Within a two-player zero-sum game under commitment, a minimizing player chooses a martingale at time 0. Probability distributions that solve distorted filtering problems serve as state variables, much like the posterior in problems without concerns about misspecification. We state conditions under which an equilibrium of the zero-sum game with commitment has a recursive representation that can be cast in terms of two risk-sensitivity operators. We apply our results to a linear quadratic example that makes contact with findings of T. Ba?ar and P. Bernhard [H^∞-Optimal Control and Related Minimax Design Problems, second ed., Birkhauser, Basel, 1995] and P. Whittle [Risk-sensitive Optimal Control, Wiley, New York, 1990].     

Another Look at the Radner-Stiglitz Nonconcavity in the Value of Information

This paper revisits the well-known result of R. Radner and J. Stiglitz (1984, in “Bayesian Models of Economic Theory,” Elsevier, Amsterdam) which shows that, under certain conditions, the value of information exhibits increasing marginal returns over some range. Their result assumes that both the number of states and the number of signal realizations are finite, assumptions which preclude most analyses of optimal information acquisition. We provide sufficient conditions that yield this nonconcavity in the value of information in a general framework; the role that these conditions play is clarified and illustrated with several examples. We also discuss the robustness of the nonconcavity result, and the difficulties involved in getting the value of information to be globally concave. Journal of Economic Literature Classification Number: D83.     

An efficiency theorem for incompletely known preferences

There are n agents who have von Neumann-Morgenstern utility functions on a finite set of alternatives A. Each agent i's utility function is known to lie in the nonempty, convex, relatively open set Ui. Suppose L is a lottery on A that is undominated, meaning that there is no other lottery that is guaranteed to Pareto dominate L no matter what the true utility functions are. Then, there exist utility functions ui∈Ui for which L is Pareto efficient. This result includes the ordinal efficiency welfare theorem as a special case.     

Greater prudence and greater downside risk aversion

Variation in the degree of downside risk aversion across decision makers has implications for efficient risk sharing. However, except for small differences in risk preferences, there is no index, analogous to the Arrow-Pratt index of risk aversion, that depends only on local properties of the utility function and indicates the degree of aversion to downside risk. A measure that does depend only on local properties of the utility function u, the index of prudence p=−u^?/u^″, is related to downside risk aversion, which is indicated by a positive value for u^?. Although we show that the degree of prudence is not an accurate indicator of the degree of downside risk aversion, we nonetheless demonstrate that a uniform increase in prudence accompanied by a uniform increase (decrease) in risk aversion is sufficient to indicate greater downside risk aversion, provided prudence is greater (less) than three times the degree of risk aversion.     

Group decision-making in the shadow of disagreement

A model of group decision-making is studied, in which one of two alternatives must be chosen. While agents differ in their preferences over alternatives, everybody prefers agreement to disagreement. Our model is distinguished by three features: private information regarding valuations, differing intensities in preferences, and the option to declare neutrality to avoid disagreement. There is always an equilibrium in which the majority is more aggressive in pushing its alternative, thus enforcing their will via both numbers and voice. However, under general conditions an aggressive minority equilibrium inevitably makes an appearance, provided that the group is large enough. Such equilibria invariably display a “tyranny of the minority”: the increased aggression of the minority always outweighs their smaller number, leading to the minority outcome being implemented with larger probability than the majority alternative. We fully characterize the asymptotic behavior of this model as group size becomes large, and show that all equilibria must converge to one of three possible limit outcomes.     

Consequences, opportunities, and generalized consequentialism and non-consequentialism

In a recent paper published in Journal of Economic Theory (101 (2001) 423-436), by Suzumura and Xu, an analytical framework was developed, which allowed us to characterize the concept of consequentialism and non-consequentialism. To simplify matters, however, the treatment in that paper was confined only to the cases where no trade-off exists between consequential considerations, which reflect the individual's preferences over alternatives, and non-consequential considerations, which reflect the individual's concern over richness of opportunities from which alternatives are chosen. The present paper represents a generalization of our previous framework, which can now accommodate the situations where consequential considerations and non-consequential considerations actively interact. The analysis covers both the cases of finite and infinite number of alternatives.     

Private Value Allocations in Large Economies with Differential Information

We consider a Radner-type (e.g., Radner, 1968, Econometrica36, 31–58) pure exchange economy with differential information and a continuum of agents. We show that under appropriate assumptions the set of Aumann–Shapley private value allocations in such an economy coincides with the set of Radner competitive equilibrium allocations. Journal of Economic Literature Classification Numbers: C71, D51, D82.     

Preference for Flexibility and Dynamic Consistency

Dekel, Lipman, and Rustichini [3] characterize preferences over menus of lotteries that can be represented by the use of a unique subjective state space and a prior. We investigate what would be the appropriate version of Dynamic Consistency in such a setup. The condition we find, which we call Flexibility Consistency, is linked to a comparative theory of preference for flexibility. When the subjective state space is finite, we show that Flexibility Consistency is equivalent to a subjective version of Dynamic Consistency and that it implies that the decision maker is a subjective state space Bayesian updater. Later we characterize when a collection of signals can be interpreted as a partition of the subjective state space of the decision maker.     

Information acquisition and full surplus extraction

It is well known that when agents' types are correlated, the mechanism designer can extract the entire surplus. This creates an incentive for agents to acquire information about other agents' types. Robust lotteries (are payment schemes that) support full extraction and partially robust lotteries support efficient implementation in the presence of information acquisition opportunities. Necessary and sufficient conditions for existence of robust and partially robust lotteries are derived. If an agent's information signal spans other agents' types then robust lotteries do not exist. However, if all agents report their signal realizations then robust lotteries exist in an extended type space.     

Learnability and rationality of choice

In this paper we study the learnability of the class of rationalizable choice functions using the basic concept of PAC-learnability from statistical learning theory. We prove that the class of rationalizable choice functions on N alternatives is learnable from O(N) examples and is optimal in terms of PAC-learnability among classes which are invariant under permutations of the elements.     

Differentiating ambiguity and ambiguity attitude

The objective of this paper is to show how ambiguity, and a decision maker (DM)'s response to it, can be modelled formally in the context of a general decision model. We introduce a relation derived from the DM's preferences, called “unambiguous preference”, and show that it can be represented by a set of probabilities. We provide such set with a simple differential characterization, and argue that it is a behavioral representation of the “ambiguity” that the DM may perceive. Given such revealed ambiguity, we provide a representation of ambiguity attitudes. We also characterize axiomatically a special case of our decision model, the “α-maxmin” expected utility model.     

A characterization of consistent collective choice rules

We characterize a class of collective choice rules such that collective preference relations are consistent. Consistency is a weakening of transitivity and a strengthening of acyclicity requiring that there be no cycles with at least one strict preference, which excludes the possibility of a “money pump.” The properties of collective choice rules used in our characterization are unrestricted domain, strong Pareto, anonymity and neutrality. If there are at most as many individuals as there are alternatives, the axioms provide an alternative characterization of the Pareto rule. If there are more individuals than alternatives, however, further rules become available.     

Optimal voting schemes with costly information acquisition

A group of individuals with identical preferences must make a decision under uncertainty about which decision is best. Before the decision is made, each agent can privately acquire a costly and imperfect signal. We discuss how to design a mechanism for eliciting and aggregating the collected information so as to maximize ex-ante social welfare.We first show that, of all mechanisms, a sequential one is optimal and works as follows. At random, one agent at a time is selected to acquire information and report the resulting signal. Agents are informed of neither their position in the sequence nor of other reports. Acquiring information when called upon and reporting truthfully is an equilibrium.We next characterize the ex-ante optimal scheme among all ex-post efficient mechanisms. In this mechanism, a decision is made when the precision of the posterior exceeds a cut-off that decreases with each additional report. The restriction to ex-post efficiency is shown to be without loss when the available signals are sufficiently imprecise. On the other hand, ex-post efficient mechanisms are shown to be suboptimal when the cost of information acquisition is sufficiently small.     

Information structures in optimal auctions

A seller wishes to sell an object to one of multiple bidders. The valuations of the bidders are privately known. We consider the joint design problem in which the seller can decide the accuracy by which bidders learn their valuation and to whom to sell at what price. We establish that optimal information structures in an optimal auction exhibit a number of properties: (i) information structures can be represented by monotone partitions, (ii) the cardinality of each partition is finite, (iii) the partitions are asymmetric across agents. We show that an optimal information structure exists.     

Veto-based delegation

In a principal-agent model with hidden information and no monetary transfers, I establish the veto-power principle: the principal can implement an optimal outcome through veto-based delegation with a properly chosen default decision. This result demonstrates the exact nature of commitment powers required by the principal: to design the default outcome and to ensure that she has almost no formal control over the agent's decisions.