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Convex Analysisの二,三の進展について

In the Hamadryas baboon, males are substantially larger than females. A troop of baboons is subdivided into a number of ‘one-male groups’, consisting of one adult male and one or more females with their young. The male prevents any of ‘his’ females from moving too far from him. Kummer (1971) performed the following experiment. Two males, A and B, previously unknown to each other, were placed in a large enclosure. Male A was free to move about the enclosure, but male B was shut in a small cage, from which he could observe A but not interfere. A female, unknown to both males, was then placed in the enclosure. Within 20 minutes male A had persuaded the female to accept his ownership. Male B was then released into the open enclosure. Instead of challenging male A , B avoided any contact, accepting A’s ownership.

Evolution and the Theory of Games

Anarchy, state, and utopia

The theory of precautionary saving is shown in this paper to be isomorphic to the Arrow-Pratt theory of risk aversion, making possible the application of a large body of knowledge about risk aversion to precautionary saving, and more generally, to the theory of optimal choice under risk In particular, a measure of the strength of precautionary saving motive analogous to the Arrow-Pratt measure of risk aversion is used to establish a number of new propositions about precautionary saving, and to give a new interpretation of the Oreze-Modigliani substitution effect

Precautionary Saving in the Small and in the Large

We propose and axiomatize a model of preferences over acts such that the decision maker evaluates acts according to the expectation (over a set of probability measures) of an increasing transformation of an act's expected utility. This expectation is calculated using a subjective probability over the set of probability measures that the decision maker thinks are relevant given her subjective information. A key feature of our model is that it achieves a separation between ambiguity, identified as a characteristic of the decision maker's subjective information, and ambiguity attitude, a characteristic of the decision maker's tastes. We show that attitudes towards risk are characterized by the shape of the von Neumann-Morgenstern utility function, as usual, while attitudes towards ambiguity are characterized by the shape of the increasing transformation applied to expected utilities. We show that the negative exponential form of this transformation is the special case of constant ambiguity aversion. Ambiguity itself is defined behaviorally and is shown to be characterized by properties of the subjective set of measures. This characterization of ambiguity is formally related to the definitions of subjective ambiguity advanced by Epstein-Zhang (2001) and Ghirardato-Marinacci (2002). One advantage of this model is that the well-developed machinery for dealing with risk attitudes can be applied as well to ambiguity attitudes. The model is also distinct from many in the literature on ambiguity in that it allows smooth, rather than kinked, indifference curves. This leads to different behavior and improved tractability, while still sharing the main features (e.g., Ellsberg's Paradox, etc.). The Maxmin EU model (e.g., Gilboa and Schmeidler (1989)) with a given set of measures may be seen as an extreme case of our model with infinite ambiguity aversion. Two illustrative applications to portfolio choice are offered.

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A Smooth Model of Decision Making under Ambiguity

This is the first of two papers where we present a formal model of unawareness. We contrast unawareness with certainty and uncertainty. A subject is certain of something when he knows that thing; he is uncertain when he does not know it, but he knows he does not: he is consciously uncertain. On the other hand, he is unaware of something when he does not know it, and he does not know he does not know, and so on ad infinitum: he does not perceive, does not have in mind, the object of knowledge. The opposite of unawareness is awareness, which includes certainty and uncertainty. This paper has three main purposes. First, we formalize the concept of awareness, and introduce a symmetry axiom which states that a subject can be aware of something, phi say, if and only if he is aware of its negation not-phi; in other words, that phi and not-phi are perceived together, or neither is. We then derive the basic properties of awareness. The second purpose is to prove a different axiomatic characterization, based on the concept of awareness of the system which underlies the model of information with partitional structures (known as S5). The third purpose of this paper is to show that without a substantial weakening of the rules of inferences normally assumed in modal logic a satisfactory model of unawareness, which includes the symmetry axiom, is impossible. This alternative approach is developed in a second paper by the same authors.

Awareness and partitional information structures

Unawareness and Partitional Information Structures

We provide comparative global conditions for downside risk aversion, which are similar to the ones studied by Ross for risk aversion. We define a coefficient of downside risk aversion, and study its local properties.

A note on comparative downside risk aversion

In a simple model of the credit market, based on Stiglitz-Weiss (1981), equilibria are computed and optimal policies to correct market failures are characterized. Some widely applied policies, notably interest-rate subsidies and investment subsidies, are compared to theoretical optimum, and an alternative optimal policy is described which we argue is more robust to model misspecification. An insight on the trade-off between credit policy and infrastructural investment is also offered. A discussion of some aspects of regional policy in Italy's Mezzogiorno is finally presented as an application of the analysis.

Salvatore Modica

Papers

Awareness and partitional information structures

Unawareness and Partitional Information Structures

A note on comparative downside risk aversion

A note on comparative downside risk aversion

Credit Market Failures and Policy