Aldo Montesano

Uncertainty Aversion and Aversion to Increasing Uncertainty

According to the original Ellsberg (1961) examples there is uncertainty version if the decision maker prefers to bet on an urn of known composition rather than on an urn of unknown composition. According to another definition (Schmeidler, 1989), there is uncertainty aversion if any convex combination of two acts is preferred to the least favorable of these acts. We show that these two definitions differ: while the first one truly refers to uncertainty aversion, the second one refers to aversion to increasing uncertainty. Besides, with reference to Choquet Expected Utility theory, uncertainty aversion means that there exists the core of a capacity, while aversion to increasing uncertainty means that the capacity is convex. Consequently, aversion to increasing uncertainty implies uncertainty aversion, but the opposite does not hold. We also show that a completely analogous situation holds for the case of risk and we define a set of risk and uncertainty premiums according to the previous analysis.

Measures of Risk Aversion with Expected and Nonexpected Utility

In the expected utility case, the risk-aversion measure is given by the Arrow-Pratt index. Three proposals of a risk-aversion measure for the nonexpected utility case are examined. The first one sets “the second derivative of the acceptance frontier as a measure of local risk aversion.” The second one takes into account the concavity in the consequences of the partial derivatives of the preference function with respect to probabilities. The third one measures risk aversion through the ratio between the risk premium and the standard deviation of the lottery. The third proposal catches the main feature of risk aversion, while the other two proposals are not always in accordance with the same crude definition of risk aversion, by which there is risk aversion when an agent prefers to get the expected value of a lottery rather than to participate in it.

The Risk Aversion Measure without the Independence Axiom

The risk premium (conveniently normalized) is defined as the measure of risk aversion. This measure does not require any relevant assumption in the theory of choice under uncertainty except the existence of a certainty equivalent. In particular, the independence axiom is not required. The measure of risk aversion of an action is provided not only for the case with one commodity and two consequences but also for the case with many commodities and consequences. The measure of mean risk aversion of all actions with given consequences is introduced and the local measure of risk aversion is obtained by making all these consequences approach the consequence under consideration. This measure is demonstrated to be zero when the von Neumann-Morgenstern utility function exists. In this case a measure of risk aversion of the second order is introduced, which turns out to be equal to the Arrow-Pratt absolute index when there is only one commodity and similar to the generalized measures proposed by several authors when there are many commodities and two consequences.

La nozione di razionalità in economia

The notion of rationality is examined with respect to its use in economics. Three kinds of rationality are distinguished. First, economics is qualified as rational when its nomological-deductive content is taken into account. Second, agents and actions are qualified as rational if a theory of choice is introduced related to a system of binary relations of preference. Third, a preference system on a set of lotteries, acts or strategies is qualified as rational if it is consistent with a theory that connects choice to its possible consequences. Moreover, in all these matters a double meaning of rationality applies, either as logical consistency or as conformity to an axiomatic model.

On the definition of risk aversion

Two definitions of risk aversion have recently been proposed for non-expected utility theories of choice under uncertainty: the former refers the measure of risk aversion (Montesano 1985, 1986 and 1988) directly to the risk premium (i.e. to the difference between the expected value of the action under consideration and its certainty equivalent); the latter defines risk aversion as a decreasing preference for an increasing risk (introduced as mean preserving spreads) (Chew, Karni and Safra 1987, Machina 1987, Röell 1987, Yaari 1987).When the von Neumann-Morgenstern utility function exists both these definitions indicate an agent as a risk averter if his or her utility function is concave. Consequently, the two definitions are equivalent. However, they are no longer equivalent when the von Neumann-Morgenstern utility function does not exist and a non-expected utility theory is assumed. Examples can be given which show how the risk aversion of the one definition can coexist with the risk attraction of the other. Indeed the two definitions consider two different questions: the risk premium definition specifically concerns risk aversion, while the mean preserving spreads definition concerns the increasing (with risk) risk aversion.The mean preserving spreads definition of risk aversion, i.e. the increasing (with risk) risk aversion, requires a special kind of concavity for the preference function (that the derivatives with respect to probabilities are concave in the respective consequences). The risk premium definition of local risk aversion requires that the probability distribution dominates on the average the distribution of the derivatives of the preference function with respect to consequences. Besides, when the local measure of the first order is zero, there is risk aversion according to the measure of the second order if the preference function is concave with respect to consequences.Yaaris (1969) measure of risk aversion is closely linked to the r.p. measure of the second order. Its sign does not indicate risk aversion (if positive) or attraction (if negative) when the measure of the first order is not zero (i.e., in Yaaris language, when subjective odds differ from the market odds).

Risk and Uncertainty Aversion on Certainty Equivalent Functions

The notion of risk aversion was originally developed with reference to the Expected Utility model. de Finetti (1952), Pratt (1964) and Arrow (1965) associated the concavity of the von Neumann-Morgenstern utility function with some relevant aspects of the decision-maker’s preferences. In particular, risk aversion can be defined in terms of risk premium (i.e., the difference between the expected value and the certainty equivalent of a lottery). With reference to the EU model the risk premium is nonnegative for all lotteries if and only if the von Neumann-Morgenstern utility function is concave. However, with reference to the EU model, other relevant aspects of the preferences also depend on the concavity of the utility function: for instance, if we compare two lotteries of which one has been obtained from the other through mean preserving spreads, the less risky lottery is (weakly) preferred for all pairs of lotteries of this kind if and only if the von Neumann-Morgenstern utility function is concave. Moreover, the EU model does not imply that a randomization of lotteries matters (for instance, according to the EU model, a lottery whose consequences are a randomization of the outcomes of two equally preferred lotteries is indifferent to them), while the possibility that a decision-maker prefers not to be involved in an additional lottery could be considered as a kind of risk aversion. Taking into consideration more general models than the EU model, it is no longer true that risk aversion only consists of positive risk premia and of aversion to riskier (in the sense of mean preserving spreads) lotteries and that these two risk aversions depend on the same characteristic of decision-maker’s preferences.

The Paretian Theory of Ophelimity in Closed and Open Cycles.

The theory of ophelimity in closed and open cycles proposed by Pareto following Volterra’s observations is examined. Although these were oriented towards identification of the integrability conditions, Pareto shows no interest in them, but in the problem of the measurement of the elementary ophelimities (i.e. of the marginal utilities) starting from the empirical data (of an ideal experiment) represented by the marginal rates of substitution and by the indifference varieties. Pareto examines both the case in which the integrability conditions are satisfied (closed cycle) and that in which they are not satisfied (open cycle) and introduces in both cases some identification conditions for the elementary ophelimities (i.e. conditions sufficient for their measurability starting from the empirical data). These conditions are commented upon and generalised.

Non-Additive Probabilities and the Measure of Uncertainty and Risk Aversion: A Proposal

Risk aversion is intuitively connected to the risk premium, which is the difference between the expected value and the certainty equivalent. The certainty equivalent is described by the preference function, which associates to every act a certain consequence indifferent to it according to the agent’s preferences. The expected value is determined once the probabilities of the events and the correspondent consequences are known.

A Measure of Risk Aversion in Terms of Preferences

The measure of risk aversion is usually given by the Arrow-Pratt index, which is referred to the neo-Bernoullian utility. But a more general measure is necessary if we accept that a preference model can be considered without assuming, for instance, the independence axiom. A new index of risk aversion is proposed in this paper. It requires only the existence of a certainty equivalent for each action. This index turns out to be zero when the von Neumann-Morgenstern axioms hold and its derivative to be proportional to the Arrow-Pratt index.

Uncertainty with Partial Information on the Possibility of the Events

The Choquet expected utility model deals with nonadditive probabilities (or capacities). Their dependence on the information the decision-maker has about the possibility of the events is taken into account. Two kinds of information are examined: interval information (for instance, the percentage of white balls in an urn is between 60% and 100%) and comparative information (for instance, the information that there are more white balls than black ones). Some implications are shown with regard to the core of the capacity and to two additive measures which can be derived from capacities: the Shapley value and the nucleolus. Interval information bounds prove to be satisfied by all probabilities in the core, but they are not necessarily satisfied by the nucleolus (when the core is empty) and the Shapley value. We must introduce the constrained nucleolus in order for these bounds to be satisfied, while the Shapley value does not seem to be adjustable. On the contrary, comparative information inequalities prove to be not necessarily satisfied by all probabilities in the core and we must introduce the constrained core in order for these inequalities be satisfied. However, both the nucleolus and the Shapley value satisfy the comparative information inequalities, and the Shapley value does it more strictly than the nucleolus.