Convex stochastic dominance with finite consequence sets

Stochastic dominance is a notion in expected-utility decision theory which has been developed to facilitate the analysis of risky or uncertain decision alternatives when the full form of the decision maker's von Neumann-Morgenstern utility function on the consequence space X is not completely specified. For example, if f and g are probability functions on X which correspond to two risky alternatives, then f first-degree stochastically dominates g if, for every consequence x in X, the chance of getting a consequence that is preferred to x is as great under f as under g. When this is true, the expected utility of f must be as great as the expected utility of g.Most work in stochastic dominance has been based on increasing utility functions on X with X an interval on the real line. The present paper, following [1], formulates appropriate notions of first-degree and second-degree stochastic dominance when X is an arbitrary finite set. The only structure imposed on X arises from the decision maker's preferences. It is shown how typical analyses with stochastic dominance can be enriched by applying the notion to convex combinations of probability functions. The potential applications of convex stochastic dominance include analyses of simple-majority voting on risky alternatives when voters have similar preference orders on the consequences.     

Mean utility preserving increases in risk for state dependent utility functions

This paper defines the concept of a mean utility preserving spread across states (MUPSAS) for state dependent utility functions and analyzes the behavioural impact of shifts in the probability distribution of wealth across states such that overall mean utility is preserved. The main result provides an alternative way of ranking state dependent utility functions according to their degree of risk aversion (thus extending Kami's theorem of comparative risk aversion) and establishes a link between increases in risk and risk aversion for state dependent preferences. In a portfolio problem where preferences and the rate of return of the risky venture are state dependent, we find sufficient conditions to determine the impact of a MUPSAS on the optimal share of the portfolio invested in the risky asset.

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Portfolio Inertia and Epsilon-Contaminations

This article analyzes investors’ portfolio selection problems in a two-period dynamic model of Knightian uncertainty. We account for the existence of portfolio inertia in this two-period framework. Furthermore, by incorporating investors’ updating behavior, we analyze how observing new information in the first period will affect investors’ behavior. By this analysis, we show that observing new information in the first period will expand portfolio inertia in the second period compared with the case in which observing new information has not been gained in the first period if the degree of Knightian uncertainty is sufficiently large.     

Ranking sets additively in decisional contexts: an axiomatic characterization

Ranking finite subsets of a given set X of elements is the formal object of analysis in this article. This problem has found a wide range of economic interpretations in the literature. The focus of the article is on the family of rankings that are additively representable. Existing characterizations are too complex and hard to grasp in decisional contexts. Furthermore, Fishburn (1996), Journal of Mathematical Psychology 40, 64–77 showed that the number of sufficient and necessary conditions that are needed to characterize such a family has no upper bound as the cardinality of X increases. In turn, this article proposes a way to overcome these difficulties and allows for the characterization of a meaningful (sub)family of additively representable rankings of sets by means of a few simple axioms. Pattanaik and Xu’s (1990), Recherches Economiques de Louvain 56, 383–390) characterization of the cardinality-based rule will be derived from our main result, and other new rules that stem from our general proposal are discussed and characterized in even simpler terms. In particular, we analyze restricted-cardinality based rules, where the set of “focal” elements is not given ex-ante; but brought out by the axioms.      

Is choice the correct primitive? On using certainty equivalents and reference levels to predict choices among gambles

Choice is viewed as a derived, not a primitive, concept. Individual gambles are assigned subjective certainty equivalents (CE₁); the choice setX has an associated reference level [RL(X)] based on the CE₁S of its members; the outcomes of each gamble are recoded as deviations from the RL(X); and new CE₂S are constructed. The gamble having the largest CE₂ is chosen. The CEs are described by the rank-and sign-dependent theory of Luce (1992b). The concept of RL is studied axiomatically. The model predicts many behavioral anomalies and is tested with data sets of Mellers, Chang, Birnbaum, and Ordóñez (1992).     

On cardinal utility

Necessary and sufficient conditions for the existence of a cardinal utility function on a set X are presented.     

The Interaction Between the Demands for Insurance and Insurable Assets

Holding more of the riskless asset and insuring the risky asset are two ways to reduce portfolio risk. These methods can be employed jointly. As a result, the amount of insurance selected to indemnify against possible losses from holding a risky asset depends, in general, on the quantities of the risky and riskless assets held in the portfolio, and vice versa. In decision models where expected utility is maximized, relatively little has been done to integrate these two decisions into a single model. Such a model is formulated in this paper and the interaction between the demand for insurance and the demand for an insurable risky asset is examined.     

Order extensions,budget correspondences,and rational choice

This article is concerned with extensions of a continuous ordering R on a set X to a subset P(X) of the power set of X. The underlying topology will be the Hausdorff metric topology. We will see that continuous extensions of R do not require that P(X) contain every nonempty finite subset of X. Therefore, the analysis can be applied to consumer theory and inverse choice functions. In analogy to these functions budget correspondences are established which relate alternatives x with certain subsets of X, according to the extended ordering.     

From fun to fraught: marketing to kids and regulating “risky foods” in Canada

ABSTRACT

In an era of industrialized food production, ultra-processed foods, “Big Food” marketing, and growing obesity rates, food has come to be framed as an object of risk – and as an object of regulation. Such reframing has fascinating implications related to issues of responsibility and decision making, especially when it comes to children’s food. This article probes the relationship between representation, regulation and “risky” consumption with respect to children’s food. I examine how child-targeted foods become framed as “risky” and what counts as “risky” food messaging under Health Canada’s commitment to restrict the marketing of unhealthy foods to children. Detailing the tension between food as a risk object and food as a child object, I suggest how issues of semantic provisioning and the politics of the unseen work to complicate and destabilize the (seemingly) straightforward process of prohibiting unhealthy food marketing to children.     

The Value of a Probability Forecast from Portfolio Theory

A probability forecast scored ex post using a probability scoring rule (e.g. Brier) is analogous to a risky financial security. With only superficial adaptation, the same economic logic by which securities are valued ex ante – in particular, portfolio theory and the capital asset pricing model (CAPM) – applies to the valuation of probability forecasts. Each available forecast of a given event is valued relative to each other and to the “market” (all available forecasts). A forecast is seen to be more valuable the higher its expected score and the lower the covariance of its score with the market aggregate score. Forecasts that score highly in trials when others do poorly are appreciated more than those with equal success in “easy” trials where most forecasts score well. The CAPM defines economically rational (equilibrium) forecast prices at which forecasters can trade shares in each other’s ex post score – or associated monetary payoff – thereby balancing forecast risk against return and ultimately forming optimally hedged portfolios. Hedging this way offers risk averse forecasters an “honest” alternative to the ruse of reporting conservative probability assessments.     

The economics of adding and subdividing independent risks: Some comparative statics results

In this paper we address the problem of determining whether adding independent risks or subdividing them is a good substitute for insurance. Despite the fact that accepting more i.i.d. risks increases total risk, it is shown that some risk-averse decision makers can rationally reduce their demand for insurance by doing so. Similarly, a better diversified portfolio of i.i.d. risky assets can rationally be more insured, even if diversification is a risk-reduction scheme. We derive conditions sufficient to obtain unambiguous comparative statics results. Assuming that absolute risk aversion is decreasing and that the fourth derivative of the utility function is positive, we show that diversification is an exceptionally good substitute for insurance. Under the same conditions, adding independent risks to wealth reduces the demand for insurance on each unit.     

‘Sex is violence’: African‐American parents' perceptions of the link between exposure to community violence and youth sexual behaviours

In Chicago, African‐American youth experience disproportionately high rates of exposure to community violence. Growing evidence documents that such exposures are related to youth sexual behaviours. This qualitative study seeks to illuminate the various ways in which violence and sex are intertwined in the lives of African‐American youth. Analysing data from four focus groups (N = 54), major themes indicated that parents linked community violence with: (i) influencing youth sexual interests and opportunities for having sex; (ii) specific notions of youth sexual identities and social status; (iii) increased gang involvement and sexual initiations; and (iv) girls' grudges and their sexual concurrency with boys. These findings suggest that addressing these specific issues is important in parent‐centred and community‐grounded interventions aimed at reducing risky sexual behaviours among African‐American youth.     

Taxation and investment behaviour under uncertainty — A multiperiod portfolio analysis

This paper deals with the effects of a proportional income tax with full loss offset provisions on optimal investment behaviour under uncertainty. For convenience we only consider investments in two assets: one risky asset with a random rate of return and one riskless asset with a secure rate of return. We shall see that for a certain class of intertemporal preferences the total tax effects on the optimal investment in the risky asset can be separated in a saving effect and a portfolio effect with the former influencing the optimal level of savings and the latter influencing the composition of the optimal savings, i.e., the choice between investments in the risky and riskless asset.The author is indebted to Terje Hansen, Jan Mossin and Agnar Sandmo for many helpful comments and suggestions.     

Nozick on Sen: A reply to Perelli-Minetti

Perelli-Minetti argues that Nozick's principle of justice in transfer requires, inter alia, Sen's liberalism condition (L) and is subject to Sen's paradox. It is shown here that a weaker liberalism condition L _W is sufficient for justice in transfer and consistent with Sen's other conditions on the social decision function. The conjunction of conditions L _W, I _P, and N is equivalent to L. It is implausible and perhaps illogical that a society applying Nozick's entitlement theory will impose both I _P and N, and if it does not, then Sen's paradox does not affect Nozick's theory of justice in transfer.     

Approaches to cardinal utility

In a previous article (see [3]) a system of axioms is proposed stating conditions which are necessary and sufficient to determine a cardinal utility function on any set, finite or infinite, of outcomes X. The present paper discusses and interprets the meaning of those axioms, and compares this new approach to cardinal utility with the utility differences approach proposed by Alt and Frisch, among others, and with the expected utility approach of von-Neuman and Morgenstern. The notion of repetition of the same choice situation is presented and its interpretation discussed. It is then argued that this notion leads naturally to the system of axioms presented in On Cardinal Utility. It is also argued that this notion must be used if we want to have a more clear understanding of the meaning of the axioms proposed by Alt and Frisch. Finally, it is remarked that since uncertainty is not present in the new approach, it is free of the paradoxes that have plagued the expected utility hypothesis.     

Ambiguity in asset pricing and portfolio choice: a review of the literature

We survey the literature that has explored the implications of decision-making under ambiguity for financial market outcomes, such as portfolio choice and equilibrium asset prices. This ambiguity literature has led to a number of significant advances in our ability to rationalize empirical features of asset returns and portfolio decisions, such as the failure of the two-fund separation theorem in portfolio decisions, the modest exposure to risky securities observed for a majority of investors, the home equity preference in international portfolio diversification, the excess volatility of asset returns, the equity premium and the risk-free rate puzzles, and the occurrence of trading break-downs.     

Determinants of risk-taking: Behavioral and economic views

The concept of risk-taking is examined from various perspectives: economic, decision theoretic, and psychological. Multiple factors are discussed as complicating the extraction of any presumed risk-taking propensity from a person's real-world behavior.Problem structuring, beliefs, andvalues (defined here as riskless as opposed to risky utility) may of course underlie differences in risk behavior. In addition,context andprocess factors can induce variance in risk-bearing. Also,portfolio effects (including cross-sectional, multiattribute, and longitudinal) may greatly complicate the measurement of risk-taking propensity. Lastly, the presence of incompletemarkets (via which risks can be partially diversified and traded) may further mask the link between intrinsic and observed risk-taking. This article examines each of these measurement obstacles and sources of variance.     

Disappointment without prior expectation: a unifying perspective on decision under risk

The central idea of Disappointment theory is that an individual forms an expectation about a risky alternative, and may experience disappointment if the outcome eventually obtained falls short of the expectation. We abandon the hypothesis of a well-defined prior expectation: disappointment feelings may arise from comparing the outcome received with anyof the gamble’s outcomes that the individual failed to get. This leads to a new, general form of Disappointment model. It encompasses Rank Dependent Utility with an explicit one-parameter probability transformation, and Risk-Value models with a generic risk measure including Variance, providing a unifying behavioral foundation for these models. JEL Classification D80 . D81     

Can good news lead to a more pessimistic choice of action?

Adapting a definition introduced by Milgrom (1981) we say that a signal about the environment is good news relative to some initial beliefs if the posterior beliefs dominate the initial beliefs in the sense of first-order stochastic dominance (the assumption being that higher values of the parameter representing the environment mean better environments). We give an example where good news leads to the adoption of a more pessimistic course of action (we say that action a ₁, reveals greater pessimism than action a ₂, if it gives higher payoff in bad environments and lower payoff in good environments). We then give sufficient conditions for a signal not to induce a more pessimistic choice of action.A first version of this paper was written when the author was Heyworth Research Fellow at Nuffield College, Oxford, and presented at the Second Annual Congress of the European Economic Association (Copenhagen, August 1987). The author is grateful to an anonymous referee for helpful comments.     

Complementary properties of binary relations

After defining the complementary relation R of a binary relation R on a set X, this paper constructs the binary relation C (is a complementary property of) on the set P of nine well known elementary properties that R might possess. It deduces some theorems about C; especially that symmetry is the only one of these possible properties of R on X that is possessed by C on P. The set P may be enlarged to contain other elementary properties of R on X without changing the truth of these theorems, when the symbols of sets are properly modified. Finally, the paper discusses the desirability of a general theory of elementary properties of binary relations for the further development of statistical decision theory.