Risk seeking with diminishing marginal utility in a non-expected utility model

The present work takes place in the framework of a non-expected utility model under risk: the RDEU theory (Rank Dependent Expected Utility, first initiated by Quiggin under the denomination of Anticipated Utility), where the decision maker's behavior is characterized by two functionsu andf. Our first result gives a condition under which the functionu characterizes the decision maker's attitude towards wealth. Then, defining a decision maker as risk averter (respectively risk seeker) when he always prefers to any random variable its expected value (weak definition of risk aversion), the second result states that a decision maker who has an increasing marginal utility of wealth (a convex functionu) can be risk averse, if his functionf issufficiently below his functionu, hence if he is sufficientlypessimistic. Obviously, he can also be risk seeking with a diminishing marginal utility of wealth. This result is noteworthy because with a stronger definition of risk aversion/risk seeking, based on mean-preserving spreads, Chew, Karni, and Safra have shown that the only way to be risk averse (in their sense) in RDEU theory is to have, simultaneously, a concave functionu and a convex functionf.     

Independence of irrelevant alternatives in the theory of voting

In social choice theory there has been, and for some authors there still is, a confusion between Arrow'sIndependence of Irrelevant Alternatives (IIA) and somechoice consistency conditions. In this paper we analyze this confusion. It is often thought that Arrow himself was confused, but we show that this is not so. What happened was that Arrow had in mind a condition we callregularity, which implies IIA, but which he could not state formally in his model because his model was not rich enough to permit certain distinctions that would have been necessary. It is the combination of regularity and IIA that he discusses, and the origin of the confusion lies in the fact that if one uses a model that does not permit a distinction between regularity and IIA, regularity looks like a consistency condition, which it is not. We also show that the famous example that proves that Arrow was confused does not prove this at all if it is correctly interpreted.     

On the Inefficiency of Bang-Bang and Stop-Loss Portfolio Strategies

We show in this article that bang-bang portfolio strategies where the investor is alternatively 100% in equity and 100% in cash are dynamically inefficient. Our proof of this result is based on a simple second-order stochastic dominance (SSD) argument. It implies that this is true for any decision criterion that satisfies SSD, not necessarily expected utility. We also examine the stop-loss strategy in which the investor is 100 percent in equity as long as the value of the portfolio exceeds a lower limit where the investor switches to 100 percent in cash. Again, we show that this strategy is inefficient under second-order risk aversion. However, a slight modification of it–in which all wealth exceeding a minimum reserve is invested in equity–is shown to be an efficient dynamic portfolio strategy. This strategy is optimal for investors with a nondifferentiable utility function.     

Aversion to one risk in the presence of others

The more risk-averse of two individuals need not have the smaller certainty equivalent for a risk \~x if another risk or combination of risks w is present. It is shown that he must, however, if either individual's conditional certainty equivalent for x is increasing in w. For independent risks, this condition follows immediately if either individual is decreasingly risk-averse, giving a natural proof of a known result. Another short proof of this result and necessary and sufficient conditions in the independent case are give. For multivariate utilities, the corresponding results do not hold, but it is proved simply that any mixture of decreasingly risk-averse utilities is decreasingly risk-averse. Also touched upon are risk aversion's relation to generalized means, concave composition, risk sharing, and interest rates, the application of the results to discounting under uncertainty and selection of investment level, and their connection to singly crossing distributions, noise, and dominance.     

Greater Downside Risk Aversion

Although investors are concerned foremost with mean and variance, they are also sensitive to downside risk. In this paper, we introduce an index of downside risk aversion to distinguish risk aversion from higher-order aspects of risk preference, including prudence. We show that the index of downside risk aversion S increases with monotonic downside risk averse transformations of utility, thereby directly linking S to the definition of downside risk aversion introduced by Menezes et al. (American Economic Review, 70, 921–932, 1980). Although the index S applies equally to risk averse and risk loving decision makers, for a given positive degree of risk aversion, S is greater when the index of prudence is greater and vice versa.     

The ostrich effect: Selective attention to information

We develop and test a model which links information acquisition decisions to the hedonic utility of information. Acquiring and attending to information increases the psychological impact of information (an impact effect), increases the speed of adjustment for a utility reference-point (a reference-point updating effect), and affects the degree of risk aversion towards randomness in news (a risk aversion effect). Given plausible parameter values, the model predicts asymmetric preferences for the timing of resolution of uncertainty: Individuals should monitor and attend to information more actively given preliminary good news but “put their heads in the sand” by avoiding additional information given adverse prior news. We test for such an “ostrich effect” in a finance context, examining the account monitoring behavior of Scandinavian and American investors in two datasets. In both datasets, investors monitor their portfolios more frequently in rising markets than when markets are flat or falling.

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.     

Risk aversion over finite domains

We investigate risk attitudes when the underlying domain of payoffs is finite and the payoffs are, in general, not numerical. In such cases, the traditional notions of absolute risk attitudes, that are designed for convex domains of numerical payoffs, are not applicable. We introduce comparative notions of weak and strong risk attitudes that remain applicable. We examine how they are characterized within the rank-dependent utility model, thus including expected utility as a special case. In particular, we characterize strong comparative risk aversion under rank-dependent utility. This is our main result. From this and other findings, we draw two novel conclusions. First, under expected utility, weak and strong comparative risk aversion are characterized by the same condition over finite domains. By contrast, such is not the case under non-expected utility. Second, under expected utility, weak (respectively: strong) comparative risk aversion is characterized by the same condition when the utility functions have finite range and when they have convex range (alternatively, when the payoffs are numerical and their domain is finite or convex, respectively). By contrast, such is not the case under non-expected utility. Thus, considering comparative risk aversion over finite domains leads to a better understanding of the divide between expected and non-expected utility, more generally, the structural properties of the main models of decision-making under risk.

     

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.     

The prevalence and composition of asset poverty in Canada: 1999, 2005, and 2012

A household is considered asset poor if its assets (financial assets or net worth, taken separately) are insufficient to maintain well‐being at a low‐income threshold for 3 months. We provide the first national‐level estimates of asset poverty for Canada, using the 1999, 2005, and 2012 cycles of the Survey of Financial Security, and juxtapose these estimates with income poverty. The analysis provides new insight into economic insecurity by showing that asset poverty rates are consistently two to three times higher than income poverty rates. In addition to the prevalence of asset poverty across socio‐demographic groups, we analyzed how the composition of the poor change over time. Age and geography shape the risk for asset poverty in distinct ways. We found that while education appears to play a comparable role in shaping both income poverty and asset poverty, immigration places Canadians at a relatively higher risk of income poverty but not asset poverty. Key Practitioner Message: ? Practitioners ought to consider assets as well as income in assessing economic vulnerability; ? Asset poverty levels are 2–3 times higher than income poverty levels; ? Certain groups (e.g., immigrants) may be income poor but maintain sufficient assets.     

Binary Relations: Finite Characterizations and Computational Complexity

A characterization of a property of binary relations is of finite type if it is stated in terms of ordered T-tuples of alternatives for some positive integer T. The concept was introduced informally by Knoblauch (2005). We give a clear, complete definition below. We prove that a characterization of finite type can be used to determine in polynomial time whether a binary relation over a finite set has the property characterized. We also prove a simple but useful nonexistence theorem and apply it to three examples.      

A strong (Ross) characterization of multivariate risk aversion

Using the addition of uncorrelated noise as a natural definition of increasing risk for multivariate lotteries, I interpret risk aversion as the willingness to pay a (possibly random) vector premium in exchange for a reduction in multivariate risk. If no restriction is placed on the sign of any co-ordinate of the vector premium then (as was the case in Kihlstrom and Mirman's (1974) analysis) only pairs of expected utility maximizers with thesame ordinal preferences for outcomes can be ranked in terms of their aversion to increasing risk. However, if we restrict the premium to be a non-negative random variable then comparisons of aversion to increasing risk may be possible between expected utility maximizers withdistinct ordinal preferences for outcomes. The relationship between their utility functions is precisely the multi-dimensional analog of Ross's (1981)global condition forstrongly more risk averse.     

Risk preferences and changes in background risk

We present two theorems that yield necessary and sufficient conditions for first- and second-degree stochastic dominance deteriorations of background risk to increase risk aversion with respect to foreground risk. We require that any change in a foreground risk that is undesirable remains so after a background risk changes in a way that is either unfair, undesirable in the sense of reducing expected utility, or undesirable in the sense of increasing expected marginal utility. Our results thus characterize utility functions that are, respectively, vulnerable, proper, or standard with respect to changes in background risk.

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.     

A geometric approach to revealed preference via Hamiltonian cycles

It is shown that a fundamental question of revealed preference theory, namely whether the weak axiom of revealed preference (WARP) implies the strong axiom of revealed preference (SARP), can be reduced to a Hamiltonian cycle problem: A set of bundles allows a preference cycle of irreducible length if and only if the convex monotonic hull of these bundles admits a Hamiltonian cycle. This leads to a new proof to show that preference cycles can be of arbitrary length for more than two but not for two commodities. For this, it is shown that a set of bundles satisfying the given condition exists if and only if the dimension of the commodity space is at least three. Preference cycles can be constructed by embedding a cyclic $(L-1)$ -polytope into a facet of a convex monotonic hull in $L$ -space, because cyclic polytopes always admit Hamiltonian cycles. An immediate corollary is that WARP only implies SARP for two commodities. The proof is intuitively appealing as this gives a geometric interpretation of preference cycles.     

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.     

The price of risk with incomplete knowledge on the utility function*

When a risk is exchanged, the exact value for the minimum price (positive or negative) that the purchaser (investor, or insurer) is willing to pay is given by the certainty equivalent wealth level, which in turn depends on his specific utility function. When this utility function is unknown, then only a sufficient condition on the price can ever be found. This paper provides methods for calculating such a sufficient condition, when only limited information on the utility function is known.     

Perceptron versus automaton in the finitely repeated prisoner’s dilemma

We study the finitely repeated prisoner’s dilemma in which the players are restricted to choosing strategies which are implementable by a machine with a bound on its complexity. One player has to use a finite automaton while the other player has to use a finite perceptron. Some examples illustrate that the sets of strategies which are induced by these two types of machines are different and not ordered by set inclusion. Repeated game payoffs are evaluated according to the limit of means. The main result establishes that a cooperation at almost all stages of the game is an equilibrium outcome if the complexity of the machines the players may use is limited enough and if the length T of the repeated game is sufficiently large. This result persists when more than T states are allowed in the player’s automaton. We further consider a variant of the model in which the two players are restricted to choosing strategies which are implementable by perceptrons and prove that the players can cooperate at most of the stages provided that the complexity of their perceptrons is sufficiently reduced.     

The sure thing principle and the value of information

This paper examines the relationship between Savage's sure thing principle and the value of information. We present two classes of results. First, we show that, under a consequentialist axiom, the sure-thing principle is neither sufficient nor necessary for perfect information to be always desirable: specifically, under consequentialism, the sure thing principle is not implied by the condition that perfect information is always valuable; moreover, the joint imposition of the sure thing principle, consequentialism and either one of two state independence axioms does not imply that perfect information is always desirable. Second, we demonstrate that, under consequentialism, the sure thing principle is necessary for a nonnegative value of possibly imperfect information (though of course the principle is still not sufficient). One implication of these results is that the sure thing principle, under consequentialism, plays a somewhat different role in ensuring dynamic consistency in decision making under uncertainty than does the independence axiom in decision making under risk.     

Honesty and method

A client must rely on the integrity of the professional money manager through the stages of the investment process: research, recommendation of specific securities, selection of investments appropriate to the investor, trading of the securities, and reporting of how successful the decisions have been. Throughout this process, the investment manager may be confronted with conflicts of interest. The Association for Investment Management and Research (AIMR) is a nonprofit association of money managers that has developed a code of ethics and standards of professional conduct. Its code contains governing principles; its standards provide the minimum rules of conduct. The AIMR Standards work on the assumption that investors will be adequately protected if the appropriate processes are in place and they provide full disclosure. Since the late 1980s, AIMR has been leading an industry effort to include in the Standards guidelines for presenting a money manager's historical performance.