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To assess the conventional treatment in evolutionary inference of alignment gaps as missing data, we propose a simple nonparametric test of the null hypothesis that the locations of alignment gaps are independent of the nucleotide substitution or amino acid replacement process. When we apply the test to 1,390 protein alignments that are informed by protein tertiary structure and use a 5% significance level, the null hypothesis of independence between amino acid replacement and gap location is rejected for ∼65% of datasets. Via simulations that include substitution and insertion–deletion, we show that the test performs well with true alignments. When we simulate according to the null hypothesis and then apply the test to optimal alignments that are inferred by each of four widely used software packages, the null hypothesis is rejected too frequently. Via further simulations and analyses, we show that the overly frequent rejections of the null hypothesis are not solely due to weaknesses of widely used software for finding optimal alignments. Instead, our evidence suggests that optimal alignments are unrepresentative of true alignments and that biased evolutionary inferences may result from relying upon individual optimal alignments.

Widely used approaches for extracting phylogenetic information from aligned sets of molecular sequences rely upon probabilistic models of nucleotide substitution or amino-acid replacement. The phylogenetic information that can be extracted depends on the number of columns in the sequence alignment and will be decreased when the alignment contains gaps due to insertion or deletion events. Motivated by the measurement of information loss, we suggest assessment of the effective sequence length (ESL) of an aligned data set. The ESL can differ from the actual number of columns in a sequence alignment because of the presence of alignment gaps. Furthermore, the estimation of phylogenetic information is affected by model misspecification. Inevitably, the actual process of molecular evolution differs from the probabilistic models employed to describe this process. This disparity means the amount of phylogenetic information in an actual sequence alignment will differ from the amount in a simulated data set of equal size, which motivated us to develop a new test for model adequacy. Via theory and empirical data analysis, we show how to disentangle the effects of gaps and model misspecification. By comparing the Fisher information of actual and simulated sequences, we identify which alignment sites and tree branches are most affected by gaps and model misspecification. [Fisher information; gaps; insertion; deletion; indel; model adequacy; goodness-of-fit test; sequence alignment.]

In referenda, projects are approved if the support for the project exceeds a particular threshold or quota. The usual threshold is the requirement of majority support. The paper provides a theoretical analysis of referenda with different threshold quotas for the provision of a binary public good. Optimal quota sizes are characterized and how the optimal quota changes as the size of the society increases is studied. While the majority quota may or may not be optimal, the conditions which determine the significance of the loss of expected welfare from the (possibly inoptimal) use of the majority quota are analyzed and interpreted. It is shown that the welfare loss from using an inoptimal majority quota will be insignificant if the ratio of the average intensity of support for the project relative to the intensity of opposition to it is positively associated with the probability of a positive net valuation of the project.

000. Within the nematode class Chromadorea, the suborder Tylenchina is an ecologically and morphologically diverse assemblage of nematodes that includes free-living microbivores, fungivores and various types of plant parasites. A recent nematode classification system based largely on SSU rDNA phylogenetic trees classified suborder Tylenchina to include four infraorders: Panagrolaimomorpha, Cephalobomorpha, Tylenchomorpha and Drilonematomorpha, and phylogenetic relationships among species of these infraorders have not always been robustly supported. In this study, we determined the complete mitochondrial genome sequences of three Tylenchina species (Aphelenchus avenae [Aphelenchidae, Tylenchomorpha], Halicephalobus gingivalis, Panagrellus redivivus [Panagrolaimomorpha]) and the partial genome sequence of Acrobeles complexus (Cephalobomorpha) and used these sequences to infer phylogenetic relationships among representatives of the Tylenchina and other nematodes. Phylogenetic analysis of amino acid sequences for 12 protein-coding genes of 100 nematode species supports monophyly of: Chromadorea, Spiruromorpha, Oxyuridomorpha, Ascarididae + Toxocaridae + Anisakidae, Meloidogynidae + Pratylenchidae + Heteroderidae and Aphelenchoidea. Bayesian and maximum-likelihood analyses also show the nested position of Diplogasteromorpha within Rhabditomorpha, and Rhigonematomorpha within Ascaridomorpha. These analyses also show non-monophyly of: clade III, Ancylostomatidae, Panagrolaimomorpha, Tylenchina and Tylenchomorpha. Reconstructed mitochondrial genome phylogeny also revealed that among two main Tylenchomorpha groups, the monophyletic group representing Aphelenchoidea species was sister to the large clade consisting of Ascaridomorpha, Diplogasteromorpha, Rhabditomorpha and Rhigonematomorpha and some Panagrolaimomorpha species, whereas Tylenchoidea species were sister to the most inclusive assemblage containing all infraordinal groups of Chromadorea, except for P. redivivus (Panagrolaimomorpha) and Acrobeles complexus (Cephalobomorpha). The monophyly of Aphelenchoidea (i.e. sister relationship between Aphelenchidae and Aphelenchoididae) recovered in this study indicates that similarity in certain aspects of pharyngeal structure between these two families appears best explained by common ancestry, rather than convergent evolution.

When inferring phylogenies, one important decision is whether and how nucleotide substitution parameters should be shared across different subsets or partitions of the data. One sort of partitioning error occurs when heterogeneous subsets are mistakenly lumped together and treated as if they share parameter values. The opposite kind of error is mistakenly treating homogeneous subsets as if they result from distinct sets of parameters. Lumping and splitting errors are not equally bad. Lumping errors can yield parameter estimates that do not accurately reflect any of the subsets that were combined whereas splitting errors yield estimates that did not benefit from sharing information across partitions. Phylogenetic partitioning decisions are often made by applying information criteria such as the Akaike information criterion (AIC). As with other information criteria, the AIC evaluates a model or partition scheme by combining the maximum log-likelihood value with a penalty that depends on the number of parameters being estimated. For the purpose of selecting an optimal partitioning scheme, we derive an adjustment to the AIC that we refer to as the AIC [Formula: see text] and that is motivated by the idea that splitting errors are less serious than lumping errors. We also introduce a similar adjustment to the Bayesian information criterion (BIC) that we refer to as the BIC [Formula: see text]. Via simulation and empirical data analysis, we contrast AIC and BIC behavior to our suggested adjustments. We discuss these results and also emphasize why we expect the probability of lumping errors with the AIC [Formula: see text] and the BIC [Formula: see text] to be relatively robust to model parameterization.

Lampisiphonia iberica gen. et sp. nov. is described on the basis of specimens collected from subtidal rocky bottom habitats on the Atlantic coast of the Iberian Peninsula. The new genus was distinguished by an erect habit, pseudodichotomous branching, and 9–11 pericentral cells. Plants were bright red to brown-red in colour, 3–6 cm high, firm at the base but ultimate divisions soft and flaccid. Cortication was thick but restricted to the base of large thalli. Plants were attached to the substratum by discoid holdfasts with the tips of decumbent branches reattaching by secondary discoid holdfasts or rhizoids cut off from pericentral cells. Trichoblasts and scar cells were absent and the branching was exogenous. Tetrasporangia were arranged in straight series of up to 15 segments. Sexual structures were not observed. Lampisiphonia was separated from other groups and genera of Polysiphonia sensu lato and the tribe Polysiphonieae by a combination of features that include the absence of trichoblasts and presen...

Antarctica is considered a relatively uncontaminated region with regard to the infectious diseases because of its extreme environment, and isolated geography. For the genetic characterization and molecular epidemiology of the newly found penguin adenovirus in Antarctica, entire genome sequencing and annual survey of penguin adenovirus were conducted. The entire genome sequences of penguin adenoviruses were completed for two Chinstrap penguins (Pygoscelis antarctica) and two Gentoo penguins (Pygoscelis papua). The whole genome lengths and G+C content of penguin adenoviruses were found to be 24,630-24,662 bp and 35.5-35.6%, respectively. Notably, the presence of putative sialidase gene was not identified in penguin adenoviruses by Rapid Amplification of cDNA Ends (RACE-PCR) as well as consensus specific PCR. The penguin adenoviruses were demonstrated to be a new species within the genus Siadenovirus, with a distance of 29.9-39.3% (amino acid, 32.1-47.9%) in DNA polymerase gene, and showed the closest relationship with turkey adenovirus 3 (TAdV-3) in phylogenetic analysis. During the 2008-2013 study period, the penguin adenoviruses were annually detected in 22 of 78 penguins (28.2%), and the molecular epidemiological study of the penguin adenovirus indicates a predominant infection in Chinstrap penguin population (12/30, 40%). Interestingly, the genome of penguin adenovirus could be detected in several internal samples, except the lymph node and brain. In conclusion, an analysis of the entire adenoviral genomes from Antarctic penguins was conducted, and the penguin adenoviruses, containing unique genetic character, were identified as a new species within the genus Siadenovirus. Moreover, it was annually detected in Antarctic penguins, suggesting its circulation within the penguin population.

Identifying causes of genetic divergence is a central goal in evolutionary biology. Although rates of nucleotide substitution vary among taxa and among genes, the causes of this variation tend to be poorly understood. In the present study, we examined the rate and pattern of molecular evolution for five DNA regions over a phylogeny of Cornus, the single genus of Cornaceae. To identify evolutionary mechanisms underlying the molecular variation, we employed Bayesian methods to estimate divergence times and to infer how absolute rates of synonymous and nonsynonymous substitutions and their ratios change over time. We found that the rates vary among genes, lineages, and through time, and differences in mutation rates, selection type and intensity, and possibly genetic drift all contributed to the variation of substitution rates observed among the major lineages of Cornus. We applied independent contrast analysis to explore whether speciation rates are linked to rates of molecular evolution. The results showed no relationships for individual genes, but suggested a possible localized link between species richness and rate of nonsynonymous nucleotide substitution for the combined cpDNA regions. Furthermore, we detected a positive correlation between rates of molecular evolution and morphological change in Cornus. This was particularly pronounced in the dwarf dogwood lineage, in which genome-wide acceleration in both molecular and morphological evolution has likely occurred.

We examine the asymptotic behavior of two strategyproof mechanisms discussed by Moulin for public goods – the conservative equal costs rule (CER) and the serial cost sharing rule (SCSR) – and compare their performance to that of the pivotal mechanism (PM) from the Clarke–Groves family. Allowing the individuals’ valuations for an excludable public project to be random variables, we show under very general assumptions that expected welfare loss generated by the CER, as the size of the population increases, becomes arbitrarily large. However, all moments of the SCSR’s random welfare loss asymptotically converge to zero. The PM does better than the SCSR, with its welfare loss converging even more rapidly to zero.

Because of the increase of genomic data, multiple genes are often available for the inference of phylogenetic relationships. The simple approach for combining multiple genes from the same taxon is to concatenate the sequences and then ignore the fact that different positions in the concatenated sequence came from different genes. Here, we discuss two criteria for inferring the optimal tree topology from data sets with multiple genes. These criteria are designed for multigene data sets where gene-specific evolutionary features are too important to ignore. One criterion is conventional and is obtained by taking the sum of log-likelihoods over all genes. The other criterion is obtained by dividing the log-likelihood for a gene by its sequence length and then taking the arithmetic mean over genes of these ratios. A similar strategy could be adopted with parsimony scores. The optimal tree is then declared to be the one for which the sum or the arithmetic mean is maximized. These criteria are justified within a two-stage hierarchical framework. The first level of the hierarchy represents gene-specific evolutionary features, and the second represents site-specific features for given genes. For testing significance of the optimal topology, we suggest a two-stage bootstrap procedure that involves resampling genes and then resampling alignment columns within resampled genes. An advantage of this procedure over concatenation is that it can effectively account for gene-specific evolutionary features. We discuss the applicability of the two-stage bootstrap idea to the Kishino–Hasegawa test and the Shimodaira–Hasegawa test.

We distinguish between absolute ability ( AATP ) to pay, which represents an absolute constraint on the amount that an individual can be charged, and relative ability to pay ( RATP ) as expressed by differential social welfare weights. We examine the implications of these two principles on free provision of public goods in the context of strategy-proof cost sharing mechanisms. We find that while the AATP considerations can lead to rich individuals being charged high prices, the high prices would be associated with the public good being provided free to fewer individuals. RATP considerations can lead to free provision of the public good to a significant fraction of the society but would imply an upper bound on the price that can be charged to any rich individual.

In the class of Clarke–Groves mechanisms, the pivotal mechanism holds a distinguished place: it is the only mechanism that solves the free rider problem and guarantees the best minimal utility level. When the pivotal mechanism is applied to the case of a pure binary public good, the asymptotic behaviour of its expected welfare loss has been thoroughly analyzed in the literature. To analyze the welfare loss from the mechanism when it is applied to an excludable public good, we first provide a closed form solution of the mechanism that is shown to be equivalent to the standard axiomatic description of the mechanism in the literature. We argue that there is no a priori logical reason to expect the asymptotic behaviour of this mechanism to be related to and derivable from the asymptotics in the case of a pure public good. We use a different computational technique to show that the welfare loss from the pivotal mechanism in the case of an excludable public good converges to zero in a very strong sense.

The paper derives a closed form solution for the maximal surplus generated by the pivotal mechanism in the case of a binary public good. The formula may be used to compare the pivotal mechanism to other mechanisms. An illustration is provided to demonstrate how such a comparison may be made.

This paper presents an application of the dual theory of choice under uncertainty to the problem of asset diversification. It is shown that when there are two or more risky assets, conditions which are sufficient for expected-utility maximizers to diversify among n assets, are also sufficient for dual agents to do so. This result is in contrast to the case of one risky and one safe asset in which dual agents invest all their funds in only one of the assets, while expected-utility maximizers usually diversify.

The phylum Cnidaria is comprised of remarkably diverse and ecologically significant taxa, such as the reef-forming corals, and occupies a basal position in metazoan evolution. The origin of this phylum and the most recent common ancestors (MRCAs) of its modern classes remain mostly unknown, although scattered fossil evidence provides some insights on this topic. Here, we investigate the molecular divergence times of the major taxonomic groups of Cnidaria (27 Hexacorallia, 16 Octocorallia, and 5 Medusozoa) on the basis of mitochondrial DNA sequences of 13 protein-coding genes. For this analysis, the complete mitochondrial genomes of seven octocoral and two scyphozoan species were newly sequenced and combined with all available mitogenomic data from GenBank. Five reliable fossil dates were used to calibrate the Bayesian estimates of divergence times. The molecular evidence suggests that cnidarians originated 741 million years ago (Ma) (95% credible region of 686–819), and the major taxa diversified prior to the Cambrian (543 Ma). The Octocorallia and Scleractinia may have originated from radiations of survivors of the Permian–Triassic mass extinction, which matches their fossil record well.

Statistical models for the evolution of molecular sequences play an important role in the study of evolutionary processes. For the evolutionary analysis of protein-coding sequences, 3 types of evolutionary models are available: 1) nucleotide, 2) amino acid, and 3) codon substitution models. Selecting appropriate models can greatly improve the estimation of phylogenies and divergence times and the detection of positive selection. Although much attention has been paid to the comparisons among the same types of models, relatively little attention has been paid to the comparisons among the different types of models. Additionally, because such models have different data structures, comparison of those models using conventional model selection criteria such as Akaike information criterion (AIC) or Bayesian information criterion (BIC) is not straightforward. Here, we suggest new procedures to convert models of the above-mentioned 3 types to 64-dimensional models with nucleotide triplet substitution. These conversion procedures render it possible to statistically compare the models of these 3 types by using AIC or BIC. By analyzing divergent and conserved interspecific mammalian sequences and intraspecific human population data, we show the superiority of the codon substitution models and discuss the advantages and disadvantages of the models of the 3 types.

This paper is an extension of Jack Meyer's paper titled “Beneficial Changes in Random Variables Under Multiple Sources of Risk and Their Comparative Statics†published in the June 1992 issue of this journal. The extension consists of showing which of the sufficient conditions in Meyer's Theorems 1 and 3 are also necessary, and which are not. In addition, conditions are provided which are necessary and sufficient for general beneficial changes to imply a decrease in the demand for insurance. The Geneva Papers on Risk and Insurance Theory (1992) 17, 171–179. doi:10.1007/BF00962713

Species identification is one of the most important issues in biological studies. Due to recent increases in the amount of genomic information available and the development of DNA sequencing technologies, the applicability of using DNA sequences to identify species (commonly referred to as “DNA barcoding”) is being tested in many areas. Several methods have been suggested to identify species using DNA sequences, including similarity scores, analysis of phylogenetic and population genetic information, and detection of species-specific sequence patterns. Although these methods have demonstrated good performance under a range of circumstances, they also have limitations, as they are subject to loss of information, require intensive computation and are sensitive to model mis-specification, and can be difficult to evaluate in terms of the significance of identification. Here, we suggest a new DNA barcoding method in which support vector machine (SVM) procedures are adopted. Our new method is nonparametric and thus is expected to be robust for a wide range of evolutionary scenarios as well as multilocus analyses. Furthermore, we describe bootstrap procedures that can be used to test the significances of species identifications. We implemented a novel conversion technique for transforming sequence data to real-valued vectors, and therefore, bootstrap procedures can be easily combined with our SVM approach. In this study, we present the results of simulation studies and empirical data analyses to demonstrate the performance of our method and discuss its properties.

In his 1984 paper in Management Science, McEntire conjectured that all concave utility functions satisfy his independence-from-irrelevant-assets (IIA) property. This paper shows that this conjecture is false. The paper provides several families of utility functions, all of whose members violate the IIA property; moreover, it is shown that the set of utility functions violating the IIA property is quite large, and that it is widespread throughout the entire space of utility functions.

This chapter explores the cost sharing implications of a number of alternative theories of justice. It looks at a two-person economy in which the individuals have quasi-linear utility functions and in which information about preferences is private information. In this framework, if the cost for a single unit of a public good is to be shared, then any strategy-proof cost sharing mechanism will not involve the use of the individuals' preferences. If the planner can observe the individuals initial holding of private goods, then cost sharing will depend on the individuals' ability-to-pay as measured by their initial endowment. In general, alternative theories of justice imply that the richer of the two individuals should pay more. Egalitarian theories further imply that the rich individual's share should increase with the difference in the individuals' initial wealth. Since in contrast to the classical derivation of the ability-to-pay principle, declining marginal utility of income plays no role, our results may be viewed as providing a distinct additional justification for both the ability-to-pay approach and for progressive taxation using norms based on theories of equity and justice. The chapter's two-person model gives rise to diagrams and simple closed form solutions for the exposition of the results in the chapter.

Codon-and amino acid-substitution models are widely used for the evolutionary analysis of protein-coding DNA sequences. Using codon models, the amounts of both nonsynonymous and synonymous DNA substitutions can be estimated. The ratio of these amounts represents the strength of selective pressure. Using amino acid models, the amount of nonsynonymous substitutions is estimated, but that of synonymous substitutions is ignored. Although amino acid models lose any information regarding synonymous substitutions, they explicitly incorporate the information for amino acid replacement, which is empirically derived from databases. It is often presumed that when the protein-coding sequences are highly divergent, synonymous substitutions might be saturated and the evolutionary analysis may be hampered by synonymous noise. However, there exists no quantitative procedure to verify whether synonymous substitutions can be ignored; therefore, amino acid models have been arbitrarily selected. In this study, we investigate the issue of a statistical comparison between codon-and amino acid-substitution models. For this purpose, we propose a new procedure to transform a 20-dimensional amino acid model to a 61-dimensional codon model. This transformation reveals that amino acid models belong to a subset of the codon models and enables us to test whether synonymous substitutions can be ignored by using the likelihood ratio. Our theoretical results and analyses of real data indicate that synonymous substitutions are very informative and substantially improve evolutionary inference, even when the sequences are highly divergent. Therefore, we note that amino acid models should be adopted only after carefully investigating and discarding the possibility that synonymous substitutions can reveal important evolutionary information.

Phylogeny estimation is extremely crucial in the study of molecular evolution. The increase in the amount of available genomic data facilitates phylogeny estimation from multilocus sequence data. Although maximum likelihood and Bayesian methods are available for phylogeny reconstruction using multilocus sequence data, these methods require heavy computation, and their application is limited to the analysis of a moderate number of genes and taxa. Distance matrix methods present suitable alternatives for analyzing huge amounts of sequence data. However, the manner in which distance methods can be applied to multilocus sequence data remains unknown. Here, we suggest new procedures to estimate molecular phylogeny using multilocus sequence data and evaluate its significance in the framework of the distance method. We found that concatenation of the multilocus sequence data may result in incorrect phylogeny estimation with an extremely high bootstrap probability (BP), which is due to incorrect estimation of the distances and intentional ignorance of the intergene variations. Therefore, we suggest that the distance matrices for multilocus sequence data be estimated separately and these matrices be subsequently combined to reconstruct phylogeny instead of phylogeny reconstruction using concatenated sequence data. To calculate the BPs of the reconstructed phylogeny, we suggest that 2-stage bootstrap procedures be adopted; in this, genes are resampled followed by resampling of the sequence columns within the resampled genes. By resampling the genes during calculation of BPs, intergene variations are properly considered. Via simulation studies and empirical data analysis, we demonstrate that our 2-stage bootstrap procedures are more suitable than the conventional bootstrap procedure that is adopted after sequence concatenation.

An extensive literature, both theoretical (see for instance, Bruno and Fischer (1990), Calvo and Leiderman (1992), Friedman (1971), Goldman (1974), Sargent and Wallace (1973)) and empirical, (see for instance, Aghevli and Khan (1977), Anderson, Bomberger, and Makinen (1988), Babcock and Makinen (1975), Cagan (1956), Christiano (1987), Easterly, Mauro, and Schmidt-Hebbel (1995), Engsted (1993), Metin and Maslu (1999), Michael, Nobay, and Peel (1994), Pickersgill (1968), Salemi and Sargent (1979), Taylor (1991)) has arisen around the special semi-logarithmic demand for money function introduced by Cagan (1956). Cagan’s motivation behind the demand for money function was mainly in terms of transactions costs and its relationship to the consumer’s ability to affect the real value of cash balances. Cagan argued that the real cost of holding cash balances fluctuates widely enough to account for the dramatic changes in the holding of cash balances observed during hyperinflation. He hypothesized that during periods of hyperinflation the demand for money is almost entirely explained by the variation in the expected rate of change in prices and that changes in expected inflation have the same effect on real balances in percentage terms regardless of the absolute amount of initial cash balances. In other words, during hyperinflations, the demand for money takes the special form: m= ke−λπ , where m is the real demand for money, π(e) is the expected rate of inflation and k, λ are positive constants. The theoretical papers using Cagan’s functional form have been written largely in the monetarist tradition, analyzing hyperinflation and the associated problem of “inflation tax.” (See, for instance, Calvo and Leiderman (1992), Friedman (1971), Sargent and Wallace (1973).) The use of Cagan’s demand for money function has, however, been “adhoc” and no attempt has been made to rationalize the function

The rate of molecular evolution can vary among lineages. Sources of this variation have differential effects on synonymous and nonsynonymous substitution rates. Changes in effective population size or patterns of natural selection will mainly alter nonsynonymous substitution rates. Changes in generation length or mutation rates are likely to have an impact on both synonymous and nonsynonymous substitution rates. By comparing changes in synonymous and nonsynonymous rates, the relative contributions of the driving forces of evolution can be better characterized. Here, we introduce a procedure for estimating the chronological rates of synonymous and nonsynonymous substitutions on the branches of an evolutionary tree. Because the widely used ratio of nonsynonymous and synonymous rates is not designed to detect simultaneous increases or simultaneous decreases in synonymous and nonsynonymous rates, the estimation of these rates rather than their ratio can improve characterization of the evolutionary process. With our Bayesian approach, we analyze cytochrome oxidase subunit I evolution in primates and infer that nonsynonymous rates have a greater tendency to change over time than do synonymous rates. Our analysis of these data also suggests that rates have been positively correlated.

Using pseudomaximum-likelihood approaches to phylogenetic inference and coalescent theory, we develop a computationally tractable method of estimating effective population size from serially sampled viral data. We show that the variance of the maximum-likelihood estimator of effective population size depends on the serial sampling design only because internal node times on a coalescent genealogy can be better estimated with some designs than with others. Given the internal node times and the number of sequences sampled, the variance of the maximum-likelihood estimator is independent of the serial sampling design. We then estimate the effective size of the HIV-1 population within nine hosts. If we assume that the mutation rate is 2.5 × 10−5 substitutions/generation and is the same in all patients, estimated generation lengths vary from 0.73 to 2.43 days/generation and the mean (1.47) is similar to the generation lengths estimated by other researchers. If we assume that generation length is 1.47 days and is the same in all patients, mutation rate estimates vary from 1.52 × 10−5 to 5.02 × 10−5. Our results indicate that effective viral population size and evolutionary rate per year are negatively correlated among HIV-1 patients.

This article shows that if Ross' definition of riskier is replaced by a more traditional definition, such as a mean-preserving spread or second-degree stochastic dominance, then the application of Ross's stronger measure of risk aversion to the portfolio problem may no longer produce the desired result. It is also shown that the stronger measure may not perform satisfactorily when applied to exponential utility functions.

In decision problems involving uncertainty, one is often interested in analyzing the effect on the optimal value of a decision variable resulting from a change in the nature of the uncertainty. One approach to this problem that has frequently been used in the literature is to subject the random variable in question to some deterministic transformation, usually of the linear type. Most researchers have chosen those transformations which bring about a mean-preserving increase in the riskiness of the random variable. Some examples in which such transformations are used are the works of Arrow [1, 104-106], Sandmo [12, 67-69], Batra and Ullah [2, 542-45], Ishii [8, 768-69], and Meyer and Ormiston [9, 181-85]. The advantage of using the transformation method for the analysis of changes in uncertainty is its simplicity, and the fact that determinate results can be obtained by imposing an additional assumption such as decreasing absolute (or increasing relative) risk aversion. However, the transformation method is not without shortcomings. The most obvious of these is the restriction imposed on the relationship between the distribution functions of the two random variables. Specifically, if the random variable X has the distribution F(x), and the transformed variable Y, where Y = t(X), has the distribution G(y), then the distribution function of Y must satisfy G(y) = F[t -(y)], assuming t is monotone. Thus, if the transformation t is linear, the two distribution functions can intersect at most once. While this feature facilitates the analysis, it clearly restricts the admissible types of changes in uncertainty, and this, in turn, may preclude certain outcomes which more general types of shifts in the distributions could bring about. Another aspect of the transformation method that could prove to be somewhat problematic in some contexts is the fact that some transformations, such as linear ones, cause a change in the effective range of the random variable. In some cases, however, the random variable has a natural bounded range (such as the unit interval, or the non-negative half-line), so that the transformed variable may have to take values outside the meaningful range. The most important aspect of the restrictiveness of the transformation method is its lack of robustness; that is, when more general changes in uncertainty are allowed, the formal propositions obtained under the transformation method may no longer be valid. The limitations of the transformation-of-variable approach to changes in uncertainty can be avoided altogether by allowing the distribution function to shift in an arbitrary fashion, at least

When the distribution of the returns of a risky asset undergoes a stochastically dominating shift, a risk-averse investor may not necessarily increase the investment in that asset. This paper provides restrictions on the investor's utility function that are necessary and sufficient for a dominating shift to bring about no decrease in the investment in the respective asset if there are two risky assets in the portfolio. These conditions are also necessary if there are n > 2 assets, and are necessary and sufficient if the utility function exhibits constant absolute risk aversion. Copyright 1990 by Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association.