Von Neumann-Morgenstern Utility Function Research Articles

A generalization of Dybvig’s result on portfolio selection with intolerance for decline in consumption

In this note we show the following result of Dybvig (1995) is valid for a general von Neumann–Morgenstern utility function: for an agent who does not tolerate a decline in consumption, the optimal investment out of discretionary wealth (in excess of the perpetuity value of current consumption) in the risky asset does not depend on the risk aversion coefficient of her felicity function locally when she does not adjust her consumption. The homotheticity assumption is not required. An implication of our result is that if an economic agent exhibits non-time-separable preference due to intertemporal linkage of consumption, her risk-taking over a short time period can be independent of her felicity function.

RISK AVERSION AND EXPECTED UTILITY THEORY: AN EXPERIMENT WITH LARGE AND SMALL STAKES

We employ a novel data set to estimate a structural econometric model of the decisions under risk of players in a game show where lotteries present payoffs in excess of half a million dollars. The decisions under risk of players in the presence of large payoffs allow us to estimate the parameters of the curvature of the von Neumann–Morgenstern utility function—not only locally, as in previous studies in the literature, but also globally. Our estimates of relative risk aversion indicate that a constant relative risk aversion parameter of about 1 captures the average of the sample population. We also find that individuals are practically risk neutral at small stakes and risk averse at large stakes—a necessary condition, according to Rabin’s calibration theorem, for expected utility to provide a unified account of individuals’ attitudes toward risk. Finally, we show that for lotteries characterized by substantial stakes, nonexpected utility theories fit the data equally as well as expected utility theory.

Discounting axioms imply risk neutrality

Although most applications of discounting occur in risky settings, the best-known axiomatic justifications are deterministic. This paper provides an axiomatic rationale for discounting in a stochastic framework. Consider a representation of time and risk preferences with a binary relation on a real vector space of vector-valued discrete-time stochastic processes on a probability space. Four axioms imply that there are unique discount factors such that preferences among stochastic processes correspond to preferences among present value random vectors. The familiar axioms are weak ordering, continuity and nontriviality. The fourth axiom, decomposition, is non-standard and key. These axioms and the converse of decomposition are assumed in previous axiomatic justifications for discounting with nonlinear intraperiod utility functions in deterministic frameworks. Thus, the results here provide the weakest known sufficient conditions for discounting in deterministic or stochastic settings. In addition to the four axioms, if there exists a von Neumann-Morgenstern utility function corresponding to the binary relation, then that function is risk neutral (i.e., affine). In this sense, discounting axioms imply risk neutrality.

Risk Aversion and Portfolio Selection in a Continuous-Time Model

The comparative statics of the optimal portfolios across individuals is carried out for the Black-Scholes market model. It turns out that the indirect utility functions inherit the order of risk aversion (in the Arrow-Pratt sense) from the von Neumann-Morgenstern utility functions, and therefore, a more risk-averse agent would invest less wealth (in absolute value) in the risky asset.

The Relevance of Utilitarianism

Harsanyi invested his Aggregation Theorem and Impartial Observer Theorem with deep utilitarian sense, but Sen redescribed them as representation with little ethical import. This negative view has gained wide acquiescence in economics. Against it, we support the utilitarian interpretation by a novel argument relative to the Aggregation Theorem. We suppose that a utilitarian observer evaluates non-risky alternatives by the sum of individual utilities and investigate his von Neumann-Morgenstern (VNM) preference on risky alternatives. Adding some technical assumptions to Harsanyi's, we conclude that (i) this observer would use the utility sum as a VNM utility function, and crucially, (ii) any social observer would evaluate both risky and non-risky alternatives in terms of a weighted utility sum. Hence, pace Sen, VNM theory can give some interesting support to utilitarianism. The argument is conveyed by means of three theorems that encapsulate Harsanyi's original one as a particular step.

Efficient Random Assignment Under a Combination of Ordinal and Cardinal Information on Preferences

Consider a collection of m indivisible objects to be allocated to n agents, where m ≥ n. Each agent falls in one of two distinct categories: either he (a) has a complete ordinal ranking over the set of individual objects, or (b) has a set of “plausible” benchmark von Neumann-Morgenstern (vNM) utility functions in whose non-negative span his “true” utility is known to lie. An allocation is undominated if there does not exist a preference-compatible profile of vNM utilities at which it is Pareto dominated by another feasible allocation. Given an undominated allocation, we use the tools of linear duality theory to construct a profile of vNM utilities at which it is ex-ante welfare maximizing. A finite set of preference-compatible vNM utility profiles is exhibited such that every undominated allocation is ex-ante welfare maximizing with respect to at least one of them. Given an arbitrary allocation, we provide an interpretation of the constructed vNM utilities as subgradients of a function which measures worst-case domination.

A game-based intrusion detection mechanism to confront internal attackers

Insiders might threaten organizations’ systems any time. By interacting with a system, an insider plays games with the security mechanisms employed to protect it. We apply game theory to model these interactions in an extensive form game that is being played repeatedly with an Intrusion Detection System (IDS). The outcomes of the game are quantified by first specifying players’ preferences, and then, by using the von Neumann–Morgenstern utility function, to assign numbers that reflect these preferences. Examining players’ best responses, the solution of the game follows by locating all the Nash Equilibria (NE). We extend the NE notion to the logit Quantal Response Equilibrium (QRE), to capture players’ bounded rationality and model insider’s behavior. The QRE results are more realistic, and show that the solution of the game might be significantly different than the corresponding NE solution. Thus, we determine how an insider will interact in the future, and how an IDS will react to protect the system. To easily exploit QRE results in ID, we propose the use of a detection mechanism. To present a possible implementation scheme of the detection mechanism, we give the application model and a detailed game-based detection algorithm. Categories and Subject Descriptors: C.2.0 [Computer-Communication Networks]: General -- Security and protection. D.4.6 [Operating Systems]: Security and Protection.

Skewness Preference, Risk Taking and Expected Utility Maximisation

Available empirical evidence suggests that skewness preference plays an important role in understanding asset pricing and gambling. This paper establishes a skewness-comparability condition on probability distributions that is necessary and sufficient for any decision-maker's preferences over the distributions to depend on their means, variances, and third moments only. Under the condition, an Expected Utility maximizer's preferences for a larger mean, a smaller variance, and a larger third moment are shown to parallel, respectively, his preferences for a first-degree stochastic dominant improvement, a mean-preserving contraction, and a downside risk decrease and are characterized in terms of the von Neumann-Morgenstern utility function in exactly the same way. By showing that all Bernoulli distributions are mutually skewness comparable, we further show that in the wide range of economic models where these distributions are used individuals’ decisions under risk can be understood as trade-offs between mean, variance, and skewness. Our results on skewness-inducing transformations of random variables can also be applied to analyze the effects of progressive tax reforms on the incentive to make risky investments.

Duality Theory for Variable Costs in Joint Production

AbstractDuality methods for incomplete systems of consumer demand equations are adapted to the dual structure of variable cost functions in joint production. This allows identification of necessary and sufficient restrictions on technology and cost so that conditional factor demands can be written as functions of input prices, fixed inputs, and cost. These are observable when variable inputs are chosen and committed to production, hence the identified restrictions allow ex ante conditional demands to be studied using observable data. This class of production technologies is consistent with all von Neumann–Morgenstern utility functions when ex post production and/or output prices are uncertain.

Comparative Risk Aversion under Background Risk Revisited

This paper determines a new sufficient condition of the (von Neumann‐Morgenstern) utility function that preserves comparative risk aversion under background risk. It is the single crossing condition of risk aversion. Because this condition requires monotonicity in the local sense, it may satisfy the U‐shaped risk aversion observed in the recent empirical literature.

Multiple Risks and Mean-Variance Preferences

We analyze comparative static effects under uncertainty when a decision maker has mean-variance preferences and faces a generic, quasi-linear decision problem with both an endogenous risk and a background risk. In terms of mean-variance preferences, we fully characterize the effects of changes in the location, scale, and concordance parameters of the stochastic environment on optimal risk taking. Presupposing compatibility between the mean-variance and the expected-utility approach, we then translate these mean-variance properties into their analogues for von Neumann-Morgenstern utility functions.

The possibility of impossible stairways: Tail events and countable player sets

In classical game theory, players have finitely many actions and evaluate outcomes of mixed strategies using a von Neumann–Morgenstern utility function. Allowing a larger, but countable, player set introduces phenomena that are impossible in finite games: Even if players have identical payoffs (no conflicts of interest), (1) this payoff may be minimized in dominant-strategy equilibria, and (2) games so alike that even the consequences of unilateral deviations are the same, may have disjoint sets of payoff-dominant equilibria. Moreover, a class of games without (pure or mixed) Nash equilibria is constructed.

Asymmetric Preferences in Investment Decisions in the Brazilian Financial Market

The main objective of this article is to test the hypothesis that utility preferences that incorporate asymmetric reactions between gains and losses generate better results than the classic Von Neumann-Morgenstern utility functions in the Brazilian market. The asymmetric behavior can be computed through the introduction of a disappointment (or loss) aversion coefficient in the classical expected utility function, which increases the impact of losses against gains. The results generated by both traditional and loss aversion utility functions are compared with real data from the Brazilian market regarding stock market participation in the investment portfolio of pension funds and individual investors.

Drèze equilibria and welfare maxima

We present an example of a production economy with incomplete markets, von Neumann–Morgenstern utility functions, and a unique Dreze equilibrium in order to illustrate and explain the following phenomenon. There exists a transfer scheme such that every shareholder’s utility after transfers and share adjustments increases the more the firm deviates from the Dreze equilibrium. However, shareholders’ welfare decreases the further the firm departs from the Dreze equilibrium. Shareholders’ welfare is defined as the sum of their utilities where every utility function is normalized such that the marginal utility of today’s consumption equals 1 at the Dreze equilibrium.

Aversão a Perdas: Comparação de Decisões de Investimento entre Investidores Individuais e Fundos de Pensão no Brasil

O objetivo deste artigo é analisar se o uso de uma função preferência que incorpora assimetria na reação do investidor frente a ganhos e perdas (aversão a perdas) permite gerar resultados mais coerentes com o comportamento real de investidores brasileiros, na seleção dos seus portfólios de investimento, do que uma função que não incorpora este comportamento (função utilidade tradicional). Os resultados quanto à participação do investimento em ações na composição do portfólio ótimo gerados pelos dois tipos de função são comparados com dados reais agregados de investimentos de dois tipos de investidores brasileiros (fundos de pensão e investidores individuais) visando verificar a capacidade de cada função em replicar o comportamento destes investidores. As simulações indicam que a função utilidade tradicional deve ser rejeitada como modelo representativo do comportamento dos investidores individuais, sendo este comportamento melhor representado por uma função que incorpora aversão a perdas.

Optimization with multivariate stochastic dominance constraints

We consider stochastic optimization problems where risk-aversion is expressed by a stochastic ordering constraint. The constraint requires that a random vector depending on our decisions stochastically dominates a given benchmark random vector. We identify a suitable multivariate stochastic order and describe its generator in terms of von Neumann–Morgenstern utility functions. We develop necessary and sufficient conditions of optimality and duality relations for optimization problems with this constraint. Assuming convexity we show that the Lagrange multipliers corresponding to dominance constraints are elements of the generator of this order, thus refining and generalizing earlier results for optimization under univariate stochastic dominance constraints. Furthermore, we obtain necessary conditions of optimality for non-convex problems under additional smoothness assumptions.

Asset Prices in an Exchange Economy with Money and Trade

We show, in an exchange economy with liquidity constraints, that state prices in a complete markets general equilibrium are a function of the supply of liquidity by the Central Bank. Our model is derived along the lines of Dubey and Geanakoplos (1992). Two agents trade goods and nominal assets (Arrow-Debreu (AD) securities) to smooth consumption across periods and future states, in the presence of cash-in-advance financing costs. We show, with Von Neumann-Morgenstern utility functions and relative risk-aversion greater than 1, that the price of AD securities, are inversely related to liquidity. A closed-from solution is obtained for a CRRA utility function, even when including aggregate uncertainty and different subjective probabilities for the two agents. The upshot of our argument is that agents' expectations computed using risk-neutral probabilities give more weight in the states with higher interest rates. This result cannot be found in a Lucas-type representative agent general equilibrium model where there is neither trade nor money. Hence, an upward yield curve can be supported in equilibrium, even though short-term interest rates are fairly stable, and even in the absence of aggregate uncertainty. The risk-premium in the term structure is therefore a pure liquidity risk premium.

The Possibility of Impossible Stairways and Greener Grass

In classical game theory, players have finitely many actions and evaluate outcomes of mixed strategies using a von Neumann-Morgenstern utility function. Allowing a larger, but countable, player set introduces a host of phenomena that are impossible in finite games. Firstly, in coordination games, all players have the same preferences: switching to a weakly dominant action makes everyone at least as well off as before. Nevertheless, there are coordina- tion games where the best outcome occurs if everyone chooses a weakly dominated action, while the worst outcome occurs if everyone chooses the weakly dominant action. Secondly, the location of payoff-dominant equilibria behaves capriciously: two coordination games that look so much alike that even the consequences of unilateral deviations are the same may nevertheless have disjoint sets of payoff-dominant equilibria. Thirdly, a large class of games has no (pure or mixed) Nash equilibria. Following the proverb \the grass is always greener on the other side of the hedge, greener-grass games model constant discontent: in one part of the strategy space, players would rather switch to its complement. Once there, they'd rather switch back.

Random Expected Utility

We develop and analyze a model of random choice and random expected utility. A decision problem is a finite set of lotteries that describe the feasible choices. A random choice rule associates with each decision problem a probability measure over choices. A random utility function is a probability measure over von Neumann-Morgenstern utility functions. We show that a random choice rule maximizes some random utility function if and only if it is mixture continuous, monotone (the probability that a lottery is chosen does not increase when other lotteries are added to the decision problem), extreme (lotteries that are not extreme points of the decision problem are chosen with probability 0), and linear (satisfies the independence axiom).

A tractable model of precautionary saving in continuous time

This letter derives an analytical solution to the following saving problem. An individual chooses the path of consumption that maximises a time-separable von Neumann–Morgenstern utility function, subject to a standard intertemporal budget constraint. Labour income follows a Poisson process. The consumer is both ‘prudent’ and ‘impatient’, and accumulates financial wealth as a buffer against the risk of income loss.