Class Of Utility Functions Research Articles

Optimal allocations with α‐MaxMin utilities, Choquet expected utilities, and prospect theory

The analysis of optimal risk sharing has been thus far largely restricted to nonexpected utility models with concave utility functions, where concavity is an expression of ambiguity aversion and/or risk aversion. This paper extends the analysis to α‐maxmin expected utility, Choquet expected utility, and cumulative prospect theory, which accommodate ambiguity seeking and risk seeking attitudes. We introduce a novel methodology of quasidifferential calculus of Demyanov and Rubinov (1986, 1992) and argue that it is particularly well suited for the analysis of these three classes of utility functions, which are neither concave nor differentiable. We provide characterizations of quasidifferentials of these utility functions, derive first‐order conditions for Pareto optimal allocations under uncertainty, and analyze implications of these conditions for risk sharing with and without aggregate risk.

Streaming adaptive submodular maximization

Adaptive submodular maximization has been extensively studied in the literature. However, most of existing studies in this field focus on pool-based setting, where one is allowed to pick items in any order, and there have been few studies for the stream-based setting where items arrive in an arbitrary order and one must immediately decide whether to select an item or not upon its arrival. In this paper, we introduce a new class of utility functions, semi-policywise submodular functions. We develop a series of effective algorithms to maximize a semi-policywise submodular function under the stream-based setting.

Two extensions of consumer surplus

AbstractWe study consumer surplus in a single market when (a) there is a lower bound in the consumption of the outside good and (b) the weights in the social welfare function given to consumers and firms are different. We assume quasilinear utility. When the lower bound constraint on the consumption of the outside good is binding, income effects arise in demand. In some cases, Cournot equilibrium output is below equilibrium output without this constraint because the constraint makes demand less elastic. When the weights given to consumers and firms are not identical, social welfare is not necessarily concave and profits might be negative at the unrestricted optimum. We characterize social welfare optimum with a bound on maximum losses in a class of utility functions. We offer a formula to find the percentage of welfare losses due to oligopoly in this case.

The Asymptotic Behavior of the Optimal Cash Holding Strategy Under a Class of Utility Functions

The present paper makes a research on the optimal cash holding problem in the continuous time under a class of utility functions, and verifies that the limit of the optimal cash holding strategy under a class of utility functions is equal to the optimal cash holding strategy under the power utility function. An empirical analysis is stated at the end of this paper.

Necessity of hyperbolic absolute risk aversion for the concavity of consumption functions

Carroll and Kimball (1996) have shown that, in the class of utility functions that are strictly increasing, strictly concave, and have nonnegative third derivatives, hyperbolic absolute risk aversion (HARA) is sufficient for the concavity of consumption functions in general consumption-saving problems. This paper shows that HARA is necessary, implying the concavity of consumption is not a robust prediction outside the HARA class.

A Generalization of Model for Portfolio Selection with Drawdown Constraint on Consumption

In this study, we generalize the results of Arun (2013) on the optimal consumption and investment problem of an infinitely lived agent who does not accept her consumption falling below a fixed proportion of her historically highest level, the so-called drawdown constraint on consumption. We extend the results to a general class of utility functions. We use the martingale method to study the dual problem, which involves the choice of a maximum consumption process. The dual problem can be formulated as a two-dimensional singular control problem, with the free boundary depending on a state variable of the maximum process. We establish the duality theorem and provide semi-closed form solutions regarding the optimal strategies. To highlight our methodology, we present some special cases of utility functions that do not allow for the dimension reduction considered in Arun (2013).

A new approach of utility function based on fractional Gini aggregation operator for intelligent access web selection

The intelligent network access selection applied by the wireless communication device is designed by a decision making problem. The technique comprises getting data related to a mobile device; the data include position data connected with the mobile device. In this paper, we propose a new single-vision and multi-vision utility processes based on the utility theory. Moreover, we apply a new approach to define a new class of utility functions by utilizing the concept of Gini aggregation operator together with the idea of ordered weighted averaging. Mathematical schemes and numerical analysis slow down the ability and the practicality of our suggested simulations.

Two Remarks on Consumer Surplus

We study consumer surplus in a single market when a) there is a lower bound in the consumption of the outside good and b) the weights in the social welfare function given to consumers and firms are different. We assume quasilinear utility. When the lower bound constraint on the consumption of the outside good is binding, income effects arise in demand. In some cases, Cournot equilibrium output is below equilibrium output without this constraint because the constraint makes demand less elastic. When the weights given to consumers and firms are not identical, social welfare is not necessarily concave and profits might be negative at the unrestricted optimum. We characterize social welfare optimum with a bound on maximum losses in a class of utility functions. We offer a formula to find the percentage of welfare losses due to oligopoly in this case.

Substitution with Satiation: A New Class of Utility Functions and a Complementary Pivot Algorithm

We introduce new classes of utility functions and production sets, called Leontief-free, which are applicable when goods are substitutes and utilities/production are subadditive (to model inter-good satiation). When goods are complements, the well studied Leontief utility functions do an adequate job; however, to the best of our knowledge, a similar concept for the case of goods that are substitutes was not known. For markets with these utility functions and production sets, we obtain the following results: Rational-valued equilibria, despite the fact that these utility functions and production sets are nonseparable. We prove that the problem of computing an equilibrium is PPAD-complete, where PPAD stands for Polynomial Parity Arguments on Directed Graphs. We obtain complementary pivot algorithms based on a suitable adaptation of Lemke’s classic algorithm. Our algorithms run in strongly polynomial time if the number of goods is a constant, despite the fact that the set of solutions is disconnected. Experimental verification confirms that our algorithms are practical.

Two stochastic dominance criteria based on tail comparisons

Actuarial risks and financial asset returns are typically heavy tailed. In this paper, we introduce 2 stochastic dominance criteria, called the right‐tail order and the left‐tail order, to compare these variables stochastically. The criteria are based on comparisons of expected utilities, for 2 classes of utility functions that give more weight to the right or the left tail (depending on the context) of the distributions. We study their properties, applications, and connections with other classical criteria, including the increasing convex and the second‐order stochastic dominance. Finally, we rank some parametric families of distributions and provide empirical evidence of the new stochastic dominance criteria with an example using real data.

Selecting Sequences of Items via Submodular Maximization

Motivated by many real world applications such as recommendations in online shopping or entertainment, we consider the problem of selecting sequences of items. In this paper we introduce a novel class of utility functions over sequences of items, strictly generalizing the commonly used class of submodular set functions. We encode the sequential dependencies between items by a directed graph underlying the utility function. Classical algorithms fail to achieve any constant factor approximation guarantees on the problem of selecting sequences of bounded length with maximum utility. We propose an efficient algorithm for this problem that comes with strong theoretical guarantees characterized by the structural properties of the underlying graph. We demonstrate the effectiveness of our algorithm in synthetic and real world experiments on a movie recommendation dataset.

Expectiles, Omega Ratios and Stochastic Ordering

In this paper we introduce the expectile order, defined by X ≤eY if eα(X) ≤eα(Y) for each α ∈ (0, 1), where eα denotes the α-expectile. We show that the expectile order is equivalent to the pointwise ordering of the Omega ratios, and we derive several necessary and sufficient conditions. In the case of equal means, the expectile order can be easily characterized by means of the stop-loss transform; in the more general case of different means we provide some sufficient conditions. In contrast with the more common stochastic orders such as ≤st and ≤cx, the expectile order is not generated by a class of utility functions and is not closed with respect to convolutions. As an illustration, we compare the ≤st, ≤icx and ≤e orders in the family of Lomax distributions and compare Lomax distributions fitted to real world data of natural disasters in the U.S. caused by different sources of weather risk like storms or floods.

The quantification of type-2 prudence in asset allocation by the trustees of a retirement fund

AbstractIn this paper, consideration is given to the normative use of expected-utility theory for the purposes of asset allocation by the trustees of retirement funds. A distinction is drawn between “type-1 prudence”, which relates to deliberate conservatism on the part of actuaries in the setting of assumptions and the determination of model parameters, and “type-2 prudence”, which relates to the risk aversion of the trustees. The intention of the research was to quantify type-2 prudence for the purposes of asset allocation, both for defined-contribution (DC) and defined-benefit (DB) funds. The authors propose new definitions of the objective variables used as the argument of the utility function: one for DC funds and another for DB funds. A new class of utility functions, referred to as the “weighted average relative risk aversion” class is proposed. Practicalities of implementation are discussed. Illustrative results of the application of the method are presented, and it is shown that the proposed approach resolves the paradox of counter-intuitive results found in the literature regarding the sensitivity of the optimal asset allocation to the funding level of a DB fund.

A Foundation of Deterministic Mechanisms

We study a general social choice environment that has multiple agents, a finite set of alternatives, and independent and diuse information. We show that for any Bayesian incentive compatible mechanism, there exists a deterministic mechanism that i) is Bayesian incentive compatible; ii) delivers the same interim expected utilities/allocation probabilities for all the agents; and iii) delivers the same ex ante expected welfare. Our result holds in settings with a rich class of utility functions, multi-dimensional types, interdependent valuations, and non-transferable utilities. More importantly, the result recovers the optimality of deterministic mechanisms (whether in terms of revenue or eciency), which sharply contrasts with the existing results in the screening literature. To prove our result, we develop a new methodology of “mutual purification”, and establish its link with the literature of mechanism design.

The Non-Existence of Representative Agents

We characterize environments in which there exists a representative agent: an agent who inherits the structure of preferences of the population that she represents. The existence of such a representative agent imposes very strong restrictions on individual utility functions -- requiring them to be linear in the allocation and additively separable in any parameter that characterizes agents (e.g., a risk aversion parameter, a discount factor, etc.). In particular, commonly used classes of utility functions (exponentially discounted utility functions, CRRA or CARA utility functions, logarithmic functions, etc.) do not admit a representative agent.

A Stochastic Flows Approach for Asset Allocation with Hidden Economic Environment

An optimal asset allocation problem for a quite general class of utility functions is discussed in a simple two-state Markovian regime-switching model, where the appreciation rate of a risky share changes over time according to the state of a hidden economy. As usual, standard filtering theory is used to transform a financial model with hidden information into one with complete information, where a martingale approach is applied to discuss the optimal asset allocation problem. Using a martingale representation coupled with stochastic flows of diffeomorphisms for the filtering equation, the integrand in the martingale representation is identified which gives rise to an optimal portfolio strategy under some differentiability conditions.

Portfolio Optimization with Ambiguous Correlation and Stochastic Volatilities

In a continuous-time economy, we investigate the asset allocation problem among a risk-free asset and two risky assets with an ambiguous correlation between the two risky assets. The portfolio selection that is robust to the uncertain correlation is formulated as the utility maximization problem over the worst-case scenario with respect to the possible choice of correlation. Thus, it becomes a maximin problem. We solve the problem under the Black-Scholes model for risky assets with an ambiguous correlation using the theory of G-Brownian motions. We then extend the problem to stochastic volatility models for risky assets with an ambiguous correlation between risky asset returns. An asymptotic closed-form solution is derived for a general class of utility functions, including CRRA and CARA utilities, when stochastic volatilities are fast mean-reverting. We propose a practical trading strategy that combines information from the option implied volatility surfaces of risky assets through the ambiguous correlation.

Power Law with Exponential Cut‐off Utility

Most of the utility functions studied earlier concentrated on properties of risk aversion. In this article, the authors have introduced a new class of utility function called the Power Law with Exponential Cut-off (PLEC) utility function, which exhibits all the absolute and relative risk aversion and risk loving preferences of individuals, under various conditions. It generalises and encompasses other systems of utility functions like that of exponential power. Certain properties of this utility function are discussed. Sensitivity analysis exhibits different portfolio allocations for various risk preferences. The analysis also shows that arbitrary risk preferences may lead to biased risk response estimates. Performance of PLEC utility function in portfolio allocation problem is demonstrated through numerical examples. This is evaluated through optimal solutions.

Violation of the Law of Demand

Following the classic work of Mitjuschin, Polterovich and Milleron, necessary and sufficient as well as sufficient conditions have been developed for when the multicommodity Law of Demand holds. We show when the widely cited Mitjuschin and Polterovich sufficient condition also becomes necessary. Using this result, violation regions for the very popular Modified Bergson (or hyperbolic absolute risk aversion) class of utility functions are fully characterized in terms of preference parameters. For a natural extension of the constant elasticity of substitution member of the Modified Bergson family that is neither homothetic nor quasihomothetic, we create the first simple, explicit example of which we are aware that (i) fully characterizes violation regions in both the preference parameter and commodity spaces and (ii) analyzes the range of relative income and price changes within which violations occur.

Gambles with Prices, Operational Measure of Riskiness and Buying and Selling Price for Risky Lotteries

In this paper I analyze operational measure of riskiness defi ned by Foster and Hart (2007). I give simple intuition behind their main result. Then I extend the concept of riskiness measure in two respects - I de fine a generalized riskiness measure based on decreasing absolute risk aversion utility function. I derive necessary and sufficient conditions for existence of such measure for DARA and CRRA class of utility functions. In addition, I show the way how to compare riskiness of gambles with negative expectation or with non negative outcomes only. To this end I use properties of buying and selling price for a lottery and their relations to riskiness measure. In particular, I show how buying and selling price for a lottery concepts may be used complementary to the concept of riskiness measure.