Nonlinear Utility Research Articles

Estimating values of travel time savings for toll roads: Avoiding a common error

Traditionally, the empirical valuation of travel time savings (VTTS) is obtained from a linear utility specification in a discrete choice model, which implicitly assumes a risk-neutral attitude. This paper draws on recent contributions by the authors that accommodate the attitude towards risk within a non-linear utility specification as a preferred framework within which to value travel time savings. The interest in the non-linear form is motivated by the evidence in Hensher et al. (2011) that mean estimates of VTTS in a proposed toll road context are significantly lower when account is taken of risk attitude. The percentage reduction in the estimate of mean VTTS is approximately (coincidentally) equal to the actual percentage error in traffic forecasts associated with the new tollroad two years after opening. If we could show that this evidence of a lower mean estimate under the non-linear treatment is found in other data settings, then we gain confidence in suggesting that the linear-utility assumption to valuing travel time savings might be a potential contributor to over-predicted tollroad traffic forecasts. The non-linear model is applied herein to two other tollroad choice data sets and we find that sampled car commuters tend to be risk taking when decision making is subject to risk (due to the presence of variability in travel times). The model produces lower mean VTTS estimates than the traditional (linear) model, providing additional evidence of a systematic over-prediction of VTTS under the linear assumption. This paper suggests that future empirical studies on valuing time savings (and variability) should address the attitude towards risk.

Optimising a nonlinear utility function in multi-objective integer programming

In this paper we develop an algorithm to optimise a nonlinear utility function of multiple objectives over the integer efficient set. Our approach is based on identifying and updating bounds on the individual objectives as well as the optimal utility value. This is done using already known solutions, linear programming relaxations, utility function inversion, and integer programming. We develop a general optimisation algorithm for use with k objectives, and we illustrate our approach using a tri-objective integer programming problem.

Are Unit Export Values Correct Measures of the Exports' Quality?

It has become common to measure the quality of exports using their unit export value (UEV). Applications of this method include studies of intra-industry trade (IIT) and analyses of industrial 'competitiveness'. This literature seems to assume that export quality and export price (the most natural interpretation of UEV) are not merely correlated but that they follow each other one-for-one. We put this assumption under scrutiny from both a theoretical and empirical point of view. In terms of theory, we formalize this assumption as a hypothesis of the proportionality of equilibrium prices and equilibrium qualities. We discuss several cases for which this hypothesis is theoretically doubtful (non-linear utility- and cost functions; strong and asymmetric horizontal product differentiation). We also suggest two methods of verifying the hypothesis for cases in which it cannot be easily rejected theoretically. These two methods are then applied to German imports in the period of 1994-2006. We find that the implications of the proportionality hypothesis are largely contradicted by the data.

Are unit export values correct measures of the exports’ quality?

It has become common to measure the quality of exports using their unit export value (UEV). Applications of this method include studies of intra-industry trade (IIT) and analyses of industrial ‘competitiveness.’ This literature seems to assume that export quality and export price (the most natural interpretation of UEV) are not merely correlated but that they follow each other one-for-one. We put this assumption under scrutiny from both a theoretical and empirical point of view. In terms of theory, we formalize this assumption as a hypothesis of the proportionality of equilibrium prices and equilibrium qualities. We discuss several cases for which this hypothesis is theoretically doubtful (nonlinear utility and cost functions; strong and asymmetric horizontal product differentiation). We also suggest a method of verifying the hypothesis for cases in which it cannot be easily rejected theoretically. This method is then applied to German imports in the period of 1994–2009. We find that the implications of the proportionality hypothesis are largely contradicted by the data.

Analysis of long-distance vacation travel demand in the United States: a multiple discrete–continuous choice framework

This study analyzes the annual vacation destination choices and related time allocation patterns of American households. More specifically, an annual vacation destination choice and time allocation model is formulated to simultaneously predict the different vacation destinations that a household visits in a year, and the time (no. of days) it allocates to each of the visited destinations. The model takes the form of a multiple discrete–continuous extreme value (MDCEV) structure. Further, a variant of the MDCEV model is proposed to reduce the prediction of unrealistically small amounts of vacation time allocation to the chosen destinations. To do so, the continuously non-linear utility functional form in the MDCEV framework is replaced with a combination of a linear and non-linear form. The empirical analysis was performed using the 1995 American Travel Survey data, with the United States divided into 210 alternative destinations. The model estimation results provide several insights into the determinants of households’ vacation destination choice and time allocation patterns. Results suggest that travel times and travel costs to the destinations, and lodging costs, leisure activity opportunities (measured by employment in the leisure industry), length of coastline, and weather conditions at the destinations influence households’ destination choices for vacations. The annual vacation destination choice model developed in this study can be incorporated into a larger national travel modeling framework for predicting the national-level, origin–destination flows for vacation travel.

Accommodating Risk Attitudes in Freight Transport Behaviour Research

Behavioural choice modelling is growing in interest as a framework to understand the decision-making of shippers, freight forwarders and other freight agents. Since the 1980s, we have witnessed an increasing number of freight behaviour studies, primarily the freight mode choice, where the roles of one or more freight decision-makers has been addressed, which was neglected in the traditional aggregate approach to freight demand modelling. Stated preference techniques have become a main approach to establishing the role of attributes that define the key drivers in the freight distribution chain. The underlying theory of choice is based on the neoclassical economic assumption that a decision-maker, in choosing, acts as if they are a utility maximizer (working to deliver a profit maximization or cost minimization outcome for the freight business), and this utility maximizing behaviour engenders a population theory of Random Utility Maximization (RUM). Despite the continuing appeal of the RUM framework in applied travel choice studies, a number of specific application assumptions have been questioned by studies in psychology and behavioural economics, arguing that the decisions made by agents are often conditioned on a number of underlying psychological components, one of which is risk attitude. This paper reviews recent freight behaviour studies established on RUM, and presents a major limitation of adopting a risk-neutral assumption through its linear utility specification. Using an existing freight stated choice data set, a nonlinear utility model is estimated which reveals risk-taking attitudes of transporters and shippers. An alternative behavioural paradigm, Rank-Dependent Utility Theory (RDUT), is introduced and incorporated to better accommodate trip time variability, a feature of growing importance in transport systems (passenger and freight). The proposed attribute-specific extended RDUT framework that accommodates the attitude towards risk and preference in freight transport behaviour modelling requires new data, which we detail.

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.

An energy management system for off-grid power systems

Next generation power management at all scales will rely on the efficient scheduling and operation of both generating units and loads to maximize efficiency and utility. The ability to schedule and modulate the demand levels of a subset of loads within a power system can lead to more efficient use of the generating units. These methods become increasingly important for systems that operate independently of the main utility, such as microgrid and off-grid systems. This work extends the principles of unit commitment and economic dispatch problems to off-grid power systems where the loads are also schedulable. We propose a general optimization framework for solving the energy management problem in these systems. An important contribution is the description of how a wide range of sources and loads, including those with discrete states, non-convex, and nonlinear cost or utility functions, can be reformulated as a convex optimization problem using, for example, a shortest path description. Once cast in this way, solution are obtainable using a sub-gradient algorithm that also lends itself to a distributed implementation. The methods are demonstrated by a simulation of an off-grid solar powered community.

Agent negotiation on resources with nonlinear utility functions

To date, researches on agent multi-issue negotiation are mostly based on linear utility functions. However, the relationship between utilities and resources is usually saturated nonlinear. To this end, we expand linear utility functions to nonlinear cases according to the law of diminishing marginal utility. Furthermore, we propose a negotiation model on multiple divisible resources with two phases to realize Pareto optimal results. The computational complexity of the proposed algorithm is polynomial order. Experimental results show that the optimized efficiency of the proposed algorithm is distinctly higher than prior work.

A Geometric Approach to Mechanism Design

We develop a novel geometric approach to mechanism design using an important result in convex analysis: the duality between a closed convex set and its support function. By deriving the support function for the set of feasible interim values we extend the well known Maskin--Riley--Matthews--Border conditions for reduced-form auctions to social choice environments. We next refine the support function to include incentive constraints using a geometric characterization of incentive compatibility. Borrowing results from majorization theory that date back to the work of Hardy, Littlewood, and P\'olya (1929) we elucidate the ironing procedure introduced by Myerson (1981) and Mussa and Rosen (1978). The inclusion of Bayesian and dominant strategy incentive constraints result in the same support function, which establishes equivalence between these implementation concepts. Using Hotelling's lemma we next derive the optimal mechanism for any social choice problem and any linear objective, including revenue and surplus maximization. We extend the approach to include general concave objectives by providing a fixed-point condition characterizing the optimal mechanism. We generalize reduced-form implementation to environments with multi-dimensional, correlated types, non-linear utilities, and interdependent values. When value interdependencies are linear we are able to include incentive constraints into the support function and provide a condition when the second-best allocation is ex post incentive compatible.

Optimal control of LQG problem with an explicit trade-off between mean and variance

For discrete-time linear-quadratic Gaussian (LQG) control problems, a utility function on the expectation and the variance of the conventional performance index is considered. The utility function is viewed as an overall objective of the system and can perform the optimal trade-off between the mean and the variance of performance index. The nonlinear utility function is first converted into an auxiliary parameters optimisation problem about the expectation and the variance. Then an optimal closed-loop feedback controller for the nonseparable mean–variance minimisation problem is designed by nonlinear mathematical programming. Finally, simulation results are given to verify the algorithm's effectiveness obtained in this article.

Return Prediction and Portfolio Selection: A Distributional Approach

The inquiries to return predictability are traditionally limited to the first two moments, mean and volatility. Analogously, literature on portfolio selection also stems from a moment-based analysis with up to the fourth moment being considered. This paper develops a distribution-based framework for both return prediction and portfolio selection. More specifically, a time-varying return distribution is modeled through quantile regression and copulas, using the quantile approach to extract information in marginal distributions and copulas to capture dependence structure. A nonlinear utility function is proposed for portfolio selection which utilizes the full underlying return distribution. An empirical application to US data highlights not only the predictability of the stock and bond return distributions, but also the additional information provided by the distributional approach which cannot be captured by the traditional moment-based methods.

Scalable and Efficient Protocols by Grouping Issues in Multiple Interdependent-Issue Negotiations

Most real-world negotiation involves multiple interdependent issues, which create agent utility functions that are nonlinear. Our research focuses on developing algorithms that enable this kind of negotiation. We present a novel bidding-based negotiation protocol that addresses the excessively high failure rates that existing approaches face when applied to highly complex nonlinear utility functions. This protocol works by using issue dependency information as follows. First, agents generate an interdependency graph by analyzing the agent's constraints. Second, a mediator identifies issue-groups based on the agents' interdependency graphs. Third, agents generate bids that are divided into these issue-groups. Finally, the mediator identifies the winning contract by finding the best combinations of bids in each issue-group. In this paper, we demonstrate that our proposed protocol is highly scalable when compared to previous efforts in a more realistic experimental setting.

The effect of nonlinear utility on behaviour in repeated prisoner’s dilemmas

The present study focuses on the effect of agents’ utility on their cooperation in indefinitely repeated two-person prisoner’s dilemma games (PDs). A game-theoretical analysis suggests that conditions for cooperation in the PDs improve with concavity of utility, with increasing risk aversion, and with increasing loss aversion in the case of PDs with both negative and positive outcomes. The hypotheses were tested in an experiment in which participants played a number of (different) indefinitely repeated PDs and their utility was elicited. The results provide no evidence that concave utility promotes cooperation in the PDs, but we do show that cooperation is promoted by risk aversion and possibly by loss aversion.

Efficient trading with nonlinear utility

In an environment in which agents have nonlinear utility and sufficiently asymmetric initial endowments, we show that efficient trading is achievable. This result is in contrast with Myerson and Satterthwaite (1983), which shows efficient trading is not possible if agents have linear utility and asymmetric initial endowments. Our result is also different from Cramton et al. (1987), in which they maintain the linear utility assumption as in Myerson and Satterthwaite but assume that traders’ initial endowments are relatively symmetric.

Non-parametric test of time consistency: Present bias and future bias

This paper reports the elicited time preference of human subjects in a laboratory setting. The model allows for non-linear utility functions, non-separability between delay and reward, and time inconsistency including future bias in addition to present bias. In particular, the experiment (1) runs a non-parametric test of time consistency and (2) estimates the form of time discount function independently of instantaneous utility functions, and then (3) the result suggests that many subjects exhibiting future bias, indicating an inverse S-curve time discount function.

Aggregation of common-metric attributes in preference revelation and implications for willingness to pay

There is a growing interest in process heterogeneity in the way that individuals evaluate packages of attributes in real or hypothetical markets and make choices. We consider the role of the relative magnitude of pairs of attributes that are defined on a common metric (e.g., minutes or dollars), to look at the extent to which attributes might be added in preference revelation, in contrast to the commonly adopted single rule of compensatory behaviour. The focus is on a choice model specification that allows for different treatments of pairs of attributes across a sample, in contrast to studies that impose a single rule on all observations, and that does not require supplementary information on whether specific individuals claimed to have added up attributes; rather we structure a non-linear utility function that permits a probabilistic aggregation of each attribute. We translate this into a willingness to pay for travel time-savings for car commuters, in the context of tolling roads in Sydney, and contrast it with the results from the additive model, and a model where self-stated attribute processing information is taken into account. The empirical evidence suggests that mean willingness to pay increases when the addition rule is accounted for. This is a potentially important message for environmental applications where two or more attributes have a common metric.

Comonotonic proper scoring rules to measure ambiguity and subjective beliefs

AbstractProper scoring rules serve to measure subjective degrees of belief. Traditional proper scoring rules are based on the assumption of expected value maximization. There are, however, many deviations from expected value, primarily due to risk aversion. Correcting techniques have been proposed in the literature for deviations due to nonlinear utility. These techniques still assumed expected utility maximization. More recently, corrections for deviations from expected utility have been proposed. The latter concerned, however, only the quadratic scoring rule, and could handle only half of the domain of subjective beliefs. Further, beliefs close to 0.5 could not be discriminated. This paper generalizes the correcting techniques to all bounded binary proper scoring rules, covers the whole domain of beliefs and, in particular, can discriminate between all degrees of belief. Thus, we fully extend the properness requirement (in the sense of identifying all degrees of subjective beliefs) to virtually all models that deviate from expected value. Copyright © 2011 John Wiley & Sons, Ltd.

Increasing marginal utility of small increases in life-expectancy?: Results from a population survey

The standard practice in cost-effectiveness analyses of health care is to assign a linear value to increasing lifetime gains. The aim of the current study was to examine the possible existence of non-linear utility for short life extensions. A representative sample of the Norwegian population, aged 40–59 years ( n = 2402), was asked to imagine that they had a limited remaining lifetime (1 year or 10 years) and were offered a treatment that would increase lifetime by a specified amount of time from 1 week to 1 year. In all scenarios, the price per week of life extension was held constant. The proportion of respondents that accepted the treatment increased with increasing extensions, indicating a convex utility function. The result suggests increasing marginal utility for life extensions up to 1 year.

Models of Mode Choice and Mobility Tool Ownership beyond 2008 Fuel Prices

A recent project addressed how travelers would react to fuel prices rising above the high levels that were reached in mid 2008. Study participants were recruited during phone interviews, in the course of which trips made on a specified day were recorded. On the basis of one of those trips and the respondents’ possession of mobility tools, stated preference (SP) experiments were constructed. The first part consisted of a mode choice situation under modified price (and travel time) settings (tactical decisions). The second part focused on long-term (strategic) choices between the current and an alternative fleet, including a redistribution of yearly mileage. From the SP data, multinomial logit models for mode and fleet choice were estimated. The mode choice models were estimated by using income- and distance-dependent nonlinear utility functions and separately for the various trip purposes (as was the practice in earlier Swiss studies on similar topics) and controlled for all relevant trip characteristics. The models for mobility tool ownership, which were formulated by using a new approach, aimed to yield trade-offs between the various attributes of the offered fleets and to forecast the distribution of annual transit passes under modified settings. The findings suggest that inertia is present in both mode choice and mobility tool ownership. Elasticities do not change much from previous studies, where more-conservative price increases were assumed. Transit pass ownership is expected to grow only when increasing fuel prices coincide with stable public transport fares.