Discounting Function Research Articles

Time ≠ Time ≠ Time: A Research Agenda Mental Temporal Accounting and the Opening of Contract Theory for Work-Leisure Trade-Off Attention

Time tacts life. While all human face the same natural constraints of 24-hours days, behavioral economics found individuals differing in discounting preference for immediate rewards over delayed gratification. Regarding monetary gains, individuals were also shown to hold mental accounts dependent on a reference point but also in regards as to how to allocate money to causes individuals care about. But what if individuals also differ in mental temporal accounts, hence regarding how to spend their time? Decision makers may have natural mental temporal accounts for how to spend 24 hours a day, 720 hours a month, 8760 hours a year or 700800 hours an average life? Could it be that individuals have implicit mental accounts for how much time to spend on their own, how much time to be allocated towards working and how much time to just enjoy life in leisure? If so, could individuals be susceptible to external cues that nudge them into certain mental time-frames that determine their mental time allocation preferences? Could this mental accounting also depend on reference points, such as the age of the individual and critical life events, such as becoming parents? If individuals differ in time spent on their own and social time but also vary in their choices of time spent working and leisure time, we would need to revise contract theory’s primary focus on incentivizing with monetary gains and call for attention to a natural mental temporal accounting trade-off decision between work and leisure time. By doing so we could show that the classical mental accounting theory (Thaler, 1999) actually describes similar processes as mental temporal accounting how to spend time but also dependent on the reference point of age. The paper is structured as follows: The theoretical part 2 describes the behavioral economics idea of mental accounting in the finance domain (Section 2.1) as well as hyperbolic discounting deviations from standard neoclassical discounting functions (Section 2.2.) in order to extend these concepts in the domain of time in Section 2.3. Part 3 proposes to study (1) whether mental accounting also occurs for time; (2) time allocation preferences based on economic, social and environmental contexts as well as age; (3) how individuals can be nudged into different time allocation preferences based on specific mindsets. Part 4 discusses the results and calls for an opening of (1) the behavioral economics idea of the classical mental accounting theory (Thaler, 1999) (2) connecting it to mental temporal accounting as well as (3) neoclassical contract theory for work-leisure time trade-offs alongside providing future research prospects. Elucidating how contexts and experiencing critical life stages influence temporal activity allocation choices promises to improve manifold decisions on education, health, asset management, career paths and common goods preservation throughout life.

An equivalent approximation approach for the Hamilton‐Jacobi‐Bellman equations in intertemporal decision problems

SummaryIn this paper, we develop an equivalent approximation approach to obtain the Hamilton‐Jacobi‐Bellman (HJB) equations of intertemporal decision problems under uncertainty. An implicit discount function is adopted in the derivation of the rate of change of value function for the HJB equations to provide the broadest coverage of time preferences. Meanwhile, we provide an illustrative example for application by revisiting the intertemporal consumption and portfolio decisions problem. Furthermore, we verify the validity of our approach in robustness tests by incorporating exponential, nonconstant, and stochastic hyperbolic discounting functions, respectively. The HJB equations and their resulting decision rules in the finite and infinite planning horizons have confirmed that our approach provides an efficient and reliable alternative to acquire the HJB equations in the intertemporal economic issues under uncertainty.

Are Farmers’ Behaviors Rational When They Pay Less for Social Endowment Insurance? Evidence from Chinese Rural Survey Data

ABSTRACTIn recent years, farmers have had high participation rate in the rural social endowment insurance in China, for which personal contribution and government subsidy are the main funding source. There have been increasingly more farmers participating into the program. However, their enthusiasm for high premium payment was rather low as most of them selected the minimum premium for insurance. In this article, the discounted utility theory from behavioral economics was adopted to analyze insurance selection behaviors of farmers; in addition, a discounted incremental utility model with a hyperbolic discounting function was also further constructed to describe their insurance decision-making processes. Based on the investigation of time preferences of farmers, their insurance participation behaviors of diverse natures were simulated. The corresponding results indicated that active insurance participation and low insurance premium payment were rational choices for most farmers; in comparison, for the elders with higher income, different choices can be made. Therefore, policy makers could formulate differentiated subsidy policies directing at farmers from different groups, so as to stimulate their enthusiasm for premium payment.

The Shape of Discount Function in Intertemporal Choice: A New Test and Experimental Data

Discounted utility theory and its generalizations (e.g., quasihyperbolic discounting, generalized hyperbolic discounting) use discount functions for weighting utilities of outcomes received in different time periods. We propose a new simple test of convexity-concavity of discount function. This test can be used with any utility function (which can be linear or not) and any preferences over risky lotteries (expected utility theory or not). The data from a controlled laboratory experiment show that about one third of experimental subjects reveal a concave discount function and another one third of subjects reveal a convex discount function (for delays up to two month).

Approximately optimizing NDCG using pair-wise loss

The Normalized Discounted Cumulative Gain (NDCG) is used to measure the performance of ranking algorithms. Much of the work on learning to rank by optimizing NDCG directly or indirectly is based on list-wise approaches. In our work, we approximately optimize a variant of NDCG called NDCGβ using pair-wise approaches. NDCGβ utilizes the linear discounting function. We first prove that the DCG error of NDCGβ is equal to the weighted pair-wise loss; then, on that basis, RankBoostndcg and RankSVMndcg are proposed to optimize the upper bound of the pair-wise 0–1 loss function. The experimental results from applying our approaches and ten other state-of-the-art methods to five public datasets show the superiority of the proposed methods, especially RankSVMndcg. In addition, RankBoostndcg are less influenced by the initial weight distribution.

Advocating a Paradigm Shift in Health-State Valuations: The Estimation of Time-Preference Corrected QALY Tariffs

BackgroundDespite evidence of nonproportional trade-offs in time trade-off exercises and the explicit incorporation of exponential discounting in health technology assessment calculations, quality-adjusted life-year (QALY) tariffs are currently still established under the assumption of linear time preferences. ObjectivesThe aim of this study was to introduce a general method of accommodating for nonlinear time preferences in discrete choice experiment (DCE) duration studies and to evaluate its impact on estimated QALY tariffs. MethodsA parsimonious utility function is proposed that accommodates any discounting function and preserves linear time preferences as a special case. Based on an efficient DCE design and 1775 respondents from a nationally representative scientific household panel, preferences and QALY tariffs for the Dutch SF-6D were estimated while accommodating for nonlinear time preferences via exponential and hyperbolic discounting functions. ResultsWhen the discount rate was estimated directly, we found strong evidence of nonlinear time preferences (with an exponential and hyperbolic discount rate of 5.7% and 16.5%, respectively). When the discount rate was estimated as a function of health state severity, we found that years lived in better health states are discounted minus years lived in impaired health states. Finally, the best statistical fit was obtained when using a hyperbolic discount function, which resulted in smaller QALY decrements and fewer health states classified as worse than immediate death. ConclusionsOur results highlight the relevance and even necessity of a paradigm shift in health valuation studies in favor of time-preference corrected QALY tariffs, with potentially important implications for health technology assessment calculations and regulatory decisions.

Heuristic algorithms for the Internet shopping optimization problem with price sensitivity discounts

PurposeRapid advancements in internet technology have made it possible to develop electronic commerce in general and internet shopping in particular. Easy access to a vast number of existing internet stores enables buyers to customize their shopping processes to minimize the total purchase cost. This paper aims to investigate a novel internet shopping problem, which consists of the diversification of a given list of products to buy among many stores and to use discounts offered by the stores.Design/methodology/approachThe adequate discrete optimization problem referred to as internet shopping optimization problem with price sensitivity discounts (ISOPwD) is investigated, which turned out to be strongly nondeterministic polynomial (NS)-hard. Two heuristic solution algorithms have been derived using the tabu search (TS) and the simulated annealing (SA) metaheuristics for having a solution in a reasonable time. The algorithms have been assessed via computational experiments, and they have been compared with another algorithm known from the literature that has been elaborated for a simpler version of ISOPwD.FindingsThe conducted evaluation has shown the advantage of both heuristic algorithms on the algorithm known from the literature. Moreover, the TS-based algorithm outperformed the other one in terms of the total cost incurred by customers and the computational time.Research limitations/implicationsThe special primary piecewise linear discounting function is only taken into account. Other possible discounts connected, for example, with bundles of products and (or) coupons are not considered.Practical implicationsThe elaborated algorithms can be recommended for internet shopping providers who want to introduce the ability to search a cost-optimized set of products in their databases or for applications that combine offers from various online retailers, e.g. internet price comparison services and auction sites.Originality/valueThe novelty of considered ISOPwD, in comparison with similar problems discussed in the literature, deals with an arbitrary number of purchased products, the possibility to buy an identical product in different stores and the consideration of the weight, the amount and the availability of goods as parameters of ISOPwD.

The Influence of Probability Discounting on Escalation in Information Technology Projects

An experimental study was conducted with 17 experienced information technology (IT) project decision-makers. Each participated in a computer based simulation where they had to choose whether to continue an ongoing IT project despite negative feedback, (called escalation of commitment), or abandon it and sell the project as is. A titration procedure for sales price was manipulated over seven probability conditions for success of the project. The switching points where each participant would choose to sell the project instead of continuing development was determined, and probability discounting factors were calculated. The median switching point of the participant's subjective values of the project fit a hyperbolic discounting function well. An escalation factor was calculated and the results indicate that several participants in the study showed signs of escalation. These data demonstrate that the discounting framework may be a viable approach to understand the phenomenon of escalation in IT projects.

Methamphetamine-induced impulsivity during chronic methamphetamine treatment in rats: Effects of the TAAR 1 agonist RO5263397

Impulsivity is an important personality trait associated with several clinical syndromes including drug abuse. While repeated drug exposure is known to increase certain behavioral responses, such as locomotion, to subsequent drug exposure, few studies have examined whether such sensitization develops for impulsive behavior. In the current study we tested the effects of methamphetamine acutely, during the course of, and upon discontinuation of chronic methamphetamine treatment on impulsive behavior in two models, the 5-choice serial reaction time task (5-CSRTT) and the delay-discounting task which measure impulsive action and impulsive choice, respectively. We also examined whether the trace amine-associated receptor 1 (TAAR1) agonist RO5263397 attenuated methamphetamine-induced effects in parallel tests. Acute methamphetamine dose-dependently increased premature responses in the 5-CSRTT and shifted the delay function upward in delay discounting. Up to 40 days of methamphetamine treatment did not significantly alter the dose-effect curve of methamphetamine-induced premature responses, but produced a significant effect in the delay-discounting task. RO5263397 attenuated acute methamphetamine-induced premature responses, but this effect became non-significant over the course of chronic treatment. RO5263397 did not significantly alter the delay-discounting performance. Discontinuation of methamphetamine treatment increased premature responses, which was attenuated by RO5263397, but did not significantly alter the delay discounting function. These results suggest that acute discontinuation from prolonged methamphetamine treatment increases impulsivity, which can be reduced by a TAAR1 agonist.

Efficiency and Risk of Investments in Nuclear Power

It is shown that together with the reduced cost of electricity it is also necessary to examine other efficiency criteria for investments in NPP – the internal profitability norm and the discounted payback time of investments – and their impact on the primary profitability criterion of an NPP project – the pure discounted income. The analytical interconnection of the pure discounted income, internal profitability norm of a project, and the discounted payback time of investments, which is obtained using an exponential discounting function for the expected cash flows, is shown. The requirements of the engineering and economic parameters of NPP that make NPP competitive on the world market are formulated from an analysis of the interrelation of investment efficiency criteria. The NPP investment risk evaluated by the Monte Carlo method with prescribed uncertainty of prediction of cash flows during construction and operation of NPP is presented.

Physical and cognitive effort discounting across different reward magnitudes: Tests of discounting models.

The effort required to obtain a rewarding outcome is an important factor in decision-making. Describing the reward devaluation by increasing effort intensity is substantial to understanding human preferences, because every action and choice that we make is in itself effortful. To investigate how reward valuation is affected by physical and cognitive effort, we compared mathematical discounting functions derived from research on discounting. Seven discounting models were tested across three different reward magnitudes. To test the models, data were collected from a total of 114 participants recruited from the general population. For one-parameter models (hyperbolic, exponential, and parabolic), the data were explained best by the exponential model as given by a percentage of explained variance. However, after introducing an additional parameter, data obtained in the cognitive and physical effort conditions were best described by the power function model. Further analysis, using the second order Akaike and Bayesian Information Criteria, which account for model complexity, allowed us to identify the best model among all tested. We found that the power function best described the data, which corresponds to conventional analyses based on the R2 measure. This supports the conclusion that the function best describing reward devaluation by physical and cognitive effort is a concave one and is different from those that describe delay or probability discounting. In addition, consistent magnitude effects were observed that correspond to those in delay discounting research.

Discounting: A practical guide to multilevel analysis of indifference data.

Multilevel modeling provides the ability to simultaneously evaluate the discounting of individuals and groups using indifference point data. After considering the conditions when weaknesses emerge in estimating individual discounting as a prelude to estimating group discounting, examples are provided that indicate that multilevel modeling improves estimation in the presence of variability and missing data, and when trying to fit two-parameter discounting functions. Concrete examples of how to fit nonlinear multilevel models are provided to help researchers in the adoption of these methods.

Observed and Normative Discount Functions in Addiction and other Diseases.

The aim of this paper is to find a suitable discount function able to describe the progression of a certain addiction or disease under treatment as a discounting process. In effect, a certain indicator related to a disease decays over time in a manner which is mathematically similar to the way in which discounting has been modeled. We analyze the discount functions observed in experiments which study addictive and other problematic behaviors as well as some alternative hyperbola-like discount functions in order to fit the patience exhibited by the subject after receiving the treatment. Additionally, it has been experimentally found that people with addiction display high rates of discount (impatience) and preference reversals (dynamic inconsistency). This excessive discounting must be correctly modeled by a suitable discount function, otherwise, it can become a trans-disease process underlying addiction and other disorders. The (generalized) exponentiated hyperbolic discount function is proposed to describe the progression of a disease with respect to the treatment, since it maintains the property of inconsistency by exhibiting a decreasing discount rate after an initial period in which the opposite occurs.

Tunable discounting and visual exploration for language models

A language model is fundamental to many applications in natural language processing. Most language models are trained on a large amount of dataset and difficult to be adapted to other domains which may have only a small dataset available. Tuning discounting parameters for smoothing is one way to adapt language models for a new domain. In this work, we present novel language models based on tunable discounting mechanisms. The language models are trained on a large dataset, but their discounting parameters can be tuned to a target dataset afterwards. We explore tunable discounting and polynomial discounting functions based on the modified Kneser–Ney (mKN) models. Specifically, we propose the tunable mKN (TmKN) model, polymomial discounting mKN (PmKN) model, and tunable and polynomial discounting mKN (TPmKN) model. We test our proposed models and compared with the mKN model, improved KN model, and the tunable mKN with the interpolation model (mKN + interp). With the implementation, our language models achieve perplexity improvements in both in-domain and out-of-domain evaluation. Experimental results indicate that our new models significantly outperform the baseline model and our models are especially suitable for adapting to new domains. In addition, we use the visualization technique to depict the relationship between parameter settings and the language model performances for guiding our parameter optimization process. The exploratory visual analysis is then used to examine the performance of the proposed language models which will reveal the strength and characteristic of the models.

Altered experiential, but not hypothetical, delay discounting in schizophrenia.

Delay discounting (DD) is a future-oriented decision-making process that refers to whether one is willing to forego a smaller, sooner reward for the sake of a larger, later reward. It can be assessed using hypothetical tasks, which involve choices between hypothetical rewards of varying amounts over delay periods of days to years, or experiential tasks, which involve receiving actual rewards in real time over delay periods of seconds to minutes. Initial studies in schizophrenia have only used hypothetical tasks and have been mixed in finding either elevated or normal levels of DD. One hundred thirty-one outpatients with schizophrenia and 70 healthy controls completed hypothetical and experiential DD tasks involving monetary rewards, and the schizophrenia group was retested after 4 weeks. Although both groups showed qualitatively similar hyperbolic discounting functions on both tasks, they showed a quantitative DD difference. The schizophrenia showed higher DD than controls on the experiential task but normal DD on the hypothetical task. This pattern was not attributable to a range of potential confounds, including smoking status, substance use disorder status, or neurocognition. It was also not attributable to differences in the test-retest reliability, which was good for both tasks. The schizophrenia group's robust pattern of altered experiential but normal hypothetical task performance points to key factors that may contribute to impaired DD in this disorder. These may include increased valuation of small (but not large) monetary rewards, or a hypersensitivity to costs associated with waiting inactively for those rewards. (PsycINFO Database Record

Social Discounting under Risk.

As a measure of how prosocial behavior depends on social distance, social discounting is defined as the decrease in generosity between the decision maker and the recipient as the social distance increases. While risk is a ubiquitous part of modern life, there is limited research on the relationship between risk and prosocial behavior. In the present experiment, we empirically test whether risk has an influence on social discounting. We use the choice titration procedure to examine this effect. Our data show that independent of risk, participants are less eager to forego money and exhibit more selfishness toward a specific person when the social distance increases; these findings are reflected in the hyperbolic model. Interestingly, risk influences the shape of the social discounting function, which is reflected in the notable different discount rates. Individuals who make decisions under risk yield a smaller discount rate than those who make decisions without risk, i.e., under risk subjects reduce less their generosity as a function of the social distance. Furthermore, this distinct type of generosity occurs typically among individuals with 10-distance recipients but not with the closest- and furthest-social-distance recipients.

Boundary Conditions of Options: A Demonstration Based on the Stochastic Discount

Abstract The aim of this paper is to provide a proof of the generally accepted boundary conditions of (call and put) financial options from a novel point of view. To do this, we will use an auxiliary discounting function which will be defined in this work. However, the financial options are derivative instruments whose function is risk hedging in contexts of uncertainty, whereby the employed discount function will be necessarily stochastic. More specifically, we will apply the classic properties of the magnitude “discount” to the so-defined discount function to obtain, in a natural way, the noteworthy boundary conditions of financial options. It is well-known that financial options (belonging to the field of stochastic finance) have been studied without any relation with the magnitude “discount” (more characteristic of the classic Financial Mathematics). Consequently, the principal contribution of this work is the construction of a stochastic discount function as a bridge connecting its associated discount and the financial options, being demonstrated that their properties can be mutually derived.

Deforming Time in a Nonadditive Discount Function

The aim of this paper is to present a methodology to generate a partially subadditive (respectively superadditive) discount function starting from an entirely subadditive (respectively superadditive) discount model. To do this, we are going to use the concept of deformation of time in a discount function, focusing on Stevens' power law. A deformation of time is a function that mathematically represents the perception from an individual or a group of individuals about how the calendar time evolves. This approach will be important when describing the treatment of addictions and other diseases in patients who show a certain degree of impulsivity in their intertemporal choice.

Syntax for calculation of discounting indices from the monetary choice questionnaire and probability discounting questionnaire.

The 27-item Monetary Choice Questionnaire (MCQ; Kirby, Petry, & Bickel, 1999) and 30-item Probability Discounting Questionnaire (PDQ; Madden, Petry, & Johnson, 2009) are widely used, validated measures of preferences for immediate versus delayed rewards and guaranteed versus risky rewards, respectively. The MCQ measures delayed discounting by asking individuals to choose between rewards available immediately and larger rewards available after a delay. The PDQ measures probability discounting by asking individuals to choose between guaranteed rewards and a chance at winning larger rewards. Numerous studies have implicated these measures in addiction and other health behaviors. Unlike typical self-report measures, the MCQ and PDQ generate inferred hyperbolic temporal and probability discounting functions by comparing choice preferences to arrays of functions to which the individual items are preconfigured. This article provides R and SPSS syntax for processing the MCQ and PDQ. Specifically, for the MCQ, the syntax generates k values, consistency of the inferred k, and immediate choice ratios; for the PDQ, the syntax generates h indices, consistency of the inferred h, and risky choice ratios. The syntax is intended to increase the accessibility of these measures, expedite the data processing, and reduce risk for error.

The hunt for the perfect discounting function and a reckoning of time perception

The hunt for the perfect discounting function and a reckoning of time perception