Linear Utility Functions Research Articles

From local utility to neural networks

We introduce and analyze two preference-based notions of local linearity in the spirit of Machina (1982). We show how the weaker among the two extends Machina’s local utility analysis, and that the stronger among the two characterizes continuous finite piecewise linear (CFPL) utility functions. We introduce a representation of the decision maker’s preference called the neural-network utility representation that is equivalent to the CFPL representation, in which the decision maker evaluates an alternative through a neural network.

Optimal Mechanism in a Dynamic Stochastic Knapsack Environment

This study introduces an optimal mechanism in a dynamic stochastic knapsack environment. The model features a single seller who has a fixed quantity of a perfectly divisible item. Impatient buyers with a piece-wise linear utility function arrive randomly and they report the two-dimensional private information: marginal value and demanded quantity. We derive a revenue-maximizing dynamic mechanism in a finite discrete time framework that satisfies incentive compatibility, individual rationality, and feasibility conditions. This is achieved by characterizing buyers' utility and utilizing the Bellman equation. Moreover, we establish the essential penalty scheme for incentive compatibility, as well as the allocation and payment policies. Lastly, we propose algorithms to approximate the optimal policy, based on the Monte Carlo simulation-based regression method and reinforcement learning.

A 0–1 linear relaxation and stochastic approximation method for real-time pricing in a smart grid with non-convex constraints

In this article, real-time pricing (RTP) for a smart grid with complicated non-convex and/or non-smooth constraints is explored. First, piecewise linear functions are adopted to approximate any kinds of utility function and RTP is formulated into bilevel non-convex programming. Then, a 0–1 relaxation method is used to linearize sparse constraints and piecewise linear utility functions, to obtain mixed integer linear programming. Finally, a distributed simultaneous perturbation stochastic approximation (DSPSA) method is designed to solve the bilevel non-convex programming. The RTP strategy is generalizable because sparse constraints and non-differentiable and/or non-concave utility functions are widespread in practice, and the piecewise linear functions approximation method is applicable to all kinds of utility function, not just to non-smooth and/or non-concave utility functions. In addition, the 0–1 linear relaxation and stochastic approximation method make the bilevel non-convex programming easily solvable by DSPSA, which converges rapidly while maintaining the high accuracy of solutions.

ECCANS: Enhanced CRITIC-based Context-Aware Network Selection algorithm for 5G HetNet

The ability of mobility management to provide seamless communication and better quality of service (QoS) to the multiservice multimode mobile node (MMMN), which can run multiple services simultaneously during roaming in ultra-dense heterogeneous networks with different radio access technologies, is a major challenge in implementing 5G and beyond. Ultra-dense heterogeneous networks (HetNet), restrictions on multiservice multimode mobile terminals, and user mobility all lead to a number of serious problems, such as the ping-pong effect, unnecessary handovers, and decreasing cell edge spectral efficiency. The dense deployment of small base stations also causes a high probability of frequent handovers in HetNet, which has led to a significant escalation in energy consumption resulting from unnecessary signaling, increased network overhead, and higher energy usage by network equipment. This increased energy requirement may lead to potentially increase the emissions of greenhouse gases to meet the energy production needs. Several handover decision algorithms have been proposed, but most of them deal with only running a single service at a time on a mobile node. In this paper, authors propose a new network selection model for MMMN which may run multiple services simultaneously. This model uses context-aware information about the user’s preferences, characteristics, and needs of the services, limitations of mobile terminal, and network conditions of an ultra-dense HetNet to choose the best network. The fundamental principle of this model is based on the comprehensive network criteria weight computed by the integration of Enhanced CRiteria Importance Through Inter-criteria Correlation (CRITIC), fuzzy analytic hierarchy process (FAHP), and group decision-making (GDM) techniques and the service priority of running services. For computing the objective weight of criteria, a novel improved CRITIC method has been proposed. At the same time, sigmoid utility functions and linear utility functions have been used to figure out the value of network criteria for running multiple services on the mobile node. Finally, the best network from HetNet for MMMN is selected using the technique for order performance by similarity to ideal solution (TOPSIS) and threshold by computing a comprehensive utility value network criteria decision matrix and comprehensive network criteria weight. The simulation results show that the proposed model increases the selection probabilities of an appropriate network according to the service requirements and network conditions. It also reduces unnecessary handovers and improves QoS while satisfying users’ preferences, service requirements, and terminal constraints, in solving the MMMN network selection problem.

A holistic failure modes and effects analysis for university plastic injection laboratory under Bayesian Network

University Research & Development (R&D) laboratories are facilities used for research and manufacturing activities in engineering and basic science fields. Various filling materials are used in plastic R&D laboratories to improve the mechanical properties of thermoplastics or to recycle waste materials. The use of filling materials causes changes in plastic production parameters. This study dealt with the risks arising from extrusion, injection and shredder in a plastic injection laboratory, which led to the inability to manufacture the appropriate product under a holistic methodology. The proposed holistic methodology merges the concepts of failure mode and effects analysis (FMEA), fuzzy best-worst method (FBWM), and a rule-based three-hierarchical Bayesian network (RB-THBN). First, 15 root risks have been identified under a three-stage hierarchy with the support of laboratory decision makers. Then, questionnaires were created using the FMEA concept. The importance values (degree of belief-DoB) of three different FMEA parameters (severity, occurrence and detection) by experts for each risk have been calculated with FBWM. And finally, the RB-THBN has been created, and the final risk values have been computed using the linear utility function for the network. The system has been analyzed via a Bayesian modeling software. A validity test and a control measures planning has also been executed. According to the results, the highest root risk has been determined as “E2-The prepared mixture is not homogeneous” with a crisp risk score of 0.639559. In addition, “Failure due to extrusion” has been determined as the highest risk category with a score of 0.564059. It was discussed in detail that these risks arise from homogenization and how the used mixture should be homogenized. A comparative study among traditional FMEA, FMEA extended under FBWM and the proposed approach is performed to observe the differences in final ranking of root risks and to highlight the arguments that strengthen the advantages of the proposed method over the other two.

Optimal insurance design under mean-variance preference with narrow framing

In this paper, we study an optimal insurance design problem under mean-variance criterion by considering the local gain-loss utility of the net payoff of insurance, namely, narrow framing. We extend the existing results in the literature to the case where the decision maker has mean-variance preference with a constraint on the expected utility of the net payoff of insurance, where the premium is determined by the mean-variance premium principle. We first show the existence and uniqueness of the optimal solution to the main problem studied in the paper. We find that the optimal indemnity function involves a deductible provided that the safety loading imposed on the “mean part” of the premium principle is strictly positive. Our main result shows that narrow framing indeed reduces the demand for insurance. The explicit optimal indemnity functions are derived under two special local gain-loss utility functions – the quadratic utility function and the piecewise linear utility function. As a spin-off result, the Bowley solution is also derived for a Stackelberg game between the decision maker and the insurer under the quadratic local gain-loss utility function. Several numerical examples are presented to further analyze the effects of narrow framing on the optimal indemnity function as well as the interests of both parties.

A deep inverse reinforcement learning approach to route choice modeling with context-dependent rewards

Route choice modeling is a fundamental task in transportation planning and demand forecasting. Classical methods generally adopt the discrete choice model (DCM) framework with linear utility functions and high-level route characteristics. While several recent studies have started to explore the applicability of deep learning for route choice modeling, they are all path-based with relatively simple model architectures and require choice set generation. Existing link-based models can capture the dynamic nature of link choices within the trip without the need for choice set generation, but still assume linear relationships and link-additive features. To address these issues, this study proposes a general deep inverse reinforcement learning (IRL) framework for link-based route choice modeling, which is capable of incorporating diverse features (of the state, action and trip context) and capturing complex relationships. Specifically, we adapt an adversarial IRL model to the route choice problem for efficient estimation of context-dependent reward functions without value iteration. Experiment results based on taxi GPS data from Shanghai, China validate the superior prediction performance of the proposed model over conventional DCMs and other imitation learning baselines, even for destinations unseen in the training data. Further analysis shows that the model exhibits competitive computational efficiency and reasonable interpretability. The proposed methodology provides a new direction for future development of route choice models. It is general and can be adaptable to other route choice problems across different modes and networks.

Happiness maximizing sets under group fairness constraints

Finding a happiness maximizing set (HMS) from a database, i.e., selecting a small subset of tuples that preserves the best score with respect to any nonnegative linear utility function, is an important problem in multi-criteria decision-making. When an HMS is extracted from a set of individuals to assist data-driven algorithmic decisions such as hiring and admission, it is crucial to ensure that the HMS can fairly represent different groups of candidates without bias and discrimination. However, although the HMS problem was extensively studied in the database community, existing algorithms do not take group fairness into account and may provide solutions that under-represent some groups. In this paper, we propose and investigate a fair variant of HMS (FairHMS) that not only maximizes the minimum happiness ratio but also guarantees that the number of tuples chosen from each group falls within predefined lower and upper bounds. Similar to the vanilla HMS problem, we show that FairHMS is NP-hard in three and higher dimensions. Therefore, we first propose an exact interval cover-based algorithm called IntCov for FairHMS on two-dimensional databases. Then, we propose a bicriteria approximation algorithm called BiGreedy for FairHMS on multi-dimensional databases by transforming it into a submodular maximization problem under a matroid constraint. We also design an adaptive sampling strategy to improve the practical efficiency of BiGreedy. Extensive experiments on real-world and synthetic datasets confirm the efficacy and efficiency of our proposal.

Risk Aversion and a Calculus for Finitely Generated Piecewise Linear Functions: A Calculus that Economists Ought to Develop?

In this note we propose a calculus for piece-wise linear functions, in order to obtain derivatives and second derivatives at points where the function is not differentiable. Such derivatives can be used to calculate coefficients of risk aversion at initial wealth for piece-wise linear utility functions for gains, which display loss aversion-and hence non differentiability at zero gains.

Extending the Multiple Discrete Continuous (MDC) modelling framework to consider complementarity, substitution, and an unobserved budget

Many decisions can be represented as interrelated discrete and continuous choices, i.e. what and how much to choose from a set of finite alternatives (incidence and quantity of consumption). In the last twenty years, several models of Karush–Kuhn–Tucker demand systems have been developed and used to study these kinds of decisions. While strongly grounded in economic theory, most of these models have two limitations: they require specifying a budget, and usually omit any complementarity effects. In this paper, we propose two extensions to the Multiple Discrete Continuous (MDC) modelling framework: (i) an MDC model including explicit complementarity and substitution effects, and (ii) an MDC model with complementarity, substitution that requires no budget definition. Model (ii) relies on the hypothesis that total expenditure on the alternatives under consideration is small compared to the overall budget. This allows using a linear utility function for the numeraire good, leading to a likelihood function without the budget or numeraire good in it. The lack of a budget is specially useful when forecasting, as it avoids cascading errors due to an inaccurate budget specifications. The inclusion of complementarity and substitution effects enriches the interpretability of the models, while the resulting functional form avoids theoretical issues present in previous formulations. Alongside the derivation of the models, we discuss their main properties and propose an efficient forecasting algorithm for (ii). We also report four applications to datasets about time use, household expenditure, supermarket scanner data, and trip generation. Free estimation code for both models is made available online.

On two-player games with pure strategies on intervals $ [a, \; b] $ and comparisons with the two-player, two-strategy matrix case

<p style='text-indent:20px;'>We consider games of two-players with utility functions which are not necessarily linear on the product of convex and compact intervals of <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{R}^2 $\end{document}</tex-math></inline-formula>. An issue is how far an analogy can be drawn with two-player, two-strategy matrix games with linear utility functions, where [0, 1] registers probabilities and equilibria are at the intersection of reaction functions. Now, the idea of <inline-formula><tex-math id="M3">\begin{document}$ \delta $\end{document}</tex-math></inline-formula> functions is exploited to construct mixed strategies to look for Nash equilibria (NE). "Reaction" functions are constructed and results are obtained graphically. They are related to topological theorems on NE. The games chosen make specific points in relation to existence conditions and properties of solutions. It is a distinguishing feature that an interval [a, b] now registers both pure and mixed strategies. For NE a choice has to be justified. Also "reaction" functions are more complicated and their intersection does not guarantee an equilibrium.</p>

Funding public projects: A case for the Nash product rule

We study a mechanism design problem where a community of agents wishes to fund public projects via voluntary monetary contributions by the community members. This serves as a model for public expenditure without an exogenously available budget, such as participatory budgeting or voluntary tax programs, as well as donor coordination when interpreting charities as public projects and donations as contributions. Our aim is to identify a mutually beneficial distribution of the individual contributions. In the preference aggregation problem that we study, agents with linear utility functions over projects report the amount of their contribution, and the mechanism determines a socially optimal distribution of the money. We identify a specific mechanism—the Nash product rule—which picks the distribution that maximizes the product of the agents’ utilities. This rule is Pareto efficient and incentivizes agents to contribute their entire budget while spending each agent’s contribution only on projects the agent finds acceptable.

Input-dependent noise can explain magnitude-sensitivity in optimal value-based decision-making

AbstractRecent work has derived the optimal policy for two-alternative value-based decisions, in which decision-makers compare the subjective expected reward of two alternatives. Under specific task assumptions — such as linear utility, linear cost of time and constant processing noise — the optimal policy is implemented by a diffusion process in which parallel decision thresholds collapse over time as a function of prior knowledge about average reward across trials. This policy predicts that the decision dynamics of each trial are dominated by the difference in value between alternatives and are insensitive to the magnitude of the alternatives (i.e., their summed values). This prediction clashes with empirical evidence showing magnitude-sensitivity even in the case of equal alternatives, and with ecologically plausible accounts of decision making. Previous work has shown that relaxing assumptions about linear utility or linear time cost can give rise to optimal magnitude-sensitive policies. Here we question the assumption of constant processing noise, in favour of input-dependent noise. The neurally plausible assumption of input-dependent noise during evidence accumulation has received strong support from previous experimental and modelling work. We show that including input-dependent noise in the evidence accumulation process results in a magnitude-sensitive optimal policy for value-based decision-making, even in the case of a linear utility function and a linear cost of time, for both single (i.e., isolated) choices and sequences of choices in which decision-makers maximise reward rate. Compared to explanations that rely on non-linear utility functions and/or non-linear cost of time, our proposed account of magnitude-sensitive optimal decision-making provides a parsimonious explanation that bridges the gap between various task assumptions and between various types of decision making.

An interactive method for multi-criteria dispatching problems with unknown preference functions

This research investigates multi-criteria dispatching problems with unknown user preference functions to help decision makers (DMs) find solutions that meet their multi-criteria preferences. The explicit consideration of the following factors distinguish this research from previous literature: (1) dispatching management with three conflicting objectives, (2) nonlinear preference functions with unknown weights, (3) indecisive threshold that models the bounded rationality of DMs, and (4) near optimal solution through a small number of interactions. In a smart factory of Industry 3.5, the proposed interactive centroid method (ICM) enhances the efficiency of human–computer interaction and improves the quality of multi-criteria manufacturing decisions significantly. Our extensive numerical study reveals that the ICM algorithm finds near optimal solutions under linear or quadratic utility functions through less than a dozen interactions with DMs. The average gap between the ICM solutions and true optimal is merely 1.537% and 0.882% under quadratic and linear utility functions, respectively.

Deep Reinforcement Learning for Trading

In this article, the authors adopt deep reinforcement learning algorithms to design trading strategies for continuous futures contracts. Both discrete and continuous action spaces are considered, and volatility scaling is incorporated to create reward functions that scale trade positions based on market volatility. They test their algorithms on 50 very liquid futures contracts from 2011 to 2019 and investigate how performance varies across different asset classes, including commodities, equity indexes, fixed income, and foreign exchange markets. They compare their algorithms against classical time-series momentum strategies and show that their method outperforms such baseline models, delivering positive profits despite heavy transaction costs. The experiments show that the proposed algorithms can follow large market trends without changing positions and can also scale down, or hold, through consolidation periods. TOPICS:Futures and forward contracts, exchanges/markets/clearinghouses, statistical methods, simulations Key Findings • In this article, the authors introduce reinforcement learning algorithms to design trading strategies for futures contracts. They investigate both discrete and continuous action spaces and improve reward functions by using volatility scaling to scale trade positions based on market volatility. • The authors discuss the connection between modern portfolio theory and the reinforcement learning reward hypothesis and show that they are equivalent if a linear utility function is used. • The authors back test their methods on 50 very liquid futures contracts from 2011 to 2019, and their algorithms deliver positive profits despite heavy transaction costs.

An Optimization Approach Considering User Utility for the PV-Storage Charging Station Planning Process

Based on the comprehensive utilization of energy storage, photovoltaic power generation, and intelligent charging piles, photovoltaic (PV)-storage charging stations can provide green energy for electric vehicles (EVs), which can significantly improve the green level of the transportation industry. However, there are many challenges in the PV-storage charging station planning process, making it theoretically and practically significant to study approaches to planning. This paper promotes a bi-level optimization planning approach for PV-storage charging stations. First, taking PV-storage charging stations and EV users as the upper- and lower-level problems, respectively, during the planning process, a bi-level optimization model for PV-storage charging stations considering user utility is established for capacity allocation and user behavior-based electricity pricing. Second, the model is converted into a single-level mixed-integer linear programming model using the piecewise linear utility function, Karush–Kuhn–Tucker (KKT) conditions, and linearization methods. Finally, to verify the validity of the proposed model and the solution algorithm, a commercial solver is used to solve the optimization model and obtain the planning scheme. The results show that the proposed bi-level optimization model can provide a more economical and reasonable planning scheme than the single-level model, and can reduce the investment cost by 8.84%, operation and maintenance cost by 13.23%, and increase net revenue by 5.11%.

Optimal Insurance under Maxmin Expected Utility

We examine a problem of demand for insurance indemnification, when the insured is sensitive to ambiguity and behaves according to the Maxmin-Expected Utility model of Gilboa and Schmeidler (1989), whereas the insurer is a (risk-averse or risk-neutral) Expected-Utility maximizer. We characterize optimal indemnity functions both with and without the customary ex ante no-sabotage requirement on feasible indemnities, and for both concave and linear utility functions for the two agents. This allows us to provide a unifying framework in which we examine the effects of the no-sabotage condition, marginal utility of wealth, belief heterogeneity, as well as ambiguity (multiplicity of priors) on the structure of optimal indemnity functions. In particular, we show how the singularity in beliefs leads to an optimal indemnity function that involves full insurance on an event to which the insurer assigns zero probability, while the decision maker assigns a positive probability. We examine several illustrative examples, and we provide numerical studies for the case of a Wasserstein and a Renyi ambiguity set.

Equilibrium Joining Strategies in the Geo/GeoK/1 Queueing System

We study strategic behavior in the G e o / G e o K / 1 queueing system under both fully observable case and fully unobservable case. Furthermore, equilibrium and socially optimal strategies are obtained according to the available information and the linear utility function. We compare the impact of system parameters on the equilibrium strategies and socially optimal strategies. At the same time, we illustrate the effects of parameters on the obtained equilibrium social benefit. Finally, some numerical examples are presented.

DER Aggregator’s Data-Driven Bidding Strategy Using the Information Gap Decision Theory in a Non-Cooperative Electricity Market

When multiple distributed energy resource (DER) aggregators exist in a non-cooperative power market, the calculation of individual aggregator’s bidding strategies could encounter significant uncertainties for considering DERs and competing market participants’ bidding strategies. In this paper, a bi-level bidding strategy optimization model is proposed for a DER aggregator which utilizes wind power, energy storage system (ESS), and curtailable load. At the upper level, the designated aggregator’s bidding strategy is optimized considering the wind power uncertainty. The wind forecast error is modeled by an ambiguity set using the data-driven approach. The information gap decision theory (IGDT) method is employed in this paper to maximize the risk level the designated aggregator can bear for a certain level of expected payoff. By detecting the worst case in wind power generation, the upper-level model is linearized as an MILP. The designated aggregator submits its bids to the market using the linear utility function acquired from linear regression. At the lower level, the market clearing is carried out using competing market participants’ bidding strategy scenarios. The scenarios and the corresponding probability are modeled through a data-driven approach. The market clearing problem is linearized using Taylor series. The price signal is iterated between the two levels as the proposed bi-level model is solved. Numerical results prove the validity and effectiveness of the proposed IGDT-based method. It is shown that the aggregator can adjust either the bidding quantities or coefficients to reach an expected payoff level. The bidding strategies are affected by uncertainties of wind power and competing bidding strategies. For an expected payoff level, when the designated aggregator is posed to consider a higher risk of wind power uncertainty, the aggregator can only bear a lower risk level from competing bidding strategies and vice versa.

Topology design of remote patient monitoring system concerning qualitative and quantitative issues

Piecewise linear functions have been used to model and solve non-linear problems in science, social science, and technology fields since the 1950′s. Multi-choice goal programming (MCGP) has been widely used to solve multiple objective decision-making (MODM) problems in which each goal mapping with multiple aspiration levels is allowed, expanding the original feasible region to obtain better solutions in the MODM problems. This paper integrates the efficient S-shaped penalty method, arbitrary piecewise linear utility functions, trapezoidal utility functions, and MCGP to solve a topology design problem in a remote patient monitoring system (RPMS) by providing universal senior citizen coverage in which quantitative (e.g., patient satisfaction-related cost) and qualitative (e.g., satisfaction) issues are considered simultaneously. In addition, some novel utility functions such as the force utility function, indicator utility function, and arbitrary utility function are proposed to improve the usefulness of MCGP in the field of management science. These proposed utility functions can be easily used to model qualitative issues in real-world problems. A topology design problem for an RPMS is demonstrated to justify the feasibility, usefulness, and compatibility of the proposed methods. Further, sensitivity analysis and managerial implications are provided using an RPMS in Taiwan.