Linear Utility Functions Research Articles

Real-Time Pricing of Smart Grid Based on Piece-Wise Linear Functions

Abstract In a power grid system, utility is a measure of the satisfaction of users’ electricity consumption; cost is a monetary value of electricity generated by the supplier. The utility and cost functions represent the satisfaction of different users and the supplier. Quadratic utility, logarithmic utility, and quadratic cost functions are widely used in social welfare maximization models of real-time pricing. These functions are not universal; they have to be discussed in detail for individual models. To overcome this problem, a piece-wise linear utility function and a piece-wise linear cost function with general properties are proposed in this paper. By smoothing the piece-wise linear utility and cost functions, a social welfare maximization model can be transformed into a differentiable convex optimization problem. A dual optimization method is used to solve the smoothed model. Through mathematical deduction and numerical simulations, the rationality of the model and the validity of the algorithm are verified as long as the elastic and cost coefficients take appropriate values. Thus, different user types and the supplier can be determined by selecting different elastic and cost coefficients.

Discrete-time mean-CVaR portfolio selection and time-consistency induced term structure of the CVaR

We investigate a discrete-time mean-risk portfolio selection problem, where risk is measured by the conditional value-at-risk (CVaR). A substantial challenge is the combination of a time-inconsistent objective with an incomplete and dynamic model for the financial market. We are able to solve this problem analytically by embedding the original, time-inconsistent problem into a family of time-consistent expected utility maximization problems with a piecewise linear utility function. The optimal investment strategy is a fully adaptive feedback policy and the cumulated amount invested in the risky assets is of a characteristic V-shaped pattern as a function of the current wealth. For the incomplete, discrete-time market considered herein, the mean-CVaR efficient frontier is a straight line in the mean-CVaR plane and thus economically meaningful. This contrasts the complete, continuous-time setting where the mean-CVaR efficient frontier is degenerate or does not exist at all. We further solve an inverse investment problem, where we investigate how mean-CVaR preferences need to adapt in order for the pre-committed optimal strategy to remain optimal at any point in time. Our result shows that a pre-committed mean-CVaR investor behaves like a naive mean-CVaR investor with a time-increasing confidence level for the CVaR, who revises the investment decision at every point in time. Finally, an empirical application of our results suggests that risk measured by the CVaR might help to understand the long-standing equity premium puzzle.

Consensus-Based Distributed Discrete Optimal Transport for Decentralized Resource Matching

Optimal transport has been extensively used in resource matching to promote the efficiency of resource usages by matching sources to targets. However, it requires a significant amount of computations and storage spaces for large-scale problems. In this paper, we take a consensus-based approach to decentralize discrete optimal transport problems and develop fully distributed algorithms with alternating direction method of multipliers. We show that our algorithms guarantee certain levels of efficiency and privacy besides the distributed nature. We further derive primal and dual algorithms by exploring the primal and dual problems of discrete optimal transport with linear utility functions and prove the equivalence between them. We verify the convergence, online adaptability, and the equivalence between the primal algorithm and the dual algorithm with numerical experiments. Our algorithms reflect the bargaining between sources and targets on the amounts and prices of transferred resources and reveal an averaging principle which can be used to regulate resource markets and improve resource efficiency.

Active Preference Learning Based on Generalized Gini Functions: Application to the Multiagent Knapsack Problem

We consider the problem of actively eliciting preferences from a Decision Maker supervising a collective decision process in the context of fair multiagent combinatorial optimization. Individual preferences are supposed to be known and represented by linear utility functions defined on a combinatorial domain and the social utility is defined as a generalized Gini Social evaluation Function (GSF) for the sake of fairness. The GSF is a non-linear aggregation function parameterized by weighting coefficients which allow a fine control of the equity requirement in the aggregation of individual utilities. The paper focuses on the elicitation of these weights by active learning in the context of the fair multiagent knapsack problem. We introduce and compare several incremental decision procedures interleaving an adaptive preference elicitation procedure with a combinatorial optimization algorithm to determine a GSF-optimal solution. We establish an upper bound on the number of queries and provide numerical tests to show the efficiency of the proposed approach.

Dynamic Credit Quality Evaluation with Social Network Data

We investigate the filtering problem where the borrower’s time varying credit quality process is estimated using continuous time observation process and her (in this paper we refer to the borrower as female and the lender as male) ego-network data. The hidden credit quality is modeled as a hidden Gaussian mean-reverting process whilst the social network is modeled as a continuous time latent space network model. At discrete times, the network data provides unbiased estimates of the current credit state of the borrower and her ego-network. Combining the continuous time observed behavioral data and network information, we provide filter equations for the hidden credit quality and show how the network information reduces information asymmetry between the borrower and the lender. Further, we consider the case when the network information arrival times are random and solve stochastic optimal control problem for a lender having linear quadratic utility function.

The semi-martingale equilibrium equity premium for risk-neutral investors

In this paper, we study the risk-neutral investor’s equilibrium equity premium in a semi-martingale market with arbitrary, normal, binomial and gamma jumps. We simulate graphs for discrete distribution of jump amplitudes in order to study the parameter effect. The equity premium for this investor remains the same regardless of [Formula: see text] and [Formula: see text] variations in the linear utility function. In fact, there is no optimal consumption for [Formula: see text]. For normal jumps, our results are consistent with the risk-averse investor’s power utility effect on the equity premium. However, the binomial and gamma amplitudes show significant variations between risk neutrality and risk aversion.

A TEORIA DO PROSPECTO: ESTIMAÇÃO DA FUNÇÃO UTILIDADE E DA FUNÇÃO PONDERAÇÃO DAS PROBABILIDADES PARA UMA AMOSTRA ESPECÍFICA

A partir da teoria do prospecto de Kahneman e Tversky (1979), este artigo investiga a função utilidade e a função ponderação das probabilidades de economistas da Pontifícia Universidade Católica do Rio Grande do Sul (PUCRS) e, portanto, seu comportamento perante o risco, comparando-os aos resultados da literatura. Para tanto, é realizado um experimento com alunos e professores da graduação e pós-graduação da PUCRS usando um método de aplicação baseado em Gonzalez e Wu (1999) e parcialmente novo, o que permite minimizar efeitos de ordem e efeitos de escala. De posse das respostas, são estimados os parâmetros da função utilidade e da função de ponderação das probabilidades de cada participante. Os resultados encontrados mostram que os economistas da PUCRS apresentam um comportamento distinto da média encontrada na literatura econômica: possuem uma função utilidade mais linear e uma discriminação das probabilidades menor que os resultados da literatura. A extensão dos resultados sugere a existência de relevantes diferenças de comportamento dentro do grupo pesquisado de acordo com suas características demográficas. Além da obtenção dos resultados finais, este artigo tem o objetivo adicional de contribuir para o debate em uma área de importância crescente e ainda pouco desenvolvida no Brasil.

A new flexible multiple discrete–continuous extreme value (MDCEV) choice model

Traditional multiple discrete–continuous (MDC) models generally predict the continuous consumption quantity component reasonably component well, but not necessarily the discrete choice component. In this paper, we propose, for the first time, a new flexible closed-form MDCEV model that breaks the tight linkage between the discrete and continuous choice dimensions of the traditional MDC models. We do so by (1) employing a linear utility function of consumption for the first outside good (which removes the continuous consumption quantity of the outside good from the discrete consumption decision, and also helps in forecasting), and (2) using separate baseline utilities for the discrete and continuous consumption decisions. In the process of proposing our new formulation, we also revisit two issues related to the traditional MDC model. The first relates to clarification regarding the identification of the scale parameter of the error terms, and the second relates to the probability of the discrete choice component of the traditional MDC model (that is, the multivariate probability of consumption or not of the alternatives). We show why the scale parameter of the error terms is indeed estimable when a γ-profile is used, as well as show how one may develop a closed-form expression for the discrete choice consumption probability. The latter contribution also presents a methodology to estimate pure multiple discrete choice models without the need for information on the continuous consumptions. Finally, we also develop forecasting procedures for our new MDC model structure.We demonstrate an application of the proposed model to the case of time-use of individuals. In a comparative empirical assessment of the fit from the proposed model and from the traditional MDCEV models, our proposed model performs better in terms of better predicting both the discrete outcome data as well as the continuous consumptions.

Discrete-Time Mean-CVaR Portfolio Selection and Time-Consistency Induced Term Structure of the CVaR

We investigate a discrete-time mean-risk portfolio selection problem, where risk is measured by the conditional value-at-risk (CVaR). By embedding this time-inconsistent problem into a family of expected utility maximization problems with a piecewise linear utility function, we solve the problem analytically. In contrast to the case of a complete, continuous-time market, the mean-CVaR efficient frontier in this generally incomplete, discrete-time setting is a straight line in the mean-CVaR plane and there is in particular a constant trade-off between risk and return. The cumulated amount invested in the risky assets under the optimal strategy is of a V -shaped pattern as a function of the current wealth. We further solve an inverse investment problem, where we investigate how mean-CVaR preferences need to adapt such that the pre-commited optimal strategy remains optimal at any point in time. Our result shows that, although conceptually distinct, a pre-commited mean-CVaR investor behaves like a naive mean-CVaR investor with a time-increasing confidence level for the CVaR, who revises her investment decision at every point in time. Finally, an empirical application of our results suggests that risk measured by the CVaR might help to understand the long-standing equity premium puzzle.

Optimal Trading Strategies within the Hamilton-Jacobi-Bellman Equation - An Application to Statistical Arbitrage

We propose optimal trading strategies based on the Hamilton-Jacobi-Bellman equation when a trader is allowed to place only market orders with a particular focus on a statistical arbitrage. A process that a placed limit order is filled is often modeled as a stochastic point process. In case of a market order, however, it is filled with a hundred percent probability meaning that it is not a stochastic process and is rather a deterministic one. They are therefore represented using the Heaviside step function and their infinitesimal generators are just time derivative and give delta functions. An integral linear utility function with an inventory penalty and a discount enables us to obtain optimal number of market order with a simple calculus and those in a limit T → ∞, i.e. a stationary limit. We show that, at least within our formulation, we cannot generate a positive PnL by trading multiple set of assets whose noises are correlated but their drifts are zero which contradicts the common statement that Given correlated multiple assets, we can exploit those correlations by trading an appropriate linear combination of However, we show that if multiple assets are cointegrated, we can create a mean-reverting portfolio and exploit them. In that case, an optimal number of market order which takes current status of market as well as a trader's preference into account is given in a simple but intuitive and reasonable form and it does generate positive PnL while controlling risk of the portfolio.

Supply Chain Contracting with Linear Utility Function

The supply chain contracting has traditionally been based on the profit maximization assumption. Recent research has shown that some behavior factors may influence the decision making of supply chain members. The authors utilize a linear utility function to depict such behavior factors and incorporate these into the newsvendor model. The linear utility function provides sufficient flexibility to better capture people's various behavior factors. By supposing the agents are concerned with behavior factors, the authors first investigate how the factors affect the supply chain under wholesale price contract, and find that they do not influence coordination condition, but can adjust the distribution of profits. Then they extend their study to other four common contracts with a similar method and systematically demonstrate that the behavior of agents in such a linear setting has no effect on the conditions of coordinating supply chain.

DEA Efficiency of Energy Consumption in China’s Manufacturing Sectors with Environmental Regulation Policy Constraints

Because overall energy consumption intensity in China’s manufacturing industry is extremely high, the study of energy efficiency in that industry, with an analysis of the policy impacts of energy intensity reduction and other key factors, will no doubt improve energy utilization in the industry and stimulate sustainable development within it. This paper uses 2004–2014 panel data of 28 manufacturing industries and a piecewise linear utility function to construct a data envelopment analysis (DEA) model of energy consumption with environmental regulations constraints. We also examine the DEA evaluation of energy efficiency in manufacturing industries. We integrate environmental regulations as qualitative variables into the energy consumption evaluation model to research the coupling effects on energy consumption intensity of energy consumption structure, opening up, environmental regulations, technological progress, and competition within industries. The research shows that energy efficiency policy intensity is not the major effect on the development of low or moderate energy-consumption industries, whereas low-energy-efficiency policy is very favorable for the development of high energy-consumption industries.

Time and Location Aware Mobile Data Pricing

Mobile users' correlated mobility and data consumption patterns often lead to severe cellular network congestion in peak hours and hot spots. This paper presents an optimal design of time and location aware mobile data pricing, which incentivizes users to smooth traffic and reduce network congestion. We derive the optimal pricing scheme through analyzing a two-stage decision process, where the operator determines the time and location aware prices by minimizing his total cost in Stage I, and each mobile user schedules his mobile traffic by maximizing his payoff (i.e., utility minus payment) in Stage II. We formulate the two-stage decision problem as a bilevel optimization problem, and propose a derivative-free algorithm to solve the problem for any increasing concave user utility functions. We further develop low complexity algorithms for the commonly used logarithmic and linear utility functions. The optimal pricing scheme ensures a win-win situation for the operator and users. Simulations show that the operator can reduce the cost by up to 97.52% in the logarithmic utility case and 98.70% in the linear utility case, and users can increase their payoff by up to 79.69% and 106.10% for the two types of utilities, respectively, comparing with a time and location independent pricing benchmark. Our study suggests that the operator should provide price discounts at less crowded time slots and locations, and the discounts need to be significant when the operator's cost of provisioning excessive traffic is high or users' willingness to delay traffic is low.

A two-resource allocation algorithm with an application to large-scale zero-sum defensive games

This paper investigates efficient computation schemes for allocating two defensive resources to multiple sites to protect against possible attacks by an adversary. The availability of the two resources is constrained and the effectiveness of each may vary over the sites. The problem is formulated as a two-person zero-sum game with particular piecewise linear utility functions: the expected damage to a site that is attacked linearly decreases in the allocated resource amounts up to a point that a site is fully protected. The utility of the attacker, equivalently the defender's disutility, is the total expected damage over all sites. A fast algorithm is devised for computing the game's Nash equilibria; it is shown to be more efficient in practice than both general purpose linear programming solvers and a specialized method developed in the mid-1980s. To develop the algorithm, optimal solution properties are explored.

Bilateral learning model in construction claim negotiations

Purpose – The learning ability on critical bargaining information contributes to accelerating construction claim negotiations in the win-win situation. The purpose of this paper is to study how to apply Zeuthen strategy and Bayesian learning to simulate the dynamic bargaining process of claim negotiations with the consideration of discount factor and risk attitude. Design/methodology/approach – The authors first adopted certainty equivalent method and curve fitting to build a party’s own curve utility function. Taking the opponent’s bottom line as the learning goal, the authors introduced Bayesian learning to refine former predicted linear utility function of the opponent according to every new counteroffer. Both parties’ utility functions were revised by taking discount factors into consideration. Accordingly, the authors developed a bilateral learning model in construction claim negotiations based on Zeuthen strategy. Findings – The consistency of Zeuthen strategy and the Nash bargaining solution model guarantees the effectiveness of the bilateral learning model. Moreover, the illustrative example verifies the feasibility of this model. Research limitations/implications – As the authors developed the bilateral learning model by mathematical deduction, scholars are expected to collect empirical cases and compare actual solutions and model solutions in order to modify the model in future studies. Practical implications – Negotiators could refer to this model to make offers dynamically, which is favorable for the parties to reach an agreement quickly and to avoid the escalation of claims into disputes. Originality/value – The proposed model provides a supplement to the existing studies on dynamic construction claim negotiations.

A hybrid choice model with a nonlinear utility function and bounded distribution for latent variables: application to purchase intention decisions of electric cars

ABSTRACTThe hybrid choice model (HCM) provides a powerful framework to account for heterogeneity across decision-makers in terms of different underlying latent attitudes. Typically, effects of the latent attitudes have been represented assuming linear utility functions. In contributing to the further elaboration of HCMs, this study suggests an extended HCM framework allowing for nonlinear utility functions of choice alternatives including not only observed but also latent variables. Box–Cox transformations are used to represent the nonlinear utility function. Johnson’s SB distribution is suggested to represent the random term of the latent variables, satisfying the constraint of the Box–Cox transformation. An empirical study using stated choice data about the intention to purchase electric cars is conducted. The empirical results show that the proposed framework can capture nonlinear effects of underlying variables including latent attitudes, thereby enhancing the explanatory power of the choice model.

Knut Wicksell on Utility and Market Aggregation

Knut Wicksell discussed in his 1893 Value, Capital, and Rent, as part of a critical comment on Launhardt and Jevons, conditions for consistent aggregation. He argued that Jevons's “trading body” concept was only valid if identical linear marginal utility functions (with ensuing parallel Engel lines for all individuals) could be assumed, which he regarded as unrealistic. Wicksell preferred Léon Walras's notion of market excess aggregate demand functions, which are functions of prices only. Such functions would not be affected by aggregation problems, if only disequilibrium transactions were assumed away. The article also addresses Wicksell's position concerning the so-called Cournot problem (that the general equilibrium system may not be able to provide precise numerical solutions), and his use of the notion of the representative agent in the formulation of the saving function.

DO INDUSTRY AND ACADEMIC EXPERTS DIFFER IN WEIGHTING CRITERIA FOR AGRICULTURAL PLANNING AND IN THEIR LOSS AVERSION?

<p>Agricultural planning models account for changing weather conditions and for high food price volatility. Applying the criteria of maximum farmer’s profit in various years enables the use of multi-criteria techniques. This paper presents the application of the Analytical Hierarchy Process (AHP) to estimate the importance of these criteria. The study contributes to the literature by a) addressing expert opinions on the importance of the criteria of profit in normal, dry years, and in years when agricultural prices rise; b) examining differences in weighting criteria between industry and academic experts; c) taking into account the loss averse behavior for these two groups of experts. Industry and academic experts were interviewed in the agricultural region of North-Eastern Israel. Changes in profit are approximated by a linear utility function with a positive slope (for both losses and gains) which is steeper for losses (profit in dry years) than for gains (profit in years when prices rise). The AHP method allows for the identification of the importance of the criteria for all respondents as a whole and for academic experts as compared to industrial experts. The share of loss averse respondents and coefficients of loss aversion are identified for both groups of experts. These coefficients are used for explaining differences between industry and academic expert opinions in estimating the criteria importance.</p>

An Expressive Mechanism for Auctions on the Web

Auctions are widely used on the Web. Applications range from sponsored search to platforms such as eBay. In these and in many other applications the auctions in use are single-/multi-item auctions with unit demand. The main drawback of standard mechanisms for this type of auctions, such as VCG and GSP, is the limited expressiveness that they offer to the bidders. The General Auction Mechanism (GAM) of Aggarwal et al. [2009] takes a first step toward addressing the problem of limited expressiveness by computing a bidder optimal, envy-free outcome for linear utility functions with identical slopes and a single discontinuity per bidder-item pair. We show that in many practical situations this does not suffice to adequately model the preferences of the bidders, and we overcome this problem by presenting the first mechanism for piecewise linear utility functions with nonidentical slopes and multiple discontinuities . Our mechanism runs in polynomial time. Like GAM it is incentive compatible for inputs that fulfill a certain nondegeneracy assumption, but our requirement is more general than the requirement of GAM. For discontinuous utility functions that are nondegenerate as well as for continuous utility functions the outcome of our mechanism is a competitive equilibrium . We also show how our mechanism can be used to compute approximately bidder optimal, envy-free outcomes for a general class of continuous utility functions via piecewise linear approximation. Finally, we prove hardness results for even more expressive settings.

Exponential utility functions aid upstream decision making

Although the concepts and mathematics of utility theory and its application to adjusting valuations to reflect the perspectives of decision makers with a range of risk preferences have been established for decades, these concepts and numerical applications remain relatively rarely applied by decision makers in the upstream gas and oil industries. Utility functions are now extensively used to assist evaluation of oil and gas hedging and trading of financial and physical commodities from the risk preferences of the parties involved. This study makes the case for more extensive use of utility functions in the upstream gas and oil sectors by presenting cases that highlight both the conceptual and valuation benefits that result from their application.Exponential utility functions adequately describe the risk preferences of risk-averse and risk-prone decision makers for a wide range of upstream gas and oil asset types and circumstances. Simple equations for the calculation of utility factors and expected utility factors, i.e., taking into account probabilities of a range of outcomes being realised, are presented and compared with the equivalent linear utility functions of a risk-neutral investor valuing assets based on unrisked discounted cash flow (i.e. net present value, NPV) and risked discounted cash flow (i.e., expected monetary value, EMV). The additional insight gained from applying utility functions is considered with examples for high-uncertainty exploration assets, decision makers constrained by various loss tolerances and selection of optimum gas field development plans from a number of distinct alternative plans. In all cases considered the utility functions provide decision makers with greater insight than just the consideration of NPV and/or EMV. A case is therefore made to justify more extensive use of utility functions by upstream decision makers.