Linear Utility Research Articles

Bi-attribute utility preference robust optimization: A continuous piecewise linear approximation approach

In this paper, we consider a bi-attribute decision making problem where the decision maker’s (DM’s) objective is to maximize the expected utility of outcomes with two attributes but where the true utility function which captures the DM’s risk preference is ambiguous. To tackle this ambiguity, we propose a maximin bi-attribute utility preference robust optimization (BUPRO) model where the optimal decision is based on the worst-case utility function in an ambiguity set of plausible utility functions constructed using partially available information such as the DM’s specific preference for certain lotteries. Specifically, we consider a BUPRO model with two attributes, where the DM’s risk attitude is bivariate risk-averse and the ambiguity set is defined by a linear system of inequalities represented by the Lebesgue–Stieltjes integrals of the DM’s utility functions. To solve the inner infinite-dimensional minimization problem, we propose a continuous piecewise linear approximation approach to approximate the DM’s unknown true utility. Unlike the univariate case, we partition the domain of the utility function into a set of small non-overlapping rectangles and then divide each rectangle into two triangles by either the main diagonal (Type-1) or the counter diagonal (Type-2). The inner minimization problem based on the piecewise linear utility function can be reformulated as a mixed-integer linear program and the outer maximization problem can be solved efficiently by the derivative-free method. In the case that all the small triangles are partitioned either in Type-1 or in Type-2, the inner minimization can be formulated as a finite dimensional linear program and the overall maximin as a single mixed-integer program. To quantify the approximation errors, we derive, under some mild conditions, the error bound for the difference between the BUPRO model and the approximate BUPRO model in terms of the ambiguity set, the optimal value and the optimal solutions. Finally, we carry out some numerical tests to examine the performance of the proposed models and computational schemes. The results demonstrate the efficiency of the computational schemes and highlight the stability of the BUPRO model against data perturbations.

Non-parametric identification and testing of quantal response equilibrium

This paper studies the falsifiability and identification of Quantal Response Equilibrium (QRE) when each player's utility and error distribution are relaxed to be unknown non-parametric functions. Using the variation of players' choices across a series of games, we first show that both the utility function and the distribution of errors are non-parametrically over-identified. This over-identification result further suggests a straightforward testing procedure for QRE which achieves the desired type-1 error and maintains a small type-2 error. To apply this methodology, we conduct an experimental study of the matching pennies game. Our non-parametric estimates strongly reject the conventional Logit choice probability. Moreover, when the utility and the error distribution are sufficiently flexible and heterogeneous, the quantal response hypothesis cannot be rejected for 70% of participants. However, strong assumptions such as linear utility, logistically distributed errors, and homogeneity lead to substantially higher rejection rates.

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.

A Dynamic Kelly Criterion for Games with a Maximum Payout

In 1956, J.L. Kelly provided the formula for the optimal portion f of one’s capital to bet on a coin toss with favorable odds. We show that when the bet can be repeated a fixed number of times, but the game has a maximum payout M, the optimal portion f becomes a function of both k, the number of games remaining, and B, the current bankroll. We describe this function f k ( B ) and quantify the advantages gained over the fixed Kelly criterion. We also discuss the impact of changing the underlying utility function from a logarithmic utility to a more linear utility, and thereby optimizing a risk-reward ratio.

A statistical interpretation of a market demand curve for a commodity obeying the law of demand

In this note we provide a statistical interpretation of the Marshallian market demand curve of a commodity that obeys the law of demand and which has a finite and positive level of satiation. A consequence of our approach is that in the context of two goods, we are able to obtain demand functions which are very similar to those obtained by “budget-constrained Cobb–Douglas utility maximization”, but now as a result of a “budget-constrained linear utility maximization” exercise, although our budget constraint is “slightly different” from the one that would be used for the former optimization problem.

Budget-Constrained Linear Utility Maximization and the Hurwicz (1951) Criterion

We show that for a trader displaying state-dependent risk-neutrality, budget constrained maximization based on the Hurwicz criterion for “non-probabilistic” uncertainty with the degree of pessimism being less than half, reduces to expected utility maximization under “the equal ignorance principle”, with all wealth being invested in positive return being available in just one state of nature and nothing at all in other states of nature. Such a portfolio is an extreme point of the set of budget-constrained expected utility maximizing portfolios, with equiprobable states of nature. With more than two uncertain states of nature, the above result extends to the case where the degree of pessimism is equal to half. We also show that if for a degree of pessimism, greater than or equal to half, an optimal solution to budget constrained maximization based on the Hurwicz criterion is not of this type, then it must be a “budget-constrained max-min utility maximizing” portfolio.

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.

Competitive Equilibria with a Constant Number of Chores

We study markets with mixed manna, where m divisible goods and chores shall be divided among n agents to obtain a competitive equilibrium. Equilibrium allocations are known to satisfy many fairness and efficiency conditions. While a lot of recent work in fair division is restricted to linear utilities and chores, we focus on a substantial generalization to separable piecewise-linear and concave (SPLC) utilities and mixed manna. We first derive polynomial-time algorithms for markets with a constant number of items or a constant number of agents. Our main result is a polynomial-time algorithm for instances with a constant number of chores (as well as any number of goods and agents) under the condition that chores dominate the utility of the agents. Interestingly, this stands in contrast to the case when the goods dominate the agents utility in equilibrium, where the problem is known to be PPAD-hard even without chores.

The Complexity of Pacing for Second-Price Auctions

Budget constraints are ubiquitous in online advertisement auctions. To manage these constraints and smooth out the expenditure across auctions, the bidders (or the platform on behalf of them) often employ pacing: each bidder is assigned a pacing multiplier between zero and one, and her bid on each item is multiplicatively scaled down by the pacing multiplier. This naturally gives rise to a game in which each bidder strategically selects a multiplier. The appropriate notion of equilibrium in this game is known as a pacing equilibrium. In this work, we show that the problem of finding an approximate pacing equilibrium is PPAD-complete for second-price auctions. This resolves an open question of Conitzer et al. [Conitzer V, Kroer C, Sodomka E, Stier-Moses NE (2022a) Multiplicative pacing equilibria in auction markets. Oper. Res. 70(2):963–989]. As a consequence of our hardness result, we show that the tâtonnement-style budget-management dynamics introduced by Borgs et al. [Borgs C, Chayes J, Immorlica N, Jain K, Etesami O, Mahdian M (2007) Dynamics of bid optimization in online advertisement auctions. Proc. 16th Internat. Conf. World Wide Web (ACM, New York), 531–540] are unlikely to converge efficiently for repeated second-price auctions. This disproves a conjecture by Borgs et al. [Borgs C, Chayes J, Immorlica N, Jain K, Etesami O, Mahdian M (2007) Dynamics of bid optimization in online advertisement auctions. Proc. 16th Internat. Conf. World Wide Web (ACM, New York), 531–540], under the assumption that the complexity class PPAD is not equal to P. Our hardness result also implies the existence of a refinement of supply-aware market equilibria which is hard to compute with simple linear utilities. Funding: This work was supported by National Science Foundation (CCF-1703925, IIS-1838154).

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.

On the functional equivalence of two perfectly competitive economies with negative exponential utility and linear utility with a quadratic holding cost

On the functional equivalence of two perfectly competitive economies with negative exponential utility and linear utility with a quadratic holding cost

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.

Exploring the Nonlinearity of Commute-Time Utility Considering Individual Ideal Preferences and Tolerance ThresholdsC: ase Study in Kunming

Exploring the nonlinearity of commute-time utility is useful in predicting urban travel demand. However, existing studies assume that commute utility decreases linearly with commute time. This ignores the influence of commuters’ ideal preferences and tolerance thresholds on commute utility. To reveal the nonlinear variation of commute utility with commute time and its influence on the choice of commute mode, the ideal commute time (ICT) and the tolerance threshold for commute time (TTCT) were introduced. Three-piecewise linear utility models (Models 2, 3, 4) were constructed and compared with the linear utility multinomial logit model (Model 1). The results of empirical study showed that: (a) the goodness of fit of these three modified models is higher than that of Model 1, indicating that the fitting effect of the commute-time utility model can be improved with either or both the ICT and TTCT; (b) there is a nonlinear relationship between the commute utility and the commute time. The commute-time utility decreases slowly before ICT, declines steadily between the ICT and the TTCT, and falls significantly after TTCT; and (c) when the commute time exceeds the TTCT, the perceived utility of commuters traveling by walking or cycling decreases significantly, and there are few changes in the perceived utility of commuters by car, which increases the probability that commuters who use active modes will transfer to commuting by car. The research results have implications for improving the prediction capacity of commute mode choice model and could guide commuters to switch to more sustainable commute modes.

Functional Equivalence Between Two Competitive Economies with Negative Exponential Utility and Linear Utility with a Quadratic Holding Cost

Functional Equivalence Between Two Competitive Economies with Negative Exponential Utility and Linear Utility with a Quadratic Holding Cost

Discrete Choice Models with Piecewise Linear Utility: Modeling, Estimation and Pricing

Discrete Choice Models with Piecewise Linear Utility: Modeling, Estimation and Pricing

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.

A Complementary Pivot Algorithm for Competitive Allocation of a Mixed Manna

We study the fair division problem of allocating a mixed manna under additively separable piecewise linear concave (SPLC) utilities. A mixed manna contains goods that everyone likes and bads (chores) that everyone dislikes as well as items that some like and others dislike. The seminal work of Bogomolnaia et al. argues why allocating a mixed manna is genuinely more complicated than a good or a bad manna and why competitive equilibrium is the best mechanism. It also provides the existence of equilibrium and establishes its distinctive properties (e.g., nonconvex and disconnected set of equilibria even under linear utilities) but leaves the problem of computing an equilibrium open. Our main results are a linear complementarity problem formulation that captures all competitive equilibria of a mixed manna under SPLC utilities (a strict generalization of linear) and a complementary pivot algorithm based on Lemke’s scheme for finding one. Experimental results on randomly generated instances suggest that our algorithm is fast in practice. Given the [Formula: see text]-hardness of the problem, designing such an algorithm is the only non–brute force (nonenumerative) option known; for example, the classic Lemke–Howson algorithm for computing a Nash equilibrium in a two-player game is still one of the most widely used algorithms in practice. Our algorithm also yields several new structural properties as simple corollaries. We obtain a (constructive) proof of existence for a far more general setting, membership of the problem in [Formula: see text], a rational-valued solution, and an odd number of solutions property. The last property also settles the conjecture of Bogomolnaia et al. in the affirmative. Furthermore, we show that, if the number of either agents or items is a constant, then the number of pivots in our algorithm is strongly polynomial when the mixed manna contains all bads. Funding: Financial support from the Division of Computing and Communication Foundations, National Science Foundation (NSF) [Grants 1942321, 1750436] is gratefully acknowledged.