Concave Utility Research Articles

Pairwise Stability in Weighted Network Formation Games: Selection and Computation

This paper is concerned with the selection and computation of pairwise stable networks when agents have differentiable and concave utility functions. We show that a pairwise stable network can be obtained by finding a Nash equilibrium of a noncooperative game played by the nodes and links in the network. Based on this observation, we introduce a logarithmic tracing procedure and a path-following algorithm for network formation games. We apply the algorithm to several models in the literature and make comparisons with two existing algorithms: a path-following algorithm based on the linear tracing procedure (LinTP) and a decompose and exhaustive search method (DaE). Numerical results indicate that the proposed method is more than four times as efficient as LinTP. Although DaE demonstrates exceptional efficiency for small-scale problems, our method outperforms it significantly for large-scale problems, where DaE may fail to find a solution. We also show that the decomposition technique of DaE can be used to further accelerate our algorithm for a special class of problems. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. Funding: This work was supported in part by the National Natural Science Foundation of China [Grants 12201289, 12122107, and 72394363/72394360], the Natural Science Foundation of Jiangsu Province [Grant BK20220754], the Guangdong Basic and Applied Basic Research Foundation [Grant 2021A1515110207], the Young Elite Scientists Sponsorship Program by CAST [Grant 2023QNRC001], and the Open Research Fund from the Guangdong Provincial Key Laboratory of Big Data Computing [Grant B10120210117-OF05]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0546 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0546 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Approximate utility

Approximating arbitrage or a perfect hedge in a limit implies approximation in utility (formally, expected utilities are continuous in L2), if and only if the utility function is bounded above and below by quadratics. We characterize when some special utility functions are continuous for all random consumptions with a common lower bound. Only linear, quadratic, and, in some cases, translated SAHARA utilities are continuous if there is no lower bound. We also characterize when a utility function defined on a convex subset can be extended to an L2-continuous nondecreasing and concave utility function on all of R.

Joint optimization of application placement and resource allocation for enhanced performance in heterogeneous multi-server systems

Efficiently placing applications remains a critical challenge across diverse multi-server environments, including web hosting centers, cloud computing, and edge computing environments. Unfortunately, most existing studies tend to overlook the crucial aspect of resource allocation, leading to suboptimal system performance. To address this gap, there is a pressing need to holistically explore both application placement and resource allocation in a unified manner. In this paper, we introduce the place and allocate problem in heterogeneous multi-server systems, a novel approach aiming at simultaneously optimizing the placement and allocation of applications to maximize the overall utility of the heterogeneous multi-server system. Our proposed methodology harnesses the interplay between application placement and resource allocation, showcasing substantial improvements in system utility. To model individual application performance concerning their allocated resources, we employ utility functions. For concave utility functions, we present an approximation algorithm that operates efficiently with a time complexity of O(mn3(logC)2), where n represents the number of applications, m is the number of servers, and C denotes the maximum available resource capacity of each server. Furthermore, we extend our approach to accommodate more general scenarios that involve applications with nonconcave utility functions and using multiple types of resources. Our study includes comprehensive experimental evaluations conducted on applications with both synthetic and real-world utility functions. Results consistently showcase that our algorithms achieve over 96.9% of optimal performance on average. Additionally, comparative analysis against several practical heuristics reveal that our algorithms outperform these methods by up to 4.3 times in total utility.

Teamwork Reinforcement Learning With Concave Utilities

Teamwork Reinforcement Learning With Concave Utilities

Optimal asset allocation for DC pension subject to allocation and terminal wealth constraints under a remuneration scheme

. We investigate the optimal investment problem faced by a defined contribution (DC) pension fund manager under simultaneous allocation and expected shortfall (ES) constraints. Under a non concave utility, a Value-at-Risk (VaR) constraint does not lead to the full prevention of moral hazard. As a widely employed risk management tool, whether an ES constraint can provide a more effective protection than a VaR constraint has been a focus point of research. We apply a dual control approach and a concavification technique to solve the ES-constrained optimization problem for a DC pension plan under an incentive scheme and derive the closed-form representations of the optimal wealth and portfolio processes. Furthermore, we compare the effect of an ES constraint on the optimal investment behavior with that under a VaR constraint in the presence of an option-like scheme for the DC pension members. Theoretical and numerical results show that for a relatively low protection level, a joint VaR and an ES constraints induce the same structure of the optimal solution, which implies that for a non concave optimization problem, the ES-based risk management has lost its advantage over the VaR-based risk management. Therefore, it needs to design a more efficient risk measure to improve the risk management for a DC pension plan.

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 structure of representative preference

In an environment of Samuelsonian aggregation, we characterize those social welfare functions which (i) map concave utility profiles to concave representative utility functions and (ii) map quasiconcave utility profiles to quasiconcave representative utility functions. Case (i) holds for any concave social welfare function, while case (ii) holds only for generalized maxmin social welfare functions. Lastly, we establish a simple duality result for computing the representative utility under a maxmin social welfare function: the representative expenditure function is the sum of the individual expenditure functions, which we use to study the aggregation of generalized Leontief utilities.

Individual risk attitudes arise from noise in neurocognitive magnitude representations.

Humans are generally risk averse, preferring smaller certain over larger uncertain outcomes. Economic theories usually explain this by assuming concave utility functions. Here, we provide evidence that risk aversion can also arise from relative underestimation of larger monetary payoffs, a perceptual bias rooted in the noisy logarithmic coding of numerical magnitudes. We confirmed this with psychophysics and functional magnetic resonance imaging, by measuring behavioural and neural acuity of magnitude representations during a magnitude perception task and relating these measures to risk attitudes during separate risky financial decisions. Computational modelling indicated that participants use similar mental magnitude representations in both tasks, with correlated precision across perceptual and risky choices. Participants with more precise magnitude representations in parietal cortex showed less variable behaviour and less risk aversion. Our results highlight that at least some individual characteristics of economic behaviour can reflect capacity limitations in perceptual processing rather than processes that assign subjective values to monetary outcomes.

Fisher Markets with Social Influence

A Fisher market is an economic model of buyer and seller interactions in which each buyer’s utility depends only on the bundle of goods she obtains. Many people’s interests, however, are affected by their social interactions with others. In this paper, we introduce a generalization of Fisher markets, namely influence Fisher markets, which captures the impact of social influence on buyers’ utilities. We show that competitive equilibria in influence Fisher markets correspond to generalized Nash equilibria in an associated pseudo-game, which implies the existence of competitive equilibria in all influence Fisher markets with continuous and concave utility functions. We then construct a monotone pseudo-game, whose variational equilibria and their duals together characterize competitive equilibria in influence Fisher markets with continuous, jointly concave, and homogeneous utility functions. This observation implies that competitive equilibria in these markets can be computed in polynomial time under standard smoothness assumptions on the utility functions. The dual of this second pseudo-game enables us to interpret the competitive equilibria of influence CCH Fisher markets as the solutions to a system of simultaneous Stackelberg games. Finally, we derive a novel first-order method that solves this Stackelberg system in polynomial time, prove that it is equivalent to computing competitive equilibrium prices via tâtonnement, and run experiments that confirm our theoretical results.

FIXP-Membership via Convex Optimization: Games, Cakes, and Markets

We introduce a new technique for proving membership of problems in FIXP - the class capturing the complexity of computing a fixed-point of an algebraic circuit. Our technique constructs a "pseudogate" which can be used as a black box when building FIXP circuits. This pseudogate, which we term the "OPT-gate", can solve most convex optimization problems. Using the OPT-gate, we prove new FIXP-membership results, and we generalize and simplify several known results from the literature on fair division, game theory and competitive markets. In particular, we prove complexity results for two classic problems: computing a market equilibrium in the Arrow-Debreu model with general concave utilities is in FIXP, and computing an envy-free division of a cake with very general valuations is FIXP-complete. We further showcase the wide applicability of our technique, by using it to obtain simplified proofs and extensions of known FIXP-membership results for equilibrium computation for various types of strategic games, as well as the pseudomarket mechanism of Hylland and Zeckhauser.

Optimal allocations with α‐MaxMin utilities, Choquet expected utilities, and prospect theory

The analysis of optimal risk sharing has been thus far largely restricted to nonexpected utility models with concave utility functions, where concavity is an expression of ambiguity aversion and/or risk aversion. This paper extends the analysis to α‐maxmin expected utility, Choquet expected utility, and cumulative prospect theory, which accommodate ambiguity seeking and risk seeking attitudes. We introduce a novel methodology of quasidifferential calculus of Demyanov and Rubinov (1986, 1992) and argue that it is particularly well suited for the analysis of these three classes of utility functions, which are neither concave nor differentiable. We provide characterizations of quasidifferentials of these utility functions, derive first‐order conditions for Pareto optimal allocations under uncertainty, and analyze implications of these conditions for risk sharing with and without aggregate risk.

Utility Maximizing Load Balancing Policies

Consider a service system where incoming tasks are instantaneously dispatched to one out of many heterogeneous server pools. Associated with each server pool is a concave utility function that depends on the class of the server pool and its current occupancy. We derive an upper bound for the mean normalized aggregate utility in stationarity and introduce two load balancing policies that achieve this upper bound in a large-scale regime. Furthermore, the transient and stationary behavior of these asymptotically optimal load balancing policies is characterized on the scale of the number of server pools in the same large-scale regime. Funding: This work was supported by the Netherlands Organization for Scientific Research (NWO) through [Gravitation Grant NETWORKS-024.002.003] and [Gravitation Grant Vici 202.068]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/stsy.2022.0103 .

Submodular maximization of concave utility functions composed with a set-union operator with applications to maximal covering location problems

We study a family of discrete optimization problems asking for the maximization of the expected value of a concave, strictly increasing, and differentiable function composed with a set-union operator. The expected value is computed with respect to a set of coefficients taking values from a discrete set of scenarios. The function models the utility function of the decision maker, while the set-union operator models a covering relationship between two ground sets, a set of items and a set of metaitems. This problem generalizes the problem introduced by Ahmed S, Atamtürk A (Mathematical programming 128(1-2):149–169, 2011), and it can be modeled as a mixed integer nonlinear program involving binary decision variables associated with the items and metaitems. Its goal is to find a subset of metaitems that maximizes the total utility corresponding to the items it covers. It has applications to, among others, maximal covering location, and influence maximization problems. In the paper, we propose a double-hypograph decomposition which allows for projecting out the variables associated with the items by separately exploiting the structural properties of the utility function and of the set-union operator. Thanks to it, the utility function is linearized via an exact outer-approximation technique, whereas the set-union operator is linearized in two ways: either (i) via a reformulation based on submodular cuts, or (ii) via a Benders decomposition. We analyze from a theoretical perspective the strength of the inequalities of the resulting reformulations, and embed them into two branch-and-cut algorithms. We also show how to extend our reformulations to the case where the utility function is not necessarily increasing. We then experimentally compare our algorithms inter se, to a standard reformulation based on submodular cuts, to a state-of-the-art global-optimization solver, and to the greedy algorithm for the maximization of a submodular function. The results reveal that, on our testbed, the method based on combining an outer approximation with Benders cuts significantly outperforms the other ones.

Are risk preferences explaining gender differences in investment behavior?

We analyze individual investment behavior among 822 young men and women that are members of 111 formal business groups in northern Ethiopia. We collected baseline data and investment data one year later combined with incentivized field experiments to obtain dis-aggregated risk preference data. We find that businesswomen on average invest significantly less at individual level than businessmen but Cohen’s d values for the gender difference are moderate in size. Women are found to have higher Constant Relative Risk Aversion (CRRA-r) coefficients (more concave utility function), to be more loss averse, but also to be more optimistic in their expectations (lower Prelec β) than men. Women were also poorer in non-land assets, came from more land-poor parents, and had lower incomes. The gender differences in risk attitudes and baseline endowments could explain some of but not all of the gender differences in investment behavior.

Optimal climate policies under fairness preferences

Integrated assessment models are tools largely used to investigate the benefit of reducing polluting emissions and limiting the anthropogenic mean temperature rise. However, they have been often criticized for their underlying assumptions, often leading to low levels of abatement. Countries and regions that are generally the actors in these models are usually depicted as having standard concave utility functions in consumption. This, however, disregards a potentially important aspect of environmental negotiations, namely its distributive implications. The present paper tries to fill this gap assuming that countries/regions have Fehr and Schmidt (The Quarterly Journal of Economics 114(3):817–868, 1999) (F&S) utility functions, specifically tailored for including inequality aversion. By adopting the RICE model, we compare its standard results with the ones obtained introducing F&S utility functions, showing that, under optimal cooperation, the level of temperature rise is lower in the last scenario. In particular, the peak temperature, reached in 2155 under standard assumptions and one or two decades later under F&S preferences with, respectively, heterogeneous and homogeneous F&S inequality aversion parameters (α and β), is of 2.86 ∘C in the former scenario and of 2.65 ∘C and 2.67 ∘C in the latter two. Furthermore, it is shown that potentially stable coalitions are easier to be achieved when F&S preferences are assumed. However, potentially stable coalitions are far from reaching environmental targets close to limiting the mean temperature rise below 2 ∘C despite the adoption of F&S utility functions. The 2 ∘C target is almost achieved in all scenarios when the payoffs in the F&S utility function are given by the per-capita consumption rather than utility of consumption, with F&S preferences and heterogeneous F&S inequality aversion parameters leading to a peak temperature rise of 2.04 ∘C.

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.

Computational markers of experience- but not description-based decision-making are associated with future depressive symptoms in young adults

BackgroundEarly prediction of high depressive symptoms is crucial for selective intervention and the minimization of functional impairment. Recent cross-sectional studies indicated decision-making deficits in depression, which may be an important contributor to the disorder. Our goal was to test whether description- and experience-based decision making, two major neuroeconomic paradigms of decision-making under uncertainty, predict future depressive symptoms in young adults. MethodsOne hundred young adults performed two decision-making tasks, one description-based, in which subjects chose between two gambling options given explicitly stated rewards and their probabilities, and the other experience-based, in which subjects were shown rewards but had to learn the probability of those rewards (or cue-outcome contingencies) via trial-and-error experience. We evaluated subjects' depressive symptoms with BDI-II at baseline (T1) and half a year later (T2). ResultsComparing subjects with low versus high levels of depressive symptoms at T2 showed that the latter performed worse on the experience- but not description-based task at T1. Computational modeling of the decision-making process suggested that subjects with high levels of depressive symptoms had a more concave utility function, indicating enhanced risk aversion. Furthermore, a more concave utility function at T1 increased the odds of high depressive symptoms at T2, even after controlling depressive symptoms at T1, perceived stress at T2, and several covariates (OR = 0.251, 95% CI [0.085, 0.741]). ConclusionsThis is the first study to demonstrate a prospective link between experience-based decision-making and depressive symptoms. Our results suggest that enhanced risk aversion in experience-based decision-making may be an important contributor to the development of depressive symptoms.

Two-party Bargaining Processes Based on Subjective Expectations: A Model and a Simulation Study

This work presents a model for a two-party bargaining process in which multiple offers are exchanged as the negotiation goes on, under a risk of breakdown. Typical game theoretical analyses of such settings assume the breakdown risk is known and the parties are able to calculate an initial offer that is immediately accepted by the other party, ending the negotiation. Aiming to develop a model that is closer to real-life situations, in which parties do exchange many offers in a bargaining process, we consider the parties are unable to compute the far-reaching consequences of their offers, and are guided by their subjective expectations of the outcome of the negotiation. This introduces a new perspective to the analysis of two-party bargaining processes: the confidence of the bargainers in terms of what they hope to achieve by bargaining with each other. The resulting model can be seen as an extension of the Zeuthen-Hicks bargaining model. We show analytically that under the assumption of concave utilities of both parties, the bargaining process converges to the nonsymmetric Nash bargaining solution, where the asymmetry is caused by differences in expectations. This result provides a new interpretation of the parameters of the nonsymmetric Nash bargaining solution, linking them to behavior in the bargaining process. As an additional contribution, we perform a simulation study to examine the role of confidence and to analyze the outcomes for non-concave utility functions.

Utility Optimal Thread Assignment and Resource Allocation in Multi-Server Systems

Achieving high performance in many multi-server systems (e.g., web hosting center, cloud) requires finding a good assignment of worker threads to servers and also effectively allocating each server’s resources to its assigned threads. The assignment and allocation components of this problem have been studied extensively but largely separately in the literature. In this paper, we introduce the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">assign and allocate (AA)</i> problem, which seeks to simultaneously find an assignment and allocation that maximizes the total utility of the threads. Assigning and allocating the threads together can result in substantially better overall utility than performing the steps separately, as is traditionally done. We model each thread by a utility function giving its performance as a function of its assigned resources. We first prove that the AA problem is NP-hard. We then present a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2 (\sqrt {2}-1) &gt; 0.828$ </tex-math></inline-formula> factor approximation algorithm for concave utility functions, which runs in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(mn^{2} + n (\log mC)^{2})$ </tex-math></inline-formula> time for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> threads and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> servers with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$C$ </tex-math></inline-formula> amount of resources each. We also give a faster algorithm with the same approximation ratio and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(n (\log mC)^{2})$ </tex-math></inline-formula> time complexity. We then extend the problem to two more general settings. First, we consider threads with nonconcave utility functions, and give a 1/2 factor approximation algorithm. Next, we give an algorithm for threads using multiple types of resources, and show the algorithm achieves good empirical performance. We conduct extensive experiments to test the performance of our algorithms on threads with both synthetic and realistic utility functions, and find that they achieve over 92% of the optimal utility on average. We also compare our algorithms with a number of practical heuristics, and find that our algorithms achieve up to 9 times higher total utility.

Nonlinear Probability Weighting in Depression and Anxiety: Insights From Healthy Young Adults.

Both depressive and anxiety disorders have been associated with excessive risk avoidant behaviors, which are considered an important contributor to the maintenance and recurrence of these disorders. However, given the high comorbidity between the two disorders, their independent association with risk preference remains unclear. Furthermore, due to the involvement of multiple cognitive computational factors in the decision-making tasks employed so far, the precise underlying mechanisms of risk preference are unknown. In the present study, we set out to investigate the common versus unique cognitive computational mechanisms of risk preference in depression and anxiety using a reward-based decision-making task and computational modeling based on economic theories. Specifically, in model-based analysis, we decomposed risk preference into utility sensitivity (a power function) and probability weighting (the one-parameter Prelec weighting function). Multiple linear regression incorporating depression (BDI-II) and anxiety (STAI state anxiety) simultaneously indicated that only depression was associated with one such risk preference parameter, probability weighting. As the symptoms of depression increased, subjects’ tendency to overweight small probabilities and underweight large probabilities decreased. Neither depression nor anxiety was associated with utility sensitivity. These associations remained even after controlling covariates or excluding anxiety-relevant items from the depression scale. To our knowledge, this is the first study to assess risk preference due to a concave utility function and nonlinear probability weighting separately for depression and anxiety using computational modeling. Our results provide a mechanistic account of risk avoidance and may improve our understanding of decision-making deficits in depression and anxiety.