Transferable Utility Research Articles

Gately values of cooperative games

AbstractWe investigate Gately’s solution concept for cooperative games with transferable utilities. Gately’s conception introduced a bargaining solution that minimises the maximal quantified “propensity to disrupt” the negotiation process of the players over the allocation of the generated collective payoffs. We show that Gately’s solution concept is well-defined for a broad class of games and that it can be interpreted as a compromise solution. We also consider a generalisation based on a parameter-based quantification of the propensity to disrupt. We provide an axiomatic characterisation of the original Gately value as well as these generalised Gately values. Furthermore, we investigate the relationship of these Gately values with the Core and the Nucleolus and show that Gately’s solution is in the Core for all regular 3-player games, but is fundamentally different from the Nucleolus. We identify exact conditions under which these Gately values are Core imputations for arbitrary regular cooperative games. Finally, we investigate the relationship of the Gately value with the Shapley value.

Resolving the Size and Charge of Small Particles: A Predictive Model of Nanopore Mechanics.

The movement of small particles and molecules through nanopore membranes is widespread and has far-reaching implications. Consequently, the development of mathematical models is essential for understanding these processes on a micro level, leading to deeper insights. In this endeavor, we suggested a model based on a set of empirical equations to predict the transport of substances through a solid-state nanopore and the associated signals generated during their translocation. This model establishes analytical relationships between the ionic current and electrical double-layer potential observed during analyte translocation and their size, charge, and mobility in an electrolyte solution. This framework allows for rapid interpretation and prediction of the nanopore system's behavior and provides a means for quantitatively determining the physical properties of molecular analytes. To illustrate the analytical capability of this model, ceria nanoparticles were investigated while undergoing oxidation or reduction within an original nanopore device. The results obtained were found to be in good agreement with predictions from physicochemical methods. This developed approach and model possess transferable utility to various porous materials, thereby expediting research efforts in membrane characterization and the advancement of nano- and ultrafiltration or electrodialysis technologies.

Moore interval subtraction and interval solutions for TU-games

AbstractStandard solutions for cooperative transferable utility (TU-) games assign to every player in a TU-game a real number representing the player’s payoff. In this paper, we introduce interval solutions for TU-games which assign to every player in a game a payoff interval. Even when the worths of coalitions are known, it might be that the individual payoff of a player is not known. According to an interval solution, every player knows at least a lower- and upper bound for its individual payoff. Therefore, interval solutions are useful when there is uncertainty about the payoff allocation even when the worths that can be earned by coalitions are known. Specifically, we consider two interval generalizations of the famous Shapley value that are based on marginal contributions in terms of intervals. To determine these marginal interval contributions, we apply the subtraction operator of Moore. We provide axiomatizations for the class of totally positive TU-games. We also show how these axiomatizations can be used to extend any linear TU-game solution to an interval solution. Finally, we illustrate these interval solutions by applying them to sequencing games.

Algorithmic solutions for maximizing shareable costs

AbstractThis article addresses the linear optimization problem to maximize the total costs that can be shared among a group of agents, while maintaining stability in the sense of the core constraints of a cooperative transferable utility game, or TU game. When maximizing total shareable costs, the cost shares must satisfy all constraints that define the core of a TU game, except for being budget balanced. The article first gives a fairly complete picture of the computational complexity of this optimization problem, its relation to optimization over the core itself, and its equivalence to other, minimal core relaxations that have been proposed earlier. We then address minimum cost spanning tree (MST) games as an example for a class of cost sharing games with non‐empty core. While submodular cost functions yield efficient algorithms to maximize shareable costs, MST games have cost functions that are subadditive, but generally not submodular. Nevertheless, it is well known that cost shares in the core of MST games can be found efficiently. In contrast, we show that the maximization of shareable costs is ‐hard for MST games and derive a 2‐approximation algorithm. Our work opens several directions for future research.

Estimating separable matching models

SummaryMost recent empirical applications of matching with transferable utility have imposed a natural restriction: that the joint surplus be separable in the sources of unobserved heterogeneity. We propose here two simple methods to estimate models in this class. The first method is a minimum distance estimator that relies on the generalized entropy of matching. The second applies to the more special but popular Choo and Siow model, which reformulates as a generalized linear model with two‐way fixed effects. Neither method requires solving for the stable matching. Both methods are easy to apply and perform very well.

Cooperative equilibria of strategy-form games with both nontransferable and transferable utilities

In this paper, we consider a class of strategy-form games with both nontransferable and transferable utilities. Inspired by NTU and TU α-core concepts, we first introduce the notion of cooperative equilibria, and prove the existence theorem in this model with finite dimensional strategy spaces. Furthermore, we extend the cooperative equilibrium existence theorem to strategy-form games with infinite dimensional strategy spaces.

The Theory of the Evolution of Economic Relations as an Approach to Generalize the Theory of Games

The Theory of Games in the classical sense uses the idea of a system of economic relations of one type, namely economic relations of a private nature when the goal of rational behavior of the subject of economic relations (ER) is to maximize the benefit (own profit). In the process of evolution, economic relations go through several stages (namely 16), and for each stage, the set of characteristics of the basic elements strictly differs from the set of other stages. The rational behavior of the subjects is largely determined by this particular set of characteristics. Thus, the corresponding Theory of Rational Behavior of Economic Subjects (TRES) is the generalization of the Theory of Games for application in economics (that is, it applies not only to one type of economic relations, aiming at private benefit). Another aspect of the generalization of the Theory of Games in conditions of global information and computer accessibility is the transition from money as a medium of exchange that was a “transferable numerical utility” to another medium of exchange - a full range of goods with the complete dynamically changing set of exchange coefficients as a “generalized medium of exchange” that organically corresponds to the economic content of the production process.

Convexity of the triple helix of innovation game

Purpose This paper aims to determine the conditions for the core of the Triple Helix game to exist. The Triple Helix of university-industry-government relationships is a three-person cooperative game with transferable utility. Then, the core, the Shapley value and the nucleolus were used as indicators of the synergy within an innovation system. Whereas the Shapley value and the nucleolus always exist, the core may not. Design/methodology/approach The core of a three-person cooperative game with transferable utility exists only if and only if the game is convex. The paper applies the convexity condition to the Triple Helix game. Findings The Triple Helix game is convex if and only if there is output within the system; it is strictly convex if and only if all the three bilateral and the trilateral relationships have an output. Practical implications Convex games are competitive situations in which there are strong incentives towards the formation of large coalitions; therefore, innovation actors must cooperate to maximise their interests. Furthermore, a Triple Helix game may be split into subgames for comprehensive analyses and several Triple Helix games may be combined for a global study. Originality/value This paper extends the meaning of the Shapley value and the nucleolus for Triple Helix innovation actors: the Shapley value indicates the quantity a player wins because of the coalitions he involves in and the nucleolus the return for solidarity of an innovation actor.

Cooperative Product Games

A cooperative product game is a transferable utility game defined by associating the product of given players’ coefficient to coalitions. This idea has been introduced by Rosales, D. ([2014] Cooperative product games, unpublished mimeo, www.academia.edu/40176429) where he investigates the core. Here, we analyze a corrected definition of a product game and cover the Shapley value to which we provide an axiomatic foundation.

On the estimation of the core for TU games

The core of a transferable utility (TU) game, if it is not empty, is prescribed by the set of all stable allocations. The exact determination of the core reaches exponential time complexity. Therefore, its exact computation is often avoided as the number of players increases. In this work, we propose an estimator for the core of a TU game based on the statistical theory of set estimation. Concretely, we provide a core reconstruction that is obtained in polynomial time for general dimension. Additionally, convergence rates for the estimation error are derived. Finally, a consistent core-center estimator is established as a geometrical application of this methodology.

Strategy-proofness and competitive equilibrium with transferable utility: Gross substitutes revisited

Strategy-proofness and competitive equilibrium with transferable utility: Gross substitutes revisited

The $$\kappa $$-core and the $$\kappa $$-balancedness of TU games

AbstractWe consider transferable utility cooperative games with infinitely many players. In particular, we generalize the notions of core and balancedness, and also the Bondareva–Shapley Theorem for infinite TU games with and without restricted cooperation, to the cases where the core consists of $$\kappa $$ κ -additive set functions. Our generalized Bondareva–Shapley Theorem extends previous results by Bondareva (Problemy Kibernetiki 10:119–139, 1963), Shapley (Naval Res Logist Q 14:453–460, 1967), Schmeidler (On balanced games with infinitely many players, The Hebrew University, Jerusalem, 1967), Faigle (Zeitschrift für Oper Res 33(6):405–422, 1989), Kannai (J Math Anal Appl 27:227–240, 1969; The core and balancedness, handbook of game theory with economic applications, North-Holland, 1992), Pintér (Linear Algebra Appl 434(3):688–693, 2011) and Bartl and Pintér (Oper Res Lett 51(2):153–158, 2023).

Coalitional Game Based Resource Allocation in D2D-Enabled V2V Communication

The joint resource block (RB) allocation and power optimization problem is studied to maximize the sum-rate of the vehicle-to-vehicle (V2V) links in the device-to-device (D2D)-enabled V2V communication system, where one feasible cellular user (FCU) can share its RB with multiple V2V pairs. The problem is first formulated as a nonconvex mixed-integer nonlinear programming (MINLP) problem with constraint of the maximum interference power in the FCU links. Using the game theory, two coalition formation algorithms are proposed to accomplish V2V link partitioning and FCU selection, where the transferable utility functions are introduced to minimize the interference among the V2V links and the FCU links for the optimal RB allocation. The successive convex approximation (SCA) is used to transform the original problem into a convex one and the Lagrangian dual method is further applied to obtain the optimal transmit power of the V2V links. Finally, numerical results demonstrate the efficiency of the proposed resource allocation algorithm in terms of the system sum-rate.

Cooperation in an Arrow-Karlin Type Supply Chain

In this paper we apply cooperative game theory concepts to analyze vertical supply chains. The bullwhip effect in a two-stage supply chain (supplier-manufacturer) in the framework of the Arrow-Karlin model with linear-convex cost functions is considered. It is assumed that both firms minimize their relevant costs, and two cases are examined: the supplier and the manufacturer minimize their relevant costs in a decentralized and in a centralized (cooperative) way. The question of how to share the savings of the decreased bullwhip effect in the centralized (cooperative) model is answered by transferable utility cooperative game theory tools.

Game Theoretic Foundations of the Gately Power Measure for Directed Networks

We introduce a new network centrality measure founded on the Gately value for cooperative games with transferable utilities. A directed network is interpreted as representing control or authority relations between players—constituting a hierarchical network. The power distribution embedded within a hierarchical network can be represented through appropriate TU-games. We investigate the properties of these TU-representations and investigate the Gately value of the TU-representation resulting in the Gately power measure. We establish when the Gately measure is a core power gauge, investigate the relationship of the Gately with the β-measure, and construct an axiomatisation of the Gately measure.

Toward Personalized Federated Learning Via Group Collaboration in IIoT

Despite the rapid growth of successful examples of Federated Learning (FL), it faces the heterogeneity of data, models, and devices in emerging applications of Industrial Internet of Things (IIoT). Existing efforts mainly focus on training multiple personalized models by adopting a global, cluster, or pairwise fashion. However, the global collaboration doesn't work well in case of the non-IID data distribution, cluster collaboration is often inefficient due to the single cluster pattern and high computational cost, and pairwise collaboration incurs the limitations of collaboration scope and communication efficiency. To address the problems, we propose a novel Personalized FL (PFL) framework with the game-theoretic insights, called group collaboration, to overcome the shortcomings of status quo. Specifically, we first formulate the group collaboration in PFL as a multi-leader multi-follower Stackelberg game, and then develop an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\epsilon$</tex-math></inline-formula> -better response to efficiently characterize its unique equilibrium through cautiously proposing a potential function. Since the existing equilibrium may not be optimal, we further design a Robin Hood mechanism by using the idea of transferable utility to improve the performance of the training model. Meanwhile, we also prove that the new mechanism is sustainable and can converge to a stable state with an upper bound of the training loss. Last, extensive experiments on a simulated dataset and four real-world datasets demonstrate the superiority of our proposed approach compared to the state-of-the-art.

On egalitarian values for cooperative games with level structures

In this paper we extend the equal division and the equal surplus division values for transferable utility cooperative games to the more general setup of transferable utility cooperative games with level structures. In the case of the equal surplus division value we propose three possible extensions, one of which has already been described in the literature. We provide axiomatic characterizations of the values considered, apply them to a particular cost sharing problem and compare them in the framework of such an application.

Revisiting Zakat with a distribution of weighted Shapley value

PurposeThis paper aims to extend the Shapley value (SV) into a discussion of Zakat, a Pillar of Islam. Lloyd Shapley was awarded the Nobel Prize in Economics in 2012. This study shows that their relationship is significant for all nations, that of levelling up. An important but neglected paper by Datta (1939) showed insights provided by the Power Law, or as it is sometimes called, the Pareto distribution, into the role of Zakat in raising the income of all above the subsistence level. The Pareto distribution describes the prevailing tendency. The SV illustrates the interdependence perspective of Zakat with the Pareto distribution, wealth, income and poverty. Payoffs apply equally to both givers and receivers. For this study’s purposes, payoffs are considered as transferable utilities. They are formed by individuals who willingly cooperate in society rather than atomistic individuals who act independently. Zakat represents the recognition that society needs to be cooperative rather than individualistic; people cooperate in groups or societies to create value. SV implications and axioms are evaluated with an illustration.Design/methodology/approachThis study extends Datta’s approach by introducing distribution weights into the SV. The authors set out the concept of weighted Shapley values that retain the elements of randomness and marginal contribution to a coalition contained in pure/true SVs and weights that follow a ley-Pareto distribution. This paper is a viewpoint work that relies primarily on the author’s qualitative interpretation.FindingsThe findings indicate that individual members of a coalition make multiple contributions that are often unrewarded. The contribution of one member of a coalition is dependent upon the contribution of others. The measure of contributions is payoffs, which have both monetary and non-monetary aspects; transferable payoffs or utilities are usually assumed. Furthermore, the significant agents in society or an organisation are stakeholders rather than the usual categories: managers, staff, shareholders, etc.Practical implicationsContextualising these concepts within the Islamic values and principles that guide Zakat administration is crucial to ensure that the distribution of Zakat funds is fair, equitable and meets the needs of all eligible recipients. By applying these concepts appropriately, Zakat administrators can ensure that the Zakat system functions effectively and fulfils its religious obligation.Originality/valueThe novelty of this paper is that it blends the SV and the idea behind Zakat by introducing the idea of alternatives of Shapley weights. The link between the institution of Zakat and SV in terms of equality, poverty elimination and wealth distribution should be at the top of the research agenda.

Average monotonic cooperative games with nontransferable utility

A non-negative transferable utility (TU) game is average monotonic if there exists a non-negative vector according to which the relative worth is not decreasing when enlarging the coalition. We generalize this definition to the nontransferable utility (NTU) case. It is shown that an average monotonic NTU game shares several properties with an average monotonic TU game. In particular it has a special core element and there exists a population monotonic allocation scheme. We show that an NTU bankruptcy game is average monotonic with respect to the claims vector.

Learning Equilibria in Matching Markets with Bandit Feedback

Large-scale, two-sided matching platforms must find market outcomes that align with user preferences while simultaneously learning these preferences from data. Classical notions of stability (Gale and Shapley, 1962; Shapley and Shubik, 1971) are, unfortunately, of limited value in the learning setting, given that preferences are inherently uncertain and destabilizing while they are being learned. To bridge this gap, we develop a framework and algorithms for learning stable market outcomes under uncertainty. Our primary setting is matching with transferable utilities, where the platform both matches agents and sets monetary transfers between them. We design an incentive-aware learning objective that captures the distance of a market outcome from equilibrium. Using this objective, we analyze the complexity of learning as a function of preference structure, casting learning as a stochastic multi-armed bandit problem. Algorithmically, we show that “optimism in the face of uncertainty,” the principle underlying many bandit algorithms, applies to a primal-dual formulation of matching with transfers and leads to near-optimal regret bounds. Our work takes a first step toward elucidating when and how stable matchings arise in large, data-driven marketplaces.