Utility Games Research Articles

Two-Layer Optimization with Utility Game and Resource Control for Federated Learning in Edge Networks

Two-Layer Optimization with Utility Game and Resource Control for Federated Learning in Edge Networks

An analysis of distribution planning under a regulatory regime: An integrated framework

AbstractDistribution system planning is a multifaceted topic involving financial, regulatory, and system level analysis. The wide nature of the topic warrants a holistic study considering all aspects of analysis. The distribution utility is a natural monopoly that is subjected to utility regulation. The regulator can impact customer experience by strategically influencing the planning decisions of the utility. Hence, this paper reviews the existing utility regulation methods in the context of the distribution system and their efficacy in improving certain reliability and efficiency objectives. A two‐bus system is used to demonstrate the impact of classical models in alleviating reliability and efficiency issues through demand response. Further, a review is conducted on distribution system planning models without a regulatory regime, and suitable models for holistic analysis are identified. A two‐person complete information regulator and utility game with a comprehensive distribution system model at the lower level is proposed. A framework based on the Mixed Integer Bilevel Linear Program (MIBLP) is discussed to find the equilibrium point of the proposed game.

Tensor approximation of cooperative games and their semivalues

We propose an algorithm to approximate the semivalues of general transferable-utility cooperative games that involve a large set of players 1,…,|N| and possibly depend on uncertain parameters. We first encode the game's utility function using a low-rank tensor decomposition, namely the tensor train (TT) model, which requires a limited number of function evaluations. The TT format casts the utility as a compressed tensor of shape 2|N| and makes it possible to efficiently work with the exponentially-sized set of all possible coalitions of players. Given a game compressed in this manner, the proposed algorithm obtains arbitrary semivalues without incurring additional error, in particular the Shapley values and Banzhaf-Coleman indices, which are two of the most important allocation rules in cooperative game theory. Our algorithm takes O(|N|R2) operations per semivalue, where R is the game's TT rank. We show experimentally that many classical games can be compressed at low error with a moderate TT rank, making our algorithm more sample-efficient than Monte Carlo-based estimation. We also give a theoretical bound for the error of the semivalues obtained through our algorithm. Last, when the game depends on randomly distributed parameters, we are able to compute the expected semivalues efficiently.

Nontransferable utility bankruptcy games

In this paper, we analyze bankruptcy problems with nontransferable utility (NTU) from a game theoretical perspective by redefining corresponding NTU-bankruptcy games in a tailor-made way. It is shown that NTU-bankruptcy games are both coalition-merge convex and ordinally convex. Generalizing the notions of core cover and compromise stability for transferable utility (TU) games to NTU-games, we also show that each NTU-bankruptcy game is compromise stable. Thus, NTU-bankruptcy games are shown to retain the two characterizing properties of TU-bankruptcy games: convexity and compromise stability. As a first example of a game theoretical NTU-bankruptcy rule, we analyze the adjusted proportional rule and show that this rule corresponds to the compromise value of NTU-bankruptcy games.

Cooperative games with nontransferable utility on antimatroids

ABSTRACTThis paper introduces a new family of nontransferable utility (NTU) games with partial cooperation. Games with transferable utility (TU) restricted by antimatroids were studied since 2004. Antimatroids are known combinatorial structures which are interpreted in games as dependency situations among a group of agents, and they generalize several other hierarchical structures studied in the literature. Now we define projected NTU games over antimatroids following the same idea. We have needed to build a new theory about projections between two coalitions. In this case, an outcome is feasible for a coalition if and only if its projection is feasible in the NTU game. This paper studies several properties of the projected games and the relations between these games and the original game for some interesting cases: TU games, hyperplane games and bargaining problems.

Generalized additive games

A transferable utility (TU) game with n players specifies a vector of \(2^n-1\) real numbers, i.e. a number for each non-empty coalition, and this can be difficult to handle for large n. Therefore, several models from the literature focus on interaction situations which are characterized by a compact representation of a TU-game, and such that the worth of each coalition can be easily computed. Sometimes, the worth of each coalition is computed from the values of single players by means of a mechanism describing how the individual abilities interact within groups of players. In this paper we introduce the class of Generalized additive games (GAGs), where the worth of a coalition \(S { \subseteq } N\) is evaluated by means of an interaction filter, that is a map \(\mathcal {M}\) which returns the valuable players involved in the cooperation among players in S. Moreover, we investigate the subclass of basic GAGs, where the filter \(\mathcal {M}\) selects, for each coalition S, those players that have friends but not enemies in S. We show that well-known classes of TU-games can be represented in terms of such basic GAGs, and we investigate the problem of computing the core and the semivalues for specific families of GAGs.

Characterizations of the core of TU and NTU games with communication structures

It is well-known that the core on several domains of cooperative transferable utility (TU) and nontransferable utility (NTU) games is characterized by various combinations of axioms containing some versions of the reduced game property, of its converse, or of the reconfirmation property with respect to the Davis–Maschler reduced game. We show that these characterizations are still valid for games with communication structures a la Myerson when using the notion of the reduced communication structure that establishes a new link between two inside players if they can communicate via outside players. Thus, it is shown that, if communication structures are present, the core may still be characterized on balanced TU games, on totally balanced TU games, on NTU games with a nonempty core, on the domains of all TU or NTU games, and on several other interesting domains of TU and NTU games. As a byproduct we construct, for any NTU game with communication structure, a certain classical NTU game with the same core that may be regarded as its Myerson restricted NTU game.

DETERMINING THE NUCLEOLUS OF COMPROMISE STABLE GAMES

The main goal is to illustrate that the so-called indirect function of a cooperative game in characteristic function form is applicable to determine the nucleolus for a subclass of coalitional games called compromise stable transferable utility (TU) games. In accordance with the Fenchel–Moreau theory on conjugate functions, the indirect function is known as the dual representation of the characteristic function of the coalitional game. The key feature of a compromise stable TU game is the coincidence of its core with a box prescribed by certain upper and lower core bounds. For the purpose of the determination of the nucleolus, we benefit from the interrelationship between the indirect function and the prekernel of coalitional TU games. The class of compromise stable TU games contains the subclasses of clan games, big boss games and $1$- and $2$-convex $n$-person TU games. As an adjunct, this paper reports the indirect function of clan games for the purpose of determining their nucleolus.

AVERAGE RULES FOR COOPERATIVE TU GAMES

In this paper, we propose a new single valued rule based on the concept of fair division for all cooperative transferable utility (TU) games. In any cooperative TU game, primarily the coalitions that are likely to form are identified and each such coalition is fixed with a payoff vector based on the notion of fairness. The value of the single valued rule is obtained from the collection of all coalition structures consisting of the coalitions that are likely to form. The uniqueness of the new rule is followed by its existence and computational simplicity for all TU games. Finally, a linear average rule is defined, and some of its properties are discussed.

USING THE ASPIRATION CORE TO PREDICT COALITION FORMATION

This work uses the defining principles of the core solution concept to determine not only payoffs but also coalition formation. Given a cooperative transferable utility (TU) game, we propose two noncooperative procedures that in equilibrium deliver a natural and nonempty core extension, the aspiration core, together with the supporting coalitions it implies. As expected, if the cooperative game is balanced, the grand coalition forms. However, if the core is empty, other coalitions arise. Following the aspiration literature, not only partitions but also overlapping coalition configurations are allowed. Our procedures interpret this fact in different ways. The first game allows players to participate with a fraction of their time in more than one coalition, while the second assigns probabilities to the formation of potentially overlapping coalitions. We use the strong Nash and subgame perfect Nash equilibrium concepts.

The construction and characterization of egalitarian solutions for multi-choice NTU games

In this article we construct a procedure to define the egalitarian solutions in the context of multi-choice non-transferable utility (NTU) games. Also, we show that in the presence of other weak axioms the egalitarian solutions are the only monotonic ones.

Game theory and time utility functions for a radio aware scheduling algorithm for WiMAX networks

In WiMAX systems the Base Station scheduler plays a key role as it controls the sharing of the radio resources among the users. The goal of the scheduler is multiple: achieve fair usage of the resources, satisfy the QoS requirements of the users, maximize goodput, and minimize power consumption, and at the same time ensuring feasible algorithm complexity and system scalability. Since most of these goals are contrasting, scheduler designers usually focus their attention on optimizing one aspect only. In this scenario, we propose a scheduling algorithm (called $$\mathrm{GTS_N}$$ ) whose goal is to contemporaneously achieve efficiency and fairness, while also taking into account the QoS requirements and the channel state. $$\mathrm{GTS_N}$$ exploits the properties of Time Utility Functions (TUFs) and Game Theory. Simulations prove that the performance of $$\mathrm{GTS_N},$$ when compared to that of several well-known schedulers, is remarkable. $$\mathrm{GTS_N}$$ provides the best compromise between the two contrasting objectives of fairness and efficiency, while QoS requirements are in most cases guaranteed. However, the exponential complexity introduced by the game theory technique makes it rather impractical and not computationally scalable for a large number of users. Thus we developed a suboptimal version, named sub- $$\mathrm{GTS_N}.$$ We show that this version retains most of the features and performance figures of its brother, but its complexity is linear with the number of users.

Two Game Models for Cooperation with Implicit Noncooperation

We propose two flexible game models to represent and analyze cases that cannot be modeled by current game models. One is called sharing creditability game (SCG) and the other is called bottomline game (BLG). The new models transform cooperative games into new games that incorporate auxiliary information (noncooperative in nature) usually neglected in previous theories. The new games will be solved only by traditional noncooperative game theory. When the new solutions are applied to the original games, the solutions can reflect the auxiliary information in addition to the original objectives of the decision makers or players. Generally, the new solutions are different from the cooperative and the noncooperative solutions of the original games. Existing transferable utility (TU) games and noncooperative games will coincide with special cases of the two new game models. Using SCG and BLG, the prisoner's dilemma can be reformulated and a richer set of decisions can be considered for the players. The two new game models have potential applications in military and socioeconomic situations.

NTU PERT games

In this paper a new class of Non-Transferable Utility (NTU) games is introduced. We associate an NTU game to the PERT problem that arises from a method of operations research called Project Evaluation and Review Technique (PERT). The goal is to divide the slacks of time among the different activities involved in the project. We model this problem as an NTU game and we study some properties. It is shown that this class of NTU games has a non-empty core. Moreover, an easy expression for the compromise value is attained and an adjusted compromise value is defined.

Values for two-stage games: Another view of the Shapley axioms

This short study reports an application of the Shapley value axioms to a new concept of “two-stage games.” In these games, the formation of a coalition in the first stage entitles its members to play a prespecified cooperative game at the second stage. The original Shapley axioms have natural equivalents in the new framework, and we show the existence of (non-unique) values and semivalues for two stage games, analogous to those defined by the corresponding axioms for the conventional (one-stage) games. However, we also prove that all semivalues (hence, perforce, all values) must give patently unacceptable solutions for some “two-stage majority games” (where the members of a majority coalition play a conventional majority game). Our reservations about these prescribed values are related to Roth's (1980) criticism of Shapley's “λ-transfer value” for non-transferable utility (NTU) games. But our analysis has wider scope than Roth's example, and the argument that it offers appears to be more conclusive. The study also indicates how the values and semivalues for two-stage games can be naturally generalized to apply for “multi-stage games.”