scholarly journals The Tractability of the Shapley Value over Bounded Treewidth Matching Games

2017 ◽  
Author(s):  
Gianluigi Greco ◽  
Francesco Lupia ◽  
Francesco Scarcello
Keyword(s):  
Polynomial Time ◽  
Shapley Value ◽  
Open Problem ◽  
Positive Answer ◽  
Coalitional Games ◽  
Solution Concepts ◽  
Bounded Treewidth ◽  
The Shapley Value ◽  
Matching Games

Matching games form a class of coalitional games that attracted much attention in the literature. Indeed, several results are known about the complexity of computing over them {solution concepts}. In particular, it is known that computing the Shapley value is intractable in general, formally #P-hard, and feasible in polynomial time over games defined on trees. In fact, it was an open problem whether or not this tractability result holds over classes of graphs properly including acyclic ones. The main contribution of the paper is to provide a positive answer to this question, by showing that the Shapley value is tractable for matching games defined over graphs having bounded treewidth. The proposed technique has been implemented and tested on classes of graphs having different sizes and treewidth at most three.

scholarly journals Efficient Computation of Semivalues for Game-Theoretic Network Centrality

2018 ◽  
Vol 63 ◽  
pp. 145-189 ◽  
Author(s):  
Mateusz K. Tarkowski ◽  
Piotr L. Szczepański ◽  
Tomasz P. Michalak ◽  
Paul Harrenstein ◽  
Michael Wooldridge
Keyword(s):  
Polynomial Time ◽  
Shapley Value ◽  
Centrality Measures ◽  
Efficient Computation ◽  
Solution Concepts ◽  
Banzhaf Index ◽  
The Shapley Value ◽  
Game Theoretic ◽  
Theoretic Solution ◽  
Network Centralities

Some game-theoretic solution concepts such as the Shapley value and the Banzhaf index have recently gained popularity as measures of node centrality in networks. While this direction of research is promising, the computational problems that surround it are challenging and have largely been left open. To date there are only a few positive results in the literature, which show that some game-theoretic extensions of degree-, closeness- and betweenness-centrality measures are computable in polynomial time, i.e., without the need to enumerate the exponential number of all possible coalitions. In this article, we show that these results can be extended to a much larger class of centrality measures that are based on a family of solution concepts known as semivalues. The family of semivalues includes, among others, the Shapley value and the Banzhaf index. To this end, we present a generic framework for defining game-theoretic network centralities and prove that all centrality measures that can be expressed in this framework are computable in polynomial time. Using our framework, we present a number of new and polynomial-time computable game-theoretic centrality measures.

scholarly journals Complexity of Computing the Shapley Value in Games with Externalities

2020 ◽  
Vol 34 (02) ◽  
pp. 2244-2251 ◽  
Author(s):  
Oskar Skibski
Keyword(s):  
Polynomial Time ◽  
Shapley Value ◽  
Marginal Contribution ◽  
The Shapley Value ◽  
Games With Externalities ◽  
Computational Properties

We study the complexity of computing the Shapley value in games with externalities. We focus on two representations based on marginal contribution nets (embedded MC-nets and weighted MC-nets) and five extensions of the Shapley value to games with externalities. Our results show that while weighted MC-nets are more concise than embedded MC-nets, they have slightly worse computational properties when it comes to computing the Shapley value: two out of five extensions can be computed in polynomial time for embedded MC-nets and only one for weighted MC-nets.

scholarly journals Shapley Value for Parallel Machine Sequencing Situation without Initial Order

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Shanshan Liu ◽  
Zhaohui Liu
Keyword(s):  
Polynomial Time ◽  
Shapley Value ◽  
Processing Time ◽  
Cost Allocation ◽  
Parallel Machine ◽  
Allocation Rule ◽  
The Shapley Value ◽  
Sequencing Game ◽  
Identical Machine

We consider the parallel identical machine sequencing situation without initial schedule. For the situation with identical job processing time, we design a cost allocation rule which gives the Shapley value of the related sequencing game in polynomial time. For the game with identical job weight, we also present a polynomial time procedure to compute the Shapley value.

Coalitional games induced by matching problems: Complexity and islands of tractability for the Shapley value

2020 ◽  
Vol 278 ◽  
pp. 103180
Author(s):  
Gianluigi Greco ◽  
Francesco Lupia ◽  
Francesco Scarcello
Keyword(s):  
Shapley Value ◽  
Coalitional Games ◽  
Matching Problems ◽  
The Shapley Value

scholarly journals Meaningful and Meaningless Solutions for Cooperative N-person Games

1996 ◽  
Vol 3 (47) ◽  
Author(s):  
Aleksandar Pekec
Keyword(s):  
Shapley Value ◽  
Game Model ◽  
Solution Concepts ◽  
Measurement Sensitivity ◽  
The Core ◽  
The Shapley Value ◽  
Person Game

<p>Game values often represent data that can be measured in more<br />than one acceptable way (e.g. monetary amounts). We point out that<br />in such a case a statement about cooperative n-person game model<br />might be "meaningless" in the sense that its truth or falsity depends on<br />the choice of an acceptable way to measure game values. In particular<br />we analyze statements about solution concepts such as the core, stable<br />sets, the nucleolus, the Shapley value (and its generalizations).</p><p>Keywords: Cooperative n-person Games, Measurement, Sensitivity<br />Analysis.</p>

scholarly journals Attaining Fairness in Communication for Omniscience

Entropy ◽  
2022 ◽  
Vol 24 (1) ◽  
pp. 109
Author(s):  
Ni Ding ◽  
Parastoo Sadeghi ◽  
David Smith ◽  
Thierry Rakotoarivelo
Keyword(s):  
Polynomial Time ◽  
Shapley Value ◽  
Data Exchange ◽  
Source Coding ◽  
Extreme Points ◽  
Mean Value ◽  
The Core ◽  
Coding Schemes ◽  
The Shapley Value ◽  
Egalitarian Solution

This paper studies how to attain fairness in communication for omniscience that models the multi-terminal compress sensing problem and the coded cooperative data exchange problem where a set of users exchange their observations of a discrete multiple random source to attain omniscience—the state that all users recover the entire source. The optimal rate region containing all source coding rate vectors that achieve omniscience with the minimum sum rate is shown to coincide with the core (the solution set) of a coalitional game. Two game-theoretic fairness solutions are studied: the Shapley value and the egalitarian solution. It is shown that the Shapley value assigns each user the source coding rate measured by their remaining information of the multiple source given the common randomness that is shared by all users, while the egalitarian solution simply distributes the rates as evenly as possible in the core. To avoid the exponentially growing complexity of obtaining the Shapley value, a polynomial-time approximation method is proposed which utilizes the fact that the Shapley value is the mean value over all extreme points in the core. In addition, a steepest descent algorithm is proposed that converges in polynomial time on the fractional egalitarian solution in the core, which can be implemented by network coding schemes. Finally, it is shown that the game can be decomposed into subgames so that both the Shapley value and the egalitarian solution can be obtained within each subgame in a distributed manner with reduced complexity.

TWISTED DUAL GAMES AND THEIR PROPERTIES

2007 ◽  
Vol 09 (02) ◽  
pp. 285-306 ◽  
Author(s):  
KENSAKU KIKUTA
Keyword(s):  
Shapley Value ◽  
Coalitional Games ◽  
Bankruptcy Problems ◽  
Self Duality ◽  
Reduced Game ◽  
The Shapley Value ◽  
Game Property ◽  
Twisted Duality

We define a duality of solutions in coalitional games that we call a twisted duality. This twisted duality extends the rule of self-duality in bankruptcy problems. After showing that the prekernel and prenucleolus exhibit twisted duality, we define a twisted reduced game property and characterize the prekernel and prenucleolus by axioms including this property. We note that the Shapley value satisfies twisted duality, and we define another twisted reduced game property by and characterize the Shapley value by axioms including this property.

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