scholarly journals Computational Aspects of the Preference Cores of Supermodular Two-Scenario Cooperative Games

2018 ◽  
Author(s):  
Daisuke Hatano ◽  
Yuichi Yoshida
Keyword(s):  
Characteristic Function ◽  
Cooperative Games ◽  
Solution Concept ◽  
Multicast Tree ◽  
Characteristic Functions ◽  
Solution Concepts ◽  
Induced Subgraph ◽  
Computational Aspects ◽  
Tree Game ◽  
Multiple Scenarios

In a cooperative game, the utility of a coalition of players is given by the characteristic function, and the goal is to find a stable value division of the total utility to the players. In real-world applications, however, multiple scenarios could exist, each of which determines a characteristic function, and which scenario is more important is unknown. To handle such situations, the notion of multi-scenario cooperative games and several solution concepts have been proposed. However, computing the value divisions in those solution concepts is intractable in general. To resolve this issue, we focus on supermodular two-scenario cooperative games in which the number of scenarios is two and the characteristic functions are supermodular and study the computational aspects of a major solution concept called the preference core. First, we show that we can compute the value division in the preference core of a supermodular two-scenario game in polynomial time. Then, we reveal the relations among preference cores with different parameters. Finally, we provide more efficient algorithms for deciding the non-emptiness of the preference core for several specific supermodular two-scenario cooperative games such as the airport game, multicast tree game, and a special case of the generalized induced subgraph game.

DYNAMIC COOPERATIVE GAMES

2000 ◽  
Vol 02 (01) ◽  
pp. 47-65 ◽  
Author(s):  
JERZY A. FILAR ◽  
LEON A. PETROSJAN
Keyword(s):  
Characteristic Function ◽  
Cooperative Game ◽  
Continuous Time ◽  
Cooperative Games ◽  
Solution Concept ◽  
Time Consistency ◽  
Standard Solution ◽  
Instantaneous Characteristic ◽  
Characteristic Function Form ◽  
Over Time

We consider dynamic cooperative games in characteristic function form in the sense that the characteristic function evolves over time in accordance with a difference or differential equation that is influenced not only by the current ("instantaneous") characteristic function but also by the solution concept used to allocate the benefits of cooperation among the players. The latter solution concept can be any one of a number of now standard solution concepts of cooperative game theory but, for demonstration purposes, we focus on the core and the Shapley value. In the process, we introduce some new mechanisms by which players may regard the evolution of cooperative game over time and analyse them with respect to the goal of attaining time consistency either in discrete or in continuous time setting. In discrete time, we illustrate the phenomena that can arise when an allocation according to a given solution concept is used to adapt the values of coalitions at successive time points. In continuous time, we introduce the notion of an "instantaneous" game and its integration over time.

Irrational-Behavior-Proof Conditions Based on Limit Characteristic Functions

2019 ◽  
Vol 7 (1) ◽  
pp. 1-16
Author(s):  
Cui Liu ◽  
Hongwei Gao ◽  
Ovanes Petrosian ◽  
Juan Xue ◽  
Lei Wang
Keyword(s):  
Characteristic Function ◽  
Cooperative Game ◽  
Shapley Value ◽  
Cooperative Games ◽  
Characteristic Functions ◽  
The Core ◽  
The Shapley Value ◽  
Irrational Behavior ◽  
The Individual

Abstract Irrational-behavior-proof (IBP) conditions are important aspects to keep stable cooperation in dynamic cooperative games. In this paper, we focus on the establishment of IBP conditions. Firstly, the relations of three kinds of IBP conditions are described. An example is given to show that they may not hold, which could lead to the fail of cooperation. Then, based on a kind of limit characteristic function, all these conditions are proved to be true along the cooperative trajectory in a transformed cooperative game. It is surprising that these facts depend only upon the individual rationalities of players for the Shapley value and the group rationalities of players for the core. Finally, an illustrative example is given.

A Solution Concept Related to “Bounded Rationality” for some Two-Echelon Models

2019 ◽  
Vol 21 (01) ◽  
pp. 1940005
Author(s):  
Joaquin Sanchez-Soriano ◽  
Natividad Llorca
Keyword(s):  
Supply Chain ◽  
Bounded Rationality ◽  
Cooperative Games ◽  
Solution Concept ◽  
Theoretical Concept ◽  
Solution Concepts ◽  
Transportation Problems ◽  
The Core ◽  
Core Catcher ◽  
Two Sided Markets

Two-echelon models describe situations in which there are two differentiated groups of agents. Some examples of these models can be found in supply chain problems, transportation problems or two-sided markets. In this paper, we deal with two-sided transportation problems which can be used to describe a wide variety of logistic and market problems. We approach the problem from the perspective of cooperative games and study some solution concepts closely related to the game theoretical concept of core, but rather than focus specifically on the core of a transportation game, we introduce and study a new solution concept, a core catcher, which can be motivated by a kind of bounded rationality which can arise in these cooperative contexts.

SOME EXTENSIONS OF A LEMMA OF KOTLARSKI

2012 ◽  
Vol 28 (4) ◽  
pp. 925-932 ◽  
Author(s):  
Kirill Evdokimov ◽  
Halbert White
Keyword(s):  
Characteristic Function ◽  
Characteristic Functions ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Weaker Conditions

This note demonstrates that the conditions of Kotlarski’s (1967, Pacific Journal of Mathematics 20(1), 69–76) lemma can be substantially relaxed. In particular, the condition that the characteristic functions of M, U1, and U2 are nonvanishing can be replaced with much weaker conditions: The characteristic function of U1 can be allowed to have real zeros, as long as the derivative of its characteristic function at those points is not also zero; that of U2 can have an isolated number of zeros; and that of M need satisfy no restrictions on its zeros. We also show that Kotlarski’s lemma holds when the tails of U1 are no thicker than exponential, regardless of the zeros of the characteristic functions of U1, U2, or M.

Certain Fourier Transforms of Distributions: II

1954 ◽  
Vol 6 ◽  
pp. 186-189 ◽  
Author(s):  
Eugene Lukacs ◽  
Otto Szász
Keyword(s):  
Characteristic Function ◽  
Fourier Transforms ◽  
Necessary Conditions ◽  
Laplace Transforms ◽  
Characteristic Functions ◽  
Necessary Condition ◽  
Multiple Roots ◽  
Not Given ◽  
Class Of Functions

In an earlier paper (1), published in this journal, a necessary condition was given which the reciprocal of a polynomial without multiple roots must satisfy in order to be a characteristic function. This condition is, however, valid for a wider class of functions since it can be shown (2, theorem 2 and corollary to theorem 3) that it holds for all analytic characteristic functions. The proof given in (1) is elementary and has some methodological interest since it avoids the use of theorems on singularities of Laplace transforms. Moreover the method used in (1) yields some additional necessary conditions which were not given in (1) and which do not seem to follow easily from the properties of analytic characteristic functions.

A new weighting scheme for arc based circle cone-beam CT reconstruction

2021 ◽  
pp. 1-19
Author(s):  
Wei Wang ◽  
Xiang-Gen Xia ◽  
Chuanjiang He ◽  
Zemin Ren ◽  
Jian Lu
Keyword(s):  
Characteristic Function ◽  
Weighting Function ◽  
Cone Beam Ct ◽  
Reconstruction Algorithm ◽  
Cone Beam ◽  
Characteristic Functions ◽  
Projection Data ◽  
Ct Reconstruction ◽  
Scanning Angle ◽  
Beam Geometry

In this paper, we present an arc based fan-beam computed tomography (CT) reconstruction algorithm by applying Katsevich’s helical CT image reconstruction formula to 2D fan-beam CT scanning data. Specifically, we propose a new weighting function to deal with the redundant data. Our weighting function ϖ ( x _ , λ ) is an average of two characteristic functions, where each characteristic function indicates whether the projection data of the scanning angle contributes to the intensity of the pixel x _ . In fact, for every pixel x _ , our method uses the projection data of two scanning angle intervals to reconstruct its intensity, where one interval contains the starting angle and another contains the end angle. Each interval corresponds to a characteristic function. By extending the fan-beam algorithm to the circle cone-beam geometry, we also obtain a new circle cone-beam CT reconstruction algorithm. To verify the effectiveness of our method, the simulated experiments are performed for 2D fan-beam geometry with straight line detectors and 3D circle cone-beam geometry with flat-plan detectors, where the simulated sinograms are generated by the open-source software “ASTRA toolbox.” We compare our method with the other existing algorithms. Our experimental results show that our new method yields the lowest root-mean-square-error (RMSE) and the highest structural-similarity (SSIM) for both reconstructed 2D and 3D fan-beam CT images.

Integral Limit Laws for Additive Functions

1973 ◽  
Vol 25 (1) ◽  
pp. 194-203
Author(s):  
J. Galambos
Keyword(s):  
Characteristic Function ◽  
Fourier Transforms ◽  
Mean Value ◽  
Characteristic Functions ◽  
Asymptotic Formulas ◽  
Mean Value Theorem ◽  
Multiplicative Functions ◽  
Additive Functions ◽  
Limit Laws ◽  
Integral Limit

In the present paper a general form of integral limit laws for additive functions is obtained. Our limit law contains Kubilius’ results [5] on his class H. In the proof we make use of characteristic functions (Fourier transforms), which reduces our problem to finding asymptotic formulas for sums of multiplicative functions. This requires an extension of previous results in order to enable us to take into consideration the parameter of the characteristic function in question. We call this extension a parametric mean value theorem for multiplicative functions and its proof is analytic on the line of [4].

The Characteristic Function, a Method-Specific Alternative to the Horwitz Function

2012 ◽  
Vol 95 (6) ◽  
pp. 1803-1806 ◽  
Author(s):  
Michael Thompson
Keyword(s):  
Detection Limit ◽  
Characteristic Function ◽  
Characteristic Functions ◽  
Mathematical Form ◽  
Essential Difference ◽  
Food Sector ◽  
Wide Range ◽  
Specific Alternative ◽  
Additional Role

Abstract The Horwitz function is compared with the characteristic function as a descriptor of the precision of individual analytical methods. The Horwitz function describes the trend of reproducibility SDs observed in collaborative trials in the food sector over a wide range of concentrations of the analyte. However, it is imperfectly adaptable for describing the precision of individual methods, which is the role of the characteristic function. An essential difference between the two functions is that the characteristic function can accommodate a detection limit. This makes it a useful alternative when the precision of a method down to a detection limit is of interest. Many characteristic functions have a simple mathematical form, the parameters of which can be estimated with the usual resources. The Horwitz function serves an additional role as a fitness-for-purpose criterion in the form of the Horwitz ratio (HorRat). This use also has some shortcomings. The functional form of the characteristic function (with suitable prescribed parameters) is better adapted to this task.

The optical imagery of curved surfaces

1957 ◽  
Vol 240 (1223) ◽  
pp. 458-461
Keyword(s):  
Characteristic Function ◽  
Functional Equations ◽  
Individual Characteristic ◽  
Optical Systems ◽  
Characteristic Functions ◽  
Curved Surfaces ◽  
Angular Characteristic ◽  
Optical Imagery ◽  
The Individual ◽  
Angular Characteristic Function

The form of Hamilton’s angular characteristic function for the aberrationless imagery of one surface of rotation on another, and the connexions between the coefficients of the surface and functional equations, are found. When several optical systems of the type considered are arranged in succession the relations between the coefficients of the individual characteristic functions and those of the combination are obtained. These connexions enable all aberrations to be computed without resorting to ray tracing.

Sign in / Sign up

Export Citation Format

Share Document