American Institute of Mathematical Sciences

Advanced
January  2020, 7(1): 21-35. doi: 10.3934/jdg.2020002

Sequencing grey games

Serap Ergün 1, , Osman Palanci 2, , Sirma Zeynep Alparslan Gök 3,, , Şule Nizamoğlu 3, and  Gerhard Wilhelm Weber 4,

1. 

Isparta University of Applied Sciences, Faculty of Technology, Department of Computer Engineering, Isparta, Turkey
2. 

Süleyman Demirel University, Faculty of Economics and Business Administration, Department of Business Administration, Isparta, Turkey
3. 

Süleyman Demirel University, Faculty of Arts and Sciences, Department of Mathematics, Isparta, Turkey
4. 

Poznan University of Technology, Chair of Marketing and Economic Engineering, Poznan, Poland

* Corresponding author: zeynepalparslan@yahoo.com

Received  December 2018 Published  December 2019

The job scheduling problem is a notoriously difficult problem in combinatorial optimization and Operational Research. In this study, we handle the job scheduling problem by using a cooperative game theoretical approach. In the sequel, sequencing situations arising grom grey uncertainty are considered. Cooperative grey game theory is applied to analyze these situations. Further, grey sequencing games are constructed and grey equal gain splitting (GEGS) rule is introduced. It is shown that cooperative grey games are convex. An application is given based on Priority Based Scheduling Algorithm. The paper ends with a conclusion.

Keywords: Sequencing games, grey data, grey games, cooperative game theory, scheduling algorithm.
Mathematics Subject Classification: Primary: 91A12.
Citation: Serap Ergün, Osman Palanci, Sirma Zeynep Alparslan Gök, Şule Nizamoğlu, Gerhard Wilhelm Weber. Sequencing grey games. Journal of Dynamics & Games, 2020, 7 (1) : 21-35. doi: 10.3934/jdg.2020002
References:
[1]

S. Z. Alparslan Gök, R. Branzei, V. Fragnelli and S. Tijs, Sequencing interval situations and related games, CEJOR Cent. Eur. J. Oper. Res., 21 (2013), 225-236.  doi: 10.1007/s10100-011-0226-3.  Google Scholar

[2]

P. Borm, H. Hamers and R. Hendrickx, Operations research games: A survey, Top, 9 (2001), 139-216.  doi: 10.1007/BF02579075.  Google Scholar

[3]

M. E. Bruni, L. D. P. Pugliese, P. Beraldi and F. Guerriero, An adjustable robust optimization model for the resource-constrained project scheduling problem with uncertain activity durations, Omega, (2016), in press. Google Scholar

[4]

P. Calleja, M. A. Estevez-Fernandez, P. Borm and H. Hamers, Job scheduling, cooperation, and control, Operations Research Letters, 34 (2006), 22-28.  doi: 10.1016/j.orl.2005.01.007.  Google Scholar

[5]

I. Curiel, G. Pederzoli and S. Tijs, Sequencing games, European Journal of Operational Research, 40 (1989), 344-351.  doi: 10.1016/0377-2217(89)90427-X.  Google Scholar

[6]

I. Curiel, H. Hamers and F. Klijn, Sequencing games: A survey, Chapters in Game Theory, Theory Decis. Lib. Ser. C Game Theory Math. Program. Oper. Res., Kluwer Acad. Publ., Boston, MA, 31 (2002), 27-50.  doi: 10.1007/0-306-47526-X_2.  Google Scholar

[7]

J.-L. Deng, Control problems of grey systems, Systems and Control Letters, 1 (1981/82), 288-294. doi: 10.1016/S0167-6911(82)80025-X.  Google Scholar

[8]

D. G. Feitelson, L. Rudolph, U. Schwiegelshohn, K. C. Sevcik and P. Wong, Theory and practice in parallel job scheduling, Workshop on Job Scheduling Strategies for Parallel Processing, Springer Berlin Heidelberg, (1997), 1–34. Google Scholar

[9]

E. Köse and J. Y.-L. Forrest, N-person grey game, Kybernetes, 44 (2015), 271-282.  doi: 10.1108/K-04-2014-0073.  Google Scholar

[10]

E. L. Lawler, J. K. Lenstra, A. H. R. Kan and D. B. Shmoys, Sequencing and scheduling: Algorithms and complexity, Handbooks in Operations Research and Management Science, 4 (1993), 445-522.   Google Scholar

[11]

I. S. Lee and S. H. Yoon, Coordinated scheduling of production and delivery stages with stage-dependent inventory holding costs, Omega, 38 (2010), 509-521.   Google Scholar

[12]

R. E. Moore, Methods and Applications of Interval Analysis, SIAM Studies in Applied Mathematics, 2. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1979.  Google Scholar

[13]

M. O. Olgun, S. Z. Alparslan Gök and G. Özdemir, Cooperative grey games and an application on economic order quantity model, Kybernetes, 45 (2016), 828-838.  doi: 10.1108/K-06-2015-0160.  Google Scholar

[14]

O. Palancı, S. Z. Alparslan Gök, S. Ergün and G.-W. Weber, Cooperative grey games and grey Shapley value, Optimization, 64 (2015), 1657-1668.  doi: 10.1080/02331934.2014.956743.  Google Scholar

[15]

R. Ramasesh, Dynamic job shop scheduling: A survey of simulation research, Omega, 18 (1990), 43-57.   Google Scholar

[16]

S. K. Roy, G. Maity and G.-W. Weber, Multi-objective two-stage grey transportation problem using utility function with goals, CEJOR Cent. Eur. J. Oper. Res., 25 (2017), 417-439.  doi: 10.1007/s10100-016-0464-5.  Google Scholar

[17]

W. E. Smith, Various optimizer for single-stage production, Naval Research Logistics Quarterly, 3 (1956), 59-66.  doi: 10.1002/nav.3800030106.  Google Scholar

[18]

W. Stallings and G. K. Paul, Operating systems: Internals and design principles, Upper Saddle River, NJ: Prentice Hall, 3 (1998). Google Scholar

[19]

H. Wu and Z. Fang, The graphical solution of zero-sum two-person grey games, Proceedings of 2007 IEEE International Conference on Grey Systems and Intelligent Services, 1/2 (2007), 1617-1620.   Google Scholar

show all references

References:
[1]

S. Z. Alparslan Gök, R. Branzei, V. Fragnelli and S. Tijs, Sequencing interval situations and related games, CEJOR Cent. Eur. J. Oper. Res., 21 (2013), 225-236.  doi: 10.1007/s10100-011-0226-3.  Google Scholar

[2]

P. Borm, H. Hamers and R. Hendrickx, Operations research games: A survey, Top, 9 (2001), 139-216.  doi: 10.1007/BF02579075.  Google Scholar

[3]

M. E. Bruni, L. D. P. Pugliese, P. Beraldi and F. Guerriero, An adjustable robust optimization model for the resource-constrained project scheduling problem with uncertain activity durations, Omega, (2016), in press. Google Scholar

[4]

P. Calleja, M. A. Estevez-Fernandez, P. Borm and H. Hamers, Job scheduling, cooperation, and control, Operations Research Letters, 34 (2006), 22-28.  doi: 10.1016/j.orl.2005.01.007.  Google Scholar

[5]

I. Curiel, G. Pederzoli and S. Tijs, Sequencing games, European Journal of Operational Research, 40 (1989), 344-351.  doi: 10.1016/0377-2217(89)90427-X.  Google Scholar

[6]

I. Curiel, H. Hamers and F. Klijn, Sequencing games: A survey, Chapters in Game Theory, Theory Decis. Lib. Ser. C Game Theory Math. Program. Oper. Res., Kluwer Acad. Publ., Boston, MA, 31 (2002), 27-50.  doi: 10.1007/0-306-47526-X_2.  Google Scholar

[7]

J.-L. Deng, Control problems of grey systems, Systems and Control Letters, 1 (1981/82), 288-294. doi: 10.1016/S0167-6911(82)80025-X.  Google Scholar

[8]

D. G. Feitelson, L. Rudolph, U. Schwiegelshohn, K. C. Sevcik and P. Wong, Theory and practice in parallel job scheduling, Workshop on Job Scheduling Strategies for Parallel Processing, Springer Berlin Heidelberg, (1997), 1–34. Google Scholar

[9]

E. Köse and J. Y.-L. Forrest, N-person grey game, Kybernetes, 44 (2015), 271-282.  doi: 10.1108/K-04-2014-0073.  Google Scholar

[10]

E. L. Lawler, J. K. Lenstra, A. H. R. Kan and D. B. Shmoys, Sequencing and scheduling: Algorithms and complexity, Handbooks in Operations Research and Management Science, 4 (1993), 445-522.   Google Scholar

[11]

I. S. Lee and S. H. Yoon, Coordinated scheduling of production and delivery stages with stage-dependent inventory holding costs, Omega, 38 (2010), 509-521.   Google Scholar

[12]

R. E. Moore, Methods and Applications of Interval Analysis, SIAM Studies in Applied Mathematics, 2. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1979.  Google Scholar

[13]

M. O. Olgun, S. Z. Alparslan Gök and G. Özdemir, Cooperative grey games and an application on economic order quantity model, Kybernetes, 45 (2016), 828-838.  doi: 10.1108/K-06-2015-0160.  Google Scholar

[14]

O. Palancı, S. Z. Alparslan Gök, S. Ergün and G.-W. Weber, Cooperative grey games and grey Shapley value, Optimization, 64 (2015), 1657-1668.  doi: 10.1080/02331934.2014.956743.  Google Scholar

[15]

R. Ramasesh, Dynamic job shop scheduling: A survey of simulation research, Omega, 18 (1990), 43-57.   Google Scholar

[16]

S. K. Roy, G. Maity and G.-W. Weber, Multi-objective two-stage grey transportation problem using utility function with goals, CEJOR Cent. Eur. J. Oper. Res., 25 (2017), 417-439.  doi: 10.1007/s10100-016-0464-5.  Google Scholar

[17]

W. E. Smith, Various optimizer for single-stage production, Naval Research Logistics Quarterly, 3 (1956), 59-66.  doi: 10.1002/nav.3800030106.  Google Scholar

[18]

W. Stallings and G. K. Paul, Operating systems: Internals and design principles, Upper Saddle River, NJ: Prentice Hall, 3 (1998). Google Scholar

[19]

H. Wu and Z. Fang, The graphical solution of zero-sum two-person grey games, Proceedings of 2007 IEEE International Conference on Grey Systems and Intelligent Services, 1/2 (2007), 1617-1620.   Google Scholar

Figure 1.  An illustration of our application
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Gantt charts of D1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Gantt charts of D2
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Gantt charts of D3
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  The properties of each jobs of D1
Job Arrival Time Execute Time Priority Service Time
J1 $\left[ 0, 1\right] $ $\left[ 2, 2\right] $ 1 $\left[ 95, 101\right] $
J2 $\left[ 1, 3\right] $ $\left[ 3, 3\right] $ 2 $\left[ 191, 198\right] $
J3 $\left[ 3, 4\right] $ $\left[ 5, 5\right] $ 3 $\left[ 288, 294\right] $
Table Options

Download as excel

Job Arrival Time Execute Time Priority Service Time
J1 $\left[ 0, 1\right] $ $\left[ 2, 2\right] $ 1 $\left[ 95, 101\right] $
J2 $\left[ 1, 3\right] $ $\left[ 3, 3\right] $ 2 $\left[ 191, 198\right] $
J3 $\left[ 3, 4\right] $ $\left[ 5, 5\right] $ 3 $\left[ 288, 294\right] $
Table 2.  The properties of each jobs of D2
Job Arrival Time Execute Time Priority Service Time
J1 $\left[ 3, 5\right] $ $\left[ 3, 5\right] $ 2 $\left[ 153, 160\right] $
J2 $\left[ 0, 2\right] $ $\left[ 4, 6\right] $ 1 $\left[ 120, 127\right] $
J3 $\left[ 6, 8\right] $ $\left[ 7, 9\right] $ 3 $\left[ 186, 193\right] $
Table Options

Download as excel

Job Arrival Time Execute Time Priority Service Time
J1 $\left[ 3, 5\right] $ $\left[ 3, 5\right] $ 2 $\left[ 153, 160\right] $
J2 $\left[ 0, 2\right] $ $\left[ 4, 6\right] $ 1 $\left[ 120, 127\right] $
J3 $\left[ 6, 8\right] $ $\left[ 7, 9\right] $ 3 $\left[ 186, 193\right] $
Table 3.  The properties of each jobs of D3
Job Arrival Time Execute Time Priority Service Time
J1 $\left[ 2, 4\right] $ $\left[ 2, 2\right] $ 2 $\left[ 124, 132\right] $
J2 $\left[ 4, 7\right] $ $\left[ 3, 3\right] $ 3 $\left[ 152, 160\right] $
J3 $\left[ 0, 3\right] $ $\left[ 4, 4\right] $ 1 $\left[ 90, 98\right] $
Table Options

Download as excel

Job Arrival Time Execute Time Priority Service Time
J1 $\left[ 2, 4\right] $ $\left[ 2, 2\right] $ 2 $\left[ 124, 132\right] $
J2 $\left[ 4, 7\right] $ $\left[ 3, 3\right] $ 3 $\left[ 152, 160\right] $
J3 $\left[ 0, 3\right] $ $\left[ 4, 4\right] $ 1 $\left[ 90, 98\right] $
Table 4.  The wait time t of each jobs of D1, D2 and D3
$ \textbf{Job (Process)}$ $\textbf{Wait Time}$
J1 of D1 $t_{11} = \left[ 95, 100\right] $
J2 of D1 $t_{12} = \left[ 180, 195\right] $
J3 of D1 $t_{13} = \left[ 285, 290\right] $
J1 of D2 $t_{21} = \left[ 150, 155\right] $
J2 of D2 $t_{22} = \left[ 120, 125\right] $
J3 of D2 $t_{23} = \left[ 180, 185\right] $
J1 of D3 $t_{31} = \left[ 120, 125\right] $
J2 of D3 $t_{32} = \left[ 150, 155\right] $
J3 of D3 $t_{33} = \left[ 90, 95\right] $
Table Options

Download as excel

$ \textbf{Job (Process)}$ $\textbf{Wait Time}$
J1 of D1 $t_{11} = \left[ 95, 100\right] $
J2 of D1 $t_{12} = \left[ 180, 195\right] $
J3 of D1 $t_{13} = \left[ 285, 290\right] $
J1 of D2 $t_{21} = \left[ 150, 155\right] $
J2 of D2 $t_{22} = \left[ 120, 125\right] $
J3 of D2 $t_{23} = \left[ 180, 185\right] $
J1 of D3 $t_{31} = \left[ 120, 125\right] $
J2 of D3 $t_{32} = \left[ 150, 155\right] $
J3 of D3 $t_{33} = \left[ 90, 95\right] $
Table 5.  The weights of c, d, n of J1 for D1, D2, D3
$ \textbf{Property of job} $ $\textbf{Compute Intensity} $ $ \textbf{Data parsing}$ $\textbf{Network}$
cost $c$ $d$ $n$
J1D1 3 2 1
J2D1 2 3 1
J3D1 1 2 3
J1D2 3 2 1
J2D2 1 3 2
J3D2 1 2 3
J1D3 3 1 2
J2D3 2 3 1
J3D3 1 1 1
Table Options

Download as excel

$ \textbf{Property of job} $ $\textbf{Compute Intensity} $ $ \textbf{Data parsing}$ $\textbf{Network}$
cost $c$ $d$ $n$
J1D1 3 2 1
J2D1 2 3 1
J3D1 1 2 3
J1D2 3 2 1
J2D2 1 3 2
J3D2 1 2 3
J1D3 3 1 2
J2D3 2 3 1
J3D3 1 1 1
Table 6.  Grey marginal vectors
$\sigma $ $m_{1}^{\sigma }\left( w^{\prime }\right) $ $m_{2}^{\sigma }\left( w^{\prime }\right) $ $m_{3}^{\sigma }\left( w^{\prime }\right) $
$\sigma _{1} = \left( 1, 2, 3\right) $ $m_{1}^{\sigma _{1}}\left( w^{\prime }\right) \in \left[ 0, 0\right] $ $m_{2}^{\sigma _{1}}\left( w^{\prime }\right) \in \left[ 0, 0\right] $ $m_{3}^{\sigma _{1}}\left( w^{\prime }\right) \in \left[ 68500, 72850\right] $
$\sigma _{2} = \left( 1, 3, 2\right) $ $m_{1}^{\sigma _{2}}\left( w^{\prime }\right) \in \left[ 0, 0\right] $ $m_{2}^{\sigma _{2}}\left( w^{\prime }\right) \in \left[ 68500, 72850\right] $ $m_{3}^{\sigma _{2}}\left( w^{\prime }\right) \in \left[ 0, 0\right] $
$\sigma _{3} = \left( 2, 1, 3\right) $ $m_{1}^{\sigma _{3}}\left( w^{\prime }\right) \in \left[ 0, 0\right] $ $m_{2}^{\sigma _{3}}\left( w^{\prime }\right) \in \left[ 0, 0\right] $ $m_{3}^{\sigma _{3}}\left( w^{\prime }\right) \in \left[ 68500, 72850\right] $
$\sigma _{4} = \left( 2, 3, 1\right) $ $m_{1}^{\sigma _{4}}\left( w^{\prime }\right) \in \left[ 26500, 28050\right] $ $m_{2}^{\sigma _{4}}\left( w^{\prime }\right) \in \left[ 0, 0\right] $ $m_{3}^{\sigma _{4}}\left( w^{\prime }\right) \in \left[ 42000, 44800\right] $
$\sigma _{5} = \left( 3, 1, 2\right) $ $m_{1}^{\sigma _{5}}\left( w^{\prime }\right) \in \left[ 0, 0\right] $ $m_{2}^{\sigma _{5}}\left( w^{\prime }\right) \in \left[ 68500, 72850\right] $ $m_{3}^{\sigma _{5}}\left( w^{\prime }\right) \in \left[ 0, 0\right] $
$\sigma _{6} = \left( 3, 2, 1\right) $ $m_{1}^{\sigma _{6}}\left( w^{\prime }\right) \in \left[ 26500, 28050\right] $ $m_{2}^{\sigma _{6}}\left( w^{\prime }\right) \in \left[ 42000, 44800\right] $ $m_{3}^{\sigma _{6}}\left( w^{\prime }\right) \in \left[ 0, 0\right] $
Table Options

Download as excel

$\sigma $ $m_{1}^{\sigma }\left( w^{\prime }\right) $ $m_{2}^{\sigma }\left( w^{\prime }\right) $ $m_{3}^{\sigma }\left( w^{\prime }\right) $
$\sigma _{1} = \left( 1, 2, 3\right) $ $m_{1}^{\sigma _{1}}\left( w^{\prime }\right) \in \left[ 0, 0\right] $ $m_{2}^{\sigma _{1}}\left( w^{\prime }\right) \in \left[ 0, 0\right] $ $m_{3}^{\sigma _{1}}\left( w^{\prime }\right) \in \left[ 68500, 72850\right] $
$\sigma _{2} = \left( 1, 3, 2\right) $ $m_{1}^{\sigma _{2}}\left( w^{\prime }\right) \in \left[ 0, 0\right] $ $m_{2}^{\sigma _{2}}\left( w^{\prime }\right) \in \left[ 68500, 72850\right] $ $m_{3}^{\sigma _{2}}\left( w^{\prime }\right) \in \left[ 0, 0\right] $
$\sigma _{3} = \left( 2, 1, 3\right) $ $m_{1}^{\sigma _{3}}\left( w^{\prime }\right) \in \left[ 0, 0\right] $ $m_{2}^{\sigma _{3}}\left( w^{\prime }\right) \in \left[ 0, 0\right] $ $m_{3}^{\sigma _{3}}\left( w^{\prime }\right) \in \left[ 68500, 72850\right] $
$\sigma _{4} = \left( 2, 3, 1\right) $ $m_{1}^{\sigma _{4}}\left( w^{\prime }\right) \in \left[ 26500, 28050\right] $ $m_{2}^{\sigma _{4}}\left( w^{\prime }\right) \in \left[ 0, 0\right] $ $m_{3}^{\sigma _{4}}\left( w^{\prime }\right) \in \left[ 42000, 44800\right] $
$\sigma _{5} = \left( 3, 1, 2\right) $ $m_{1}^{\sigma _{5}}\left( w^{\prime }\right) \in \left[ 0, 0\right] $ $m_{2}^{\sigma _{5}}\left( w^{\prime }\right) \in \left[ 68500, 72850\right] $ $m_{3}^{\sigma _{5}}\left( w^{\prime }\right) \in \left[ 0, 0\right] $
$\sigma _{6} = \left( 3, 2, 1\right) $ $m_{1}^{\sigma _{6}}\left( w^{\prime }\right) \in \left[ 26500, 28050\right] $ $m_{2}^{\sigma _{6}}\left( w^{\prime }\right) \in \left[ 42000, 44800\right] $ $m_{3}^{\sigma _{6}}\left( w^{\prime }\right) \in \left[ 0, 0\right] $
[1]

Eduardo Espinosa-Avila, Pablo Padilla Longoria, Francisco Hernández-Quiroz. Game theory and dynamic programming in alternate games. Journal of Dynamics & Games, 2017, 4 (3) : 205-216. doi: 10.3934/jdg.2017013
[2]

J-F. Clouët, R. Sentis. Milne problem for non-grey radiative transfer. Kinetic & Related Models, 2009, 2 (2) : 345-362. doi: 10.3934/krm.2009.2.345
[3]

Deng-Feng Li, Yin-Fang Ye, Wei Fei. Extension of generalized solidarity values to interval-valued cooperative games. Journal of Industrial & Management Optimization, 2020, 16 (2) : 919-931. doi: 10.3934/jimo.2018185
[4]

Ekaterina Gromova, Ekaterina Marova, Dmitry Gromov. A substitute for the classical Neumann–Morgenstern characteristic function in cooperative differential games. Journal of Dynamics & Games, 2020, 7 (2) : 105-122. doi: 10.3934/jdg.2020007
[5]

Zeyang Wang, Ovanes Petrosian. On class of non-transferable utility cooperative differential games with continuous updating. Journal of Dynamics & Games, 2020  doi: 10.3934/jdg.2020020
[6]

Fabián Crocce, Ernesto Mordecki. A non-iterative algorithm for generalized pig games. Journal of Dynamics & Games, 2018, 5 (4) : 331-341. doi: 10.3934/jdg.2018020
[7]

Leon Petrosyan, David Yeung. Shapley value for differential network games: Theory and application. Journal of Dynamics & Games, 2020  doi: 10.3934/jdg.2020021
[8]

Kuang Huang, Xuan Di, Qiang Du, Xi Chen. A game-theoretic framework for autonomous vehicles velocity control: Bridging microscopic differential games and macroscopic mean field games. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020131
[9]

Serap Ergün, Bariş Bülent Kırlar, Sırma Zeynep Alparslan Gök, Gerhard-Wilhelm Weber. An application of crypto cloud computing in social networks by cooperative game theory. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1927-1941. doi: 10.3934/jimo.2019036
[10]

Mohamed A. Tawhid, Ahmed F. Ali. A simplex grey wolf optimizer for solving integer programming and minimax problems. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 301-323. doi: 10.3934/naco.2017020
[11]

Serap Ergün, Sirma Zeynep Alparslan Gök, Tuncay Aydoǧan, Gerhard Wilhelm Weber. Performance analysis of a cooperative flow game algorithm in ad hoc networks and a comparison to Dijkstra's algorithm. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1085-1100. doi: 10.3934/jimo.2018086
[12]

Jiahua Zhang, Shu-Cherng Fang, Yifan Xu, Ziteng Wang. A cooperative game with envy. Journal of Industrial & Management Optimization, 2017, 13 (4) : 2049-2066. doi: 10.3934/jimo.2017031
[13]

Alan Beggs. Learning in monotone bayesian games. Journal of Dynamics & Games, 2015, 2 (2) : 117-140. doi: 10.3934/jdg.2015.2.117
[14]

Konstantin Avrachenkov, Giovanni Neglia, Vikas Vikram Singh. Network formation games with teams. Journal of Dynamics & Games, 2016, 3 (4) : 303-318. doi: 10.3934/jdg.2016016
[15]

Hassan Najafi Alishah, Pedro Duarte. Hamiltonian evolutionary games. Journal of Dynamics & Games, 2015, 2 (1) : 33-49. doi: 10.3934/jdg.2015.2.33
[16]

Yonghui Zhou, Jian Yu, Long Wang. Topological essentiality in infinite games. Journal of Industrial & Management Optimization, 2012, 8 (1) : 179-187. doi: 10.3934/jimo.2012.8.179
[17]

Carlos Hervés-Beloso, Emma Moreno-García. Market games and walrasian equilibria. Journal of Dynamics & Games, 2020, 7 (1) : 65-77. doi: 10.3934/jdg.2020004
[18]

Dario Bauso, Thomas W.L. Norman. Approachability in population games. Journal of Dynamics & Games, 2020  doi: 10.3934/jdg.2020019
[19]

Elena Travaglia, Valentina La Morgia, Ezio Venturino. Poxvirus, red and grey squirrel dynamics: Is the recovery of a common predator affecting system equilibria? Insights from a predator-prey ecoepidemic model. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2023-2040. doi: 10.3934/dcdsb.2019200
[20]

Daniel Brinkman, Christian Ringhofer. A kinetic games framework for insurance plans. Kinetic & Related Models, 2017, 10 (1) : 93-116. doi: 10.3934/krm.2017004

 Impact Factor: 

Tools

Metrics

  • PDF downloads (116)
  • HTML views (252)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences