American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Control systems of interacting objects modeled as a game against nature under a mean field approach
  • JDG Home
  • This Issue
  • Next Article
    Global analysis of solutions on the Cournot-Theocharis duopoly with variable marginal costs
January  2017, 4(1): 41-58. doi: 10.3934/jdg.2017003

Price of anarchy for graph coloring games with concave payoff

Lasse Kliemann , , Elmira Shirazi Sheykhdarabadi and  Anand Srivastav

Department of Computer Science, Kiel University, Christian-Albrechts-Platz 4,24118 Kiel, Germany

* Corresponding author: Lasse Kliemann lki@informatik.uni-kiel.de

Received  February 2016 Revised  November 2016 Published  December 2016

Fund Project: L. Kliemann is supported by DFG grants KL 2087/1-1 and SR 7/15-1. E. Shirazi Sheykhdarabadi is supported by a Federal State Scholarship at Kiel University.

We study the price of anarchy in graph coloring games (a subclass of polymatrix common-payoff games). Players are vertices of an undirected graph, and the strategies for each player are the colors $\left\{ {1, \ldots ,k} \right\}$. A tight bound of $\frac{k}{k-1}$ is known (Hoefer 2007, Kun et al. 2013), if each player's payoff is the number of neighbors with different color than herself.

In our generalization, payoff is computed by determining the distance of the player's color to the color of each neighbor, applying a concave function $f$ to each distance, and then summing up the resulting values. This is motivated, e. g., by spectrum sharing, and includes the payoff functions suggested by Kun et al. (2013) for future work as special cases.

Denote $f^*$ the maximum value that $f$ attains on $\left\{ {0, \ldots ,k - 1} \right\}$. We prove an upper bound of $2$ on the price of anarchy if $f$ is non-decreasing or assumes $f^*$ somewhere in $\left\{ {0, \ldots ,{\frac{k}{2}}} \right\}$. Matching lower bounds are given for the monotone case and if $f^*$ is assumed in $\frac{k}{2}$ for even $k$. For general concave $f$, we prove an upper bound of $3$. We use a new technique that works by an appropriate splitting $\lambda = \lambda_1 + \ldots + \lambda_k$ of the bound $\lambda$ we are proving.

Keywords: Graph coloring, strategic game, price of anarchy, concave function, spectrum sharing.
Mathematics Subject Classification: Primary: 91A43, 91A06; Secondary: 05C15.
Citation: Lasse Kliemann, Elmira Shirazi Sheykhdarabadi, Anand Srivastav. Price of anarchy for graph coloring games with concave payoff. Journal of Dynamics & Games, 2017, 4 (1) : 41-58. doi: 10.3934/jdg.2017003
References:
[1]

K. I. Aardal, S. P. M. van Hoesel, A. M. C. A. Koster, C. Mannino and A. Sassano, Models and solution techniques for frequency assignment problems, Annals of Operations Research, 153 (2007), 79-129.  doi: 10.1007/s10479-007-0178-0.  Google Scholar

[2]

S. M. Allen, D. H. Smith and S. Hurley, Generation of lower bounds for minimum span frequency assignment, Discrete Applied Mathematics, 119 (2002), 59-78.  doi: 10.1016/S0166-218X(01)00265-7.  Google Scholar

[3]

K. Apt, M. Rahn, G. Schäfer and S. Simon, Coordination games on graphs, International Journal of Game Theory, (2016), 1-27, available from http://arxiv.org/abs/1501.07388. doi: 10.1007/s00182-016-0560-8.  Google Scholar

[4]

L. Barenboim and M. Elkin, Distributed Graph Coloring: Fundamentals and Recent Developments Morgan & Claypool Publishers, 2013. doi: 10.2200/S00520ED1V01Y201307DCT011.  Google Scholar

[5]

S. Bosio, A. Eisenblätter, H. -F. Geerdes, I. Siomina and D. Yuan, Mathematical optimization models for WLAN planning, Graphs and Algorithms in Communication Networks (Arie M. C. A. Koster and Xavier Muñoz, eds. ), Springer-Verlag Berlin Heidelberg, 2010,283-309. doi: 10.1007/978-3-642-02250-0_11.  Google Scholar

[6]

R. Leonard Brooks, On colouring the nodes of a network, Mathematical Proceedings of the Cambridge Philosophical Society, 37 (1941), 194-197.  doi: 10.1017/S030500410002168X.  Google Scholar

[7]

Y. Cai and C. Daskalakis, On minmax theorems for multiplayer games, Proceedings of the 22th Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, CA, USA, January 2011 (SODA 2011), 2011,217{234, available from http://hdl.handle.net/1721.1/73129.  Google Scholar

[8]

T. Calamoneri, The L(h, k)-labelling problem: An updated survey and annotated bibliography, The Computer Journal, 54 (2011), 1344-1371.  doi: 10.1093/comjnl/bxr037.  Google Scholar

[9]

I. Chatzigiannakis, C. Koninis, P. N. Panagopoulou and P. G. Spirakis, Distributed gametheoretic vertex coloring, Principles of Distributed Systems, Volume 6490 of the series Lecture Notes in Computer Science, 2010,103-118. doi: 10.1007/978-3-642-17653-1_9.  Google Scholar

[10]

K. Chaudhuri1, F. Chung Graham and M. S. Jamall, A network coloring game, Proceedings of the 4th International Workshop on Internet and Network Economics, Shanghai, China, December 2008 (WINE 2008), 5385 (2008), 522-530. doi: 10.1007/978-3-540-92185-1_58.  Google Scholar

[11]

E. Driouch and W. Ajib, Greedy spectrum sharing for cognitive MIMO networks, Proceedings of the 4th IEEE International Conference on Communications and Information Technology, Hammamet, Tunisia, June 2012 (IEEE ICCIT 2012), 2012,139-143. doi: 10.1109/ICCITechnol.2012.6285777.  Google Scholar

[12]

B. Escoffier and J. Monnot and L. Gourvès, Strategic coloring of a graph, Algorithms and complexity, Lecture Notes in Comput. Sci., Springer, Berlin, 6078 (2010), 155-166. doi: 10.1007/978-3-642-13073-1_15.  Google Scholar

[13]

F. H. P. Fitzek and M. D. Katz, Cognitive Wireless Networks: Concepts, Methodologies and Visions Inspiring the Age of Enlightenment of Wireless Communications Springer Netherlands, 2007. doi: 10.1007/978-1-4020-5979-7.  Google Scholar

[14]

A. Gamst, Some lower bounds for a class of frequency assignment problems, IEEE Transactions on Vehicular Technology, 35 (1986), 8-14.  doi: 10.1109/T-VT.1986.24063.  Google Scholar

[15]

L. Gourvès and J. Monnot, On strong equilibria in the max cut game, Internet and Network Economics, Volume 5929 of the series Lecture Notes in Computer Science, (2009), 608-615. doi: 10.1007/978-3-642-10841-9_62.  Google Scholar

[16]

W. K. Hale, Frequency assignment: Theory and applications, Proceedings of the IEEE, 68 (1980), 1497-1514.  doi: 10.1109/PROC.1980.11899.  Google Scholar

[17]

M. M. Halldórsson, J. Y. Halpern, L. E. Li and V. S. Mirrokni, On spectrum sharing games, Distributed Computing, 22 (2010), 235-248, Conference version at PODC 2004.  doi: 10.1007/s00446-010-0098-0.  Google Scholar

[18]

T. Hasunuma, T. Ishii, H. Ono and Y. Uno, Algorithmic aspects of distance constrained labeling: A survey, International Journal of Networking and Computing, 4 (2014), 251-259, available from http://www.ijnc.org/index.php/ijnc/article/view/85. doi: 10.15803/ijnc.4.2_251.  Google Scholar

[19]

M. Hoefer, Cost Sharing and Clustering Under Distributed Competition, Ph. D. thesis, Department of Computer and Information Science, University of Konstanz, 2007, available from https://people.mpi-inf.mpg.de/~mhoefer/05-07/diss.pdf. Google Scholar

[20]

J. C. M. Janssen, Channel Assignment and Graph Labeling Handbook of Wireless Networks and Mobile Computing, John Wiley & Sons, Inc., New York, USA, 2002. doi: 10.1002/0471224561.ch5.  Google Scholar

[21]

D. R. Karger, R. Motwani and M. Sudan, Approximate graph coloring by semidefinite programming, Journal of the ACM, 45 (1998), 246-265, Conference version at FOCS 1994.  doi: 10.1145/274787.274791.  Google Scholar

[22]

R. M. Karp, Reducibility among combinatorial problems, Complexity of Computer Computations, Springer US, 1972, 85-103, available from http://cgi.di.uoa.gr/~sgk/teaching/grad/handouts/karp.pdf. doi: 10.1007/978-1-4684-2001-2_9.  Google Scholar

[23]

M. Kearns, S. Suri and N. Montfort, An experimental study of the coloring problem on human subject networks, Science, 313 (2006), 824-827.  doi: 10.1126/science.1127207.  Google Scholar

[24]

Elias Koutsoupias and Christos H. Papadimitriou, Worst-case equilibria, Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science, Trier, Germany, March 1999 (STACS 1999), 1563 (1999), 404-413.  doi: 10.1007/3-540-49116-3_38.  Google Scholar

[25]

J. Kun, B. Powers and L. Reyzin, Anti-coordination games and stable graph colorings, Proceedings of the 6th Annual ACM-SIAM Symposium on Algorithmic Game Theory, Aachen, Germany, October 2013 (SAGT 2013) (Berthold Vöcking, ed. ), Lecture Notes in Computer Science, 8146 (2013), 122-133. doi: 10.1007/978-3-642-41392-6_11.  Google Scholar

[26]

K. Leyton-Brown and Y. Shoham, Essentials of Game Theory: A Concise Multidisceplanary Introduction Morgan & Claypool Publishers, 2008, available from http://www.morganclaypool.com/doi/abs/10.2200/S00108ED1V01Y200802AIM003.  Google Scholar

[27]

L. Lovász, Three short proofs in graph theory, Journal of Combinatorial Theory, Series B, 19 (1975), 269-271.  doi: 10.1016/0095-8956(75)90089-1.  Google Scholar

[28]

P. N. Panagopoulou and P. G. Spirakis, A game theoretic approach for efficient graph coloring, Proceedings of the 19th International Symposium on Algorithms and Computation, Gold Coast, Australia, December 2008 (ISAAC 2008), 5369 (2008), 183-195.  doi: 10.1007/978-3-540-92182-0_19.  Google Scholar

[29]

C. H. Papadimitriou, Algorithms, games, and the Internet, Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, Crete, Greece, July 2001 (STOC 2001), 2001, Extended abstract at ICALP, 2001,749-753. doi: 10.1145/380752.380883.  Google Scholar

[30]

C. Peng, H. Zheng and B. Y. Zhao, Utilization and fairness in spectrum assignment for opportunistic spectrum access, Mobile Networks and Applications, 11 (2006), 555-576.  doi: 10.1007/s11036-006-7322-y.  Google Scholar

[31]

M. Rahn and G. Schäfer, Efficient equilibria in polymatrix coordination games, Mathematical foundations of computer science 2015, Part Ⅱ, 529-541, Lecture Notes in Comput. Sci., 9235, Springer, Heidelberg, 2015, available from http://arxiv.org/abs/1504.07518. doi: 10.1007/978-3-662-48054-0_44.  Google Scholar

[32]

F. S. Roberts, T-colorings of graphs: Recent results and open problems, Discrete Mathematics, 93 (1991), 229-245.  doi: 10.1016/0012-365X(91)90258-4.  Google Scholar

[33]

O. Schink, Der Price of Anarchy und Die Komplexität von Stabilen Graphfärbungen, Master's thesis, Christian-Albrechts-Universität Kiel, Mathe-matisches Seminar, 2014. Google Scholar

[34]

K. Smith and M. Palaniswami, Static and dynamic channel assignment using neural networks, IEEE Journal on Selected Areas in Communications, 15 (2002), 238-249.  doi: 10.1109/49.552073.  Google Scholar

[35]

J. van den Heuvel, R. A. Leese and M. A. Shepherd, Graph labeling and radio channel assignment, Journal of Graph Theory, 29 (1998), 263-283.  doi: 10.1002/(SICI)1097-0118(199812)29:4<263::AID-JGT5>3.0.CO;2-V.  Google Scholar

[36]

E. Borisovna Yanovskaya, Equilibrium points in polymatrix games, Litovskii Matematicheskii Sbornik, 8 (1968), 381-384, In Russian.   Google Scholar

[37]

R. K. Yeh, A survey on labeling graphs with a condition at distance two, Discrete Mathematics, 306 (2006), 1217-1231.  doi: 10.1016/j.disc.2005.11.029.  Google Scholar

show all references

References:
[1]

K. I. Aardal, S. P. M. van Hoesel, A. M. C. A. Koster, C. Mannino and A. Sassano, Models and solution techniques for frequency assignment problems, Annals of Operations Research, 153 (2007), 79-129.  doi: 10.1007/s10479-007-0178-0.  Google Scholar

[2]

S. M. Allen, D. H. Smith and S. Hurley, Generation of lower bounds for minimum span frequency assignment, Discrete Applied Mathematics, 119 (2002), 59-78.  doi: 10.1016/S0166-218X(01)00265-7.  Google Scholar

[3]

K. Apt, M. Rahn, G. Schäfer and S. Simon, Coordination games on graphs, International Journal of Game Theory, (2016), 1-27, available from http://arxiv.org/abs/1501.07388. doi: 10.1007/s00182-016-0560-8.  Google Scholar

[4]

L. Barenboim and M. Elkin, Distributed Graph Coloring: Fundamentals and Recent Developments Morgan & Claypool Publishers, 2013. doi: 10.2200/S00520ED1V01Y201307DCT011.  Google Scholar

[5]

S. Bosio, A. Eisenblätter, H. -F. Geerdes, I. Siomina and D. Yuan, Mathematical optimization models for WLAN planning, Graphs and Algorithms in Communication Networks (Arie M. C. A. Koster and Xavier Muñoz, eds. ), Springer-Verlag Berlin Heidelberg, 2010,283-309. doi: 10.1007/978-3-642-02250-0_11.  Google Scholar

[6]

R. Leonard Brooks, On colouring the nodes of a network, Mathematical Proceedings of the Cambridge Philosophical Society, 37 (1941), 194-197.  doi: 10.1017/S030500410002168X.  Google Scholar

[7]

Y. Cai and C. Daskalakis, On minmax theorems for multiplayer games, Proceedings of the 22th Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, CA, USA, January 2011 (SODA 2011), 2011,217{234, available from http://hdl.handle.net/1721.1/73129.  Google Scholar

[8]

T. Calamoneri, The L(h, k)-labelling problem: An updated survey and annotated bibliography, The Computer Journal, 54 (2011), 1344-1371.  doi: 10.1093/comjnl/bxr037.  Google Scholar

[9]

I. Chatzigiannakis, C. Koninis, P. N. Panagopoulou and P. G. Spirakis, Distributed gametheoretic vertex coloring, Principles of Distributed Systems, Volume 6490 of the series Lecture Notes in Computer Science, 2010,103-118. doi: 10.1007/978-3-642-17653-1_9.  Google Scholar

[10]

K. Chaudhuri1, F. Chung Graham and M. S. Jamall, A network coloring game, Proceedings of the 4th International Workshop on Internet and Network Economics, Shanghai, China, December 2008 (WINE 2008), 5385 (2008), 522-530. doi: 10.1007/978-3-540-92185-1_58.  Google Scholar

[11]

E. Driouch and W. Ajib, Greedy spectrum sharing for cognitive MIMO networks, Proceedings of the 4th IEEE International Conference on Communications and Information Technology, Hammamet, Tunisia, June 2012 (IEEE ICCIT 2012), 2012,139-143. doi: 10.1109/ICCITechnol.2012.6285777.  Google Scholar

[12]

B. Escoffier and J. Monnot and L. Gourvès, Strategic coloring of a graph, Algorithms and complexity, Lecture Notes in Comput. Sci., Springer, Berlin, 6078 (2010), 155-166. doi: 10.1007/978-3-642-13073-1_15.  Google Scholar

[13]

F. H. P. Fitzek and M. D. Katz, Cognitive Wireless Networks: Concepts, Methodologies and Visions Inspiring the Age of Enlightenment of Wireless Communications Springer Netherlands, 2007. doi: 10.1007/978-1-4020-5979-7.  Google Scholar

[14]

A. Gamst, Some lower bounds for a class of frequency assignment problems, IEEE Transactions on Vehicular Technology, 35 (1986), 8-14.  doi: 10.1109/T-VT.1986.24063.  Google Scholar

[15]

L. Gourvès and J. Monnot, On strong equilibria in the max cut game, Internet and Network Economics, Volume 5929 of the series Lecture Notes in Computer Science, (2009), 608-615. doi: 10.1007/978-3-642-10841-9_62.  Google Scholar

[16]

W. K. Hale, Frequency assignment: Theory and applications, Proceedings of the IEEE, 68 (1980), 1497-1514.  doi: 10.1109/PROC.1980.11899.  Google Scholar

[17]

M. M. Halldórsson, J. Y. Halpern, L. E. Li and V. S. Mirrokni, On spectrum sharing games, Distributed Computing, 22 (2010), 235-248, Conference version at PODC 2004.  doi: 10.1007/s00446-010-0098-0.  Google Scholar

[18]

T. Hasunuma, T. Ishii, H. Ono and Y. Uno, Algorithmic aspects of distance constrained labeling: A survey, International Journal of Networking and Computing, 4 (2014), 251-259, available from http://www.ijnc.org/index.php/ijnc/article/view/85. doi: 10.15803/ijnc.4.2_251.  Google Scholar

[19]

M. Hoefer, Cost Sharing and Clustering Under Distributed Competition, Ph. D. thesis, Department of Computer and Information Science, University of Konstanz, 2007, available from https://people.mpi-inf.mpg.de/~mhoefer/05-07/diss.pdf. Google Scholar

[20]

J. C. M. Janssen, Channel Assignment and Graph Labeling Handbook of Wireless Networks and Mobile Computing, John Wiley & Sons, Inc., New York, USA, 2002. doi: 10.1002/0471224561.ch5.  Google Scholar

[21]

D. R. Karger, R. Motwani and M. Sudan, Approximate graph coloring by semidefinite programming, Journal of the ACM, 45 (1998), 246-265, Conference version at FOCS 1994.  doi: 10.1145/274787.274791.  Google Scholar

[22]

R. M. Karp, Reducibility among combinatorial problems, Complexity of Computer Computations, Springer US, 1972, 85-103, available from http://cgi.di.uoa.gr/~sgk/teaching/grad/handouts/karp.pdf. doi: 10.1007/978-1-4684-2001-2_9.  Google Scholar

[23]

M. Kearns, S. Suri and N. Montfort, An experimental study of the coloring problem on human subject networks, Science, 313 (2006), 824-827.  doi: 10.1126/science.1127207.  Google Scholar

[24]

Elias Koutsoupias and Christos H. Papadimitriou, Worst-case equilibria, Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science, Trier, Germany, March 1999 (STACS 1999), 1563 (1999), 404-413.  doi: 10.1007/3-540-49116-3_38.  Google Scholar

[25]

J. Kun, B. Powers and L. Reyzin, Anti-coordination games and stable graph colorings, Proceedings of the 6th Annual ACM-SIAM Symposium on Algorithmic Game Theory, Aachen, Germany, October 2013 (SAGT 2013) (Berthold Vöcking, ed. ), Lecture Notes in Computer Science, 8146 (2013), 122-133. doi: 10.1007/978-3-642-41392-6_11.  Google Scholar

[26]

K. Leyton-Brown and Y. Shoham, Essentials of Game Theory: A Concise Multidisceplanary Introduction Morgan & Claypool Publishers, 2008, available from http://www.morganclaypool.com/doi/abs/10.2200/S00108ED1V01Y200802AIM003.  Google Scholar

[27]

L. Lovász, Three short proofs in graph theory, Journal of Combinatorial Theory, Series B, 19 (1975), 269-271.  doi: 10.1016/0095-8956(75)90089-1.  Google Scholar

[28]

P. N. Panagopoulou and P. G. Spirakis, A game theoretic approach for efficient graph coloring, Proceedings of the 19th International Symposium on Algorithms and Computation, Gold Coast, Australia, December 2008 (ISAAC 2008), 5369 (2008), 183-195.  doi: 10.1007/978-3-540-92182-0_19.  Google Scholar

[29]

C. H. Papadimitriou, Algorithms, games, and the Internet, Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, Crete, Greece, July 2001 (STOC 2001), 2001, Extended abstract at ICALP, 2001,749-753. doi: 10.1145/380752.380883.  Google Scholar

[30]

C. Peng, H. Zheng and B. Y. Zhao, Utilization and fairness in spectrum assignment for opportunistic spectrum access, Mobile Networks and Applications, 11 (2006), 555-576.  doi: 10.1007/s11036-006-7322-y.  Google Scholar

[31]

M. Rahn and G. Schäfer, Efficient equilibria in polymatrix coordination games, Mathematical foundations of computer science 2015, Part Ⅱ, 529-541, Lecture Notes in Comput. Sci., 9235, Springer, Heidelberg, 2015, available from http://arxiv.org/abs/1504.07518. doi: 10.1007/978-3-662-48054-0_44.  Google Scholar

[32]

F. S. Roberts, T-colorings of graphs: Recent results and open problems, Discrete Mathematics, 93 (1991), 229-245.  doi: 10.1016/0012-365X(91)90258-4.  Google Scholar

[33]

O. Schink, Der Price of Anarchy und Die Komplexität von Stabilen Graphfärbungen, Master's thesis, Christian-Albrechts-Universität Kiel, Mathe-matisches Seminar, 2014. Google Scholar

[34]

K. Smith and M. Palaniswami, Static and dynamic channel assignment using neural networks, IEEE Journal on Selected Areas in Communications, 15 (2002), 238-249.  doi: 10.1109/49.552073.  Google Scholar

[35]

J. van den Heuvel, R. A. Leese and M. A. Shepherd, Graph labeling and radio channel assignment, Journal of Graph Theory, 29 (1998), 263-283.  doi: 10.1002/(SICI)1097-0118(199812)29:4<263::AID-JGT5>3.0.CO;2-V.  Google Scholar

[36]

E. Borisovna Yanovskaya, Equilibrium points in polymatrix games, Litovskii Matematicheskii Sbornik, 8 (1968), 381-384, In Russian.   Google Scholar

[37]

R. K. Yeh, A survey on labeling graphs with a condition at distance two, Discrete Mathematics, 306 (2006), 1217-1231.  doi: 10.1016/j.disc.2005.11.029.  Google Scholar

[1]

Evan Greif, Daniel Kaplan, Robert S. Strichartz, Samuel C. Wiese. Spectrum of the Laplacian on regular polyhedra. Communications on Pure & Applied Analysis, 2021, 20 (1) : 193-214. doi: 10.3934/cpaa.2020263
[2]

Sushil Kumar Dey, Bibhas C. Giri. Coordination of a sustainable reverse supply chain with revenue sharing contract. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020165
[3]

Gökhan Mutlu. On the quotient quantum graph with respect to the regular representation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020295
[4]

Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1469-1482. doi: 10.3934/dcds.2020325
[5]

Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321
[6]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033
[7]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296
[8]

Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, Stock price fluctuation prediction method based on time series analysis. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 915-915. doi: 10.3934/dcdss.2019061
[9]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047
[10]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443
[11]

Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117
[12]

David W. K. Yeung, Yingxuan Zhang, Hongtao Bai, Sardar M. N. Islam. Collaborative environmental management for transboundary air pollution problems: A differential levies game. Journal of Industrial & Management Optimization, 2021, 17 (2) : 517-531. doi: 10.3934/jimo.2019121
[13]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, 2021, 14 (1) : 115-148. doi: 10.3934/krm.2020051
[14]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347
[15]

Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017
[16]

Alain Bensoussan, Xinwei Feng, Jianhui Huang. Linear-quadratic-Gaussian mean-field-game with partial observation and common noise. Mathematical Control & Related Fields, 2021, 11 (1) : 23-46. doi: 10.3934/mcrf.2020025
[17]

Qingfeng Zhu, Yufeng Shi. Nonzero-sum differential game of backward doubly stochastic systems with delay and applications. Mathematical Control & Related Fields, 2021, 11 (1) : 73-94. doi: 10.3934/mcrf.2020028
[18]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070
[19]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021011
[20]

Madhurima Mukhopadhyay, Palash Sarkar, Shashank Singh, Emmanuel Thomé. New discrete logarithm computation for the medium prime case using the function field sieve. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020119

 Impact Factor: 

Tools

Metrics

  • PDF downloads (73)
  • HTML views (164)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences