Abstract
The chapter deals with strategic form games on digraphs, and examines maximin solution concepts based on different types of digraph substructures Ungureanu (Computer Science Journal of Moldova 6 3(18): 313–337, 1998, [1]), Ungureanu (ROMAI Journal 12(1): 133–161, 2016, [2]). Necessary and sufficient conditions for maximin solution existence in digraph matrix games with pure strategies are formulated and proved. Some particular games are considered. Algorithms for finding maximin substructures are suggested. Multi-player simultaneous games and dynamical/hierarchical games on digraphs are considered too.
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References
Ungureanu, V. 1998. Games on digraphs and constructing maximin structures. Computer Science Journal of Moldova 6 3(18): 313–337.
Ungureanu, V. 2016. Strategic games on digraphs. ROMAI Journal 12 (1): 133–161.
Christofides, N. 1975. Graph Theory: An Algorithmic Approach, 415. London: Academic Press.
Papadimitriou, C., and K. Steiglitz. 1982. Combinatorial Optimization: Algorithms and Complexity, 510. Englewood Cliffs: Prentice-Hall Inc.
Berge, C. 1962. The Theory of Graphs (and its Applications). London: Methuen & Co.; New York: Wiley, 272pp.
Van den Nouweland, A., P. Borm, W. van Golstein Brouwers, R. Groot Bruinderink, and S. Tijs. 1996. A game theoretic approach to problems in telecommunications. Management Science 42 (2): 294–303.
Altman, E., T. Boulogne, R. El-Azouzi, T. JimTnez, and L. Wynter. 2006. A survey on networking games in telecommunications. Computers & Operations Research 33 (2): 286–311.
Tardos, E. 2004. Network Games, Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing (STOC?04), 341–342. Chicago, Illinois, USA, June 13–15 2004.
Altman, E., and L. Wynter. 2004. Crossovers between transportation planning and telecommunications, networks and spatial economics. Editors 4 (1): 5–124.
Jackson, M.O., and Y. Zenou. 2015. Games on networks. In Handbook of Game Theory with Economic Applications, vol. 4, ed. H. Peyton Young, and Sh Zamir, 95–163. Amsterdam: North-Holland.
Roughgarden, T. 2002. The price of anarchy is independent of the network topology. Journal of Computer and System Sciences 67 (2): 341–364.
Suri, S., C. Toth, and Y. Zhou. 2005. Selfish load balancing and atomic congestion games, Proceedings of the 16th Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), 188–195.
Christodoulou, G., and E. Koutsoupias. 2005. The price of anarchy of finite congestion games, Proceedings of the Thirty-Seven Annual ACM Symposium on Theory of Computing (STOC?05), 67–73. New York, USA.
Czumaj, A., and B. Vöcking. 2007. Tight bounds for worst-case equilibria. ACM Transactions on Algorithms 3 (1): 11. Article 4.
Vetta, A. 2002. Nash equilibria in competitive societies, with applications to facility location, traffic routing and auctions, Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science (FOCS’02), 416pp.
Akella, A., S. Seshan, R. Karp, S. Shenker, and C. Papadimitriou. 2002. Selfish behavior and stability of the Internet: A game-theoretic analysis of TCP, ACM SIGCOMM Computer Communication Review - Proceedings of the 2002 SIGCOMM Conference, 32(4): 117–130.
Dutta, D., A. Goel, and J. Heidemann. 2002. Oblivious AQM and nash equilibria, ACM SIGCOMM Computer Communication Review - Proceedings of the 2002 SIGCOMM conference, 32(3): 20.
Fabrikant, A., A. Luthra, E. Maneva, C.H. Papadimitriou and S. Shenker. 2003. On a network creation game, Proceedings of the Twenty-Second Annual Symposium on Principles of Distributed Computing (PODC ’03), 347–351.
Anshelevich, E., A. Dasgupta, E. Tardos, and T. Wexler. 2008. Near-optimal network design with selfish agents. Theory of Computing 4: 77–109.
Kodialam, M., and T.V. Lakshman. 2003. Detecting network intrusions via sampling: A game theoretic approach. IEEE INFOCOM 1–10.
Han, Z., D. Niyato, W. Saad, T. Başar, and A. Hjørungnes. 2012. Game Theory in Wireless and Communication Networks: Theory, Models, and Applications. Cambridge: Cambridge University Press, XVIII+535pp.
Zhang, Y., and M. Guizani (eds.). 2011. Game Theory for Wireless Communications and Networking. Boca Raton: CRC Press, XIV+571pp.
Kim, S. 2014. Game Theory Applications in Network Design. Hershey: IGI Global, XXII+500pp.
Mazalov, V. 2014. Mathematical Game Theory and its Applications. Tokyo: Wiley, XIV+414pp.
Antoniou, J., and A. Pitsillides. 2013. Game Theory in Communication Networks: Cooperative Resolution of Interactive Networking Scenarios. Boca Raton: CRC Press, X+137pp.
Aubin, J.-P. 2005. Dynamical connectionist network and cooperative games. In Dynamic Games: Theory and Applications, ed. A. Haurie, and G. Zaccour, 1–36. US: Springer.
El Azouzi, R., E. Altman, and O. Pourtallier. 2005. Braess paradox and properties of wardrop equilibrium in some multiservice networks. In Dynamic Games: Theory and Applications, ed. A. Haurie, and G. Zaccour, 57–77. US: Springer.
Pavel, L. 2012. Game Theory for Control of Optical Networks. New York: Birkhäuser, XIII+261pp.
Gurvitch, V., A. Karzanov, and L. Khatchiyan. 1988. Cyclic games: Finding of minimax mean cycles in digraphs. Journal of Computational Mathematics and Mathematical Physics 9 (28): 1407–1417. (in Russian).
Von Stackelberg, H. 1934. Marktform und Gleichgewicht (Market Structure and Equilibrium). Vienna: Springer, (in German), XIV+134pp.
Wolfram, S. 2002. A New Kind of Science, 1197. Champaign, IL: Wolfram Media Inc.
Wolfram, S. 2016. An Elementary Introduction to the Wolfram Language. Champaign, IL: Wolfram Media, Inc., XV+324pp.
Boliac, R., and D. Lozovanu. 1996. Finding of minimax paths tree in weighted digraph. Buletinul Academiei de Ştiinţe a Republicii Moldova 3 (66): 74–82. (in Russian).
Biggs, N.L., E.K. Lloyd, and R.J. Wilson. 1976. Graph Theory, 1736–1936, 255. Oxford: Clarendon Press.
Schrijver, A. 2005. On the history of combinatorial optimization (Till 1960). Handbooks in Operations Research and Management Science 12: 1–68.
Gutin, G., and P.P. Abraham (eds.). 2004. The Traveling Salesman Problem and its Variation, vol. 2, New-York: : Kluwer Academic Publishers, 831pp.
Lawler, E.L., J.K. Lenstra, A.H.G. Rinooy Kan, and D.B. Shmoys (eds.). 1985. The Traveling Salesman Problem. Chichester, UK: Wiley.
Reinelt, G. 1994. The Traveling Salesman: Computational Solutions for TSP Applications, 230. Berlin: Springer.
Fleishner, H. 2000. Traversing graphs: The Eulerian and Hamiltonian theme. In Arc Routing: Theory, Solutions, and Applications, ed. M. Dror, 19–87. The Netherlands: Kluwer Academic Publishers.
Golden, B., S. Raghavan, and E. Wasil (eds.). 2008. The Vehicle Routing Problem: Latest Advances and New Challenges. New York: Springer.
Ball, M.O., T.L. Magnanti, C.L. Monma, and G.L. Nemhauser (eds.). 1995. Network Routing, 779. Elsevier: Amsterdam.
Garey, M.R., and D.S. Johnson. 1979. Computers and Intractability: A Guide to the Theory of NP Completeness, 351. San Francisco: W.H. Freeman.
Sipser, M. 2006. Introduction to the Theory of Computation, 2nd ed, Boston, Massachusetts: Thomson Course Technology, XIX+431pp.
Nielsen, M.A., and I.L. Chuang. 2010. Quantum Computation and Quantum Information, 10th Anniversary ed. Cambridge, UK: Cambridge University Press, XXXII+676pp.
Applegate, D.L., R.E. Bixby, V. Chvtal, J. William, and W.J. Cook. 2006. The Traveling Salesman Problem: A Computational Study, 606. Princeton: Princeton University Press.
Hitchcock, F.L. 1941. The distribution of product from several sources to numerous localities. Journal of Mathematical Physics 20 (2): 217–224.
Díaz-Parra, O., J.A. Ruiz-Vanoye, B.B. Loranca, A. Fuentes-Penna, and R.A. and Barrera-Cámara. 2014. A survey of transportation problems. Journal of Applied Mathematics 2014: 17. Article ID 848129.
Hoffman, K.L., and M. Padberg. 1991. LP-based combinatorial problem solving. Annals of Operations Research 4: 145–194.
Padberg, M., and G. Rinaldi. 1991. A branch-and-cut algorithm for the resolution of large-scale traveling salesman problem. SIAM Review 33: 60–100.
Villani, C. 2003. Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58, 382. Providence: American Mathematical Society.
Villani, C. 2008. Optimal Transport, Old and New, 1000. Berlin: Springer.
Ungureanu, V. 2006. Traveling salesman problem with transportation. Computer Science Journal of Moldova 14 2(41): 202–206.
Caric, T., and H. Gold (eds.). 2008. Vehicle Routing Problem, 152. InTech: Croatia.
Toth, P., and D. Vigo. 2002. The Vehicle Routing Problem, Society for Industrial and Applied Mathematics, 386pp.
Labadie, N., and C. Prodhon. 2014. A survey on multi-criteria analysis in logistics: Focus on vehicle routing problems. In Applications of Multi-Criteria and Game Theory Approaches: Manufacturing and Logistics, ed. L. Benyoucef, J.-C. Hennet, and M.K. Tiwari, 3–29. New Jersey: Springer.
Bellman, R. 1957. Dynamic Programming, 365. New Jersey: Princeton University Press.
Golshtein, E., and D. Yudin. 1966. New Directions in Linear Programming. Moscow: Sovetskoe Radio, 527pp. (in Russian).
Ungureanu, V. 1997. Minimizing a concave quadratic function over a hypercube. Buletinul Academiei de Ştiinţe a Republicii Moldova: Mathematics Series 2 (24): 69–76. (in Romanian).
Yanovskaya, E.B. 1968. Equilibrium points in polymatrix games. Lithuanian Mathematical Collection (Litovskii Matematicheskii Sbornik) 8 (2): 381–384. (in Russian).
Howson Jr., J.T. 1972. Equilibria of polymatrix games. Management Science 18: 312–318.
Eaves, B.C. 1973. Polymatrix games with joint constraints. SIAM Journal of Applied Mathematics 24: 418–423.
Von Neumann, J. 1928. Zur Theorie der Gesellschaftsspiele. Mathematische Annalen 100: 295–320. (in German).
Kuhn, H.W. 1950. Extensive games, Proceedings of the National Academy of Sciences U.S.A., Vol. 36, 570–576.
Kuhn, H.W. 1953. Extensive games and the problem of information. Contributions to the Theory of Games, Vol. II, vol. 28, 217–243., Annals of Mathematics Study Princeton: Princeton University Press.
Kuhn, H.W. 2003. Lectures on the Theory of Games, vol. 37, 118. Annals of Mathematics Study Princeton: Princeton University Press.
Alós-Ferrer, C., and K. Ritzberger. 2016. The Theory of Extensive Form Games. Berlin: Springer, XVI+239pp.
Nisan, N., T. Roughgarden, E. Tardos, and V.V. Vazirani (eds.). 2007. Algorithmic Game Theory. Cambridge, UK: Cambridge University Press, 775pp.
Shoham, Y., and K. Leyton-Brown. 2009. Multi-Agent Systems: Algorithmic, Game-Theoretic, and Logical Foundations, 532. Cambridge: Cambridge University Press.
Easley, D., and D. Kleinberg. 2010. Networks, Crowds, and Markets: Reasoning about a Highly Connected World, 833. Cambridge: Cambridge University Press.
Menache, I., and A. Ozdaglar. 2011. Network Games: Theory, Models, and Dynamics. Synthesis Lectures on Communication Networks California: Morgan & Claypool Publishers, XIV+143pp.
Lozovanu, D. 1993. A strongly polynomial time algorithm for finding minimax paths in network and solving cyclic games. Cybernetics and System Analysis 5: 145–151. (in Russian).
Mertens, J.F. 1987. Repeated Games, Proceedings of the International Congress of Mathematicians, Berkeley, Providence: American Mathematical Society, 1528–1577.
Aumann, R.J., M.B. Maschler, and R.E. Stearns. 1995. Repeated Games with Incomplete Information, 360. Cambridge, Massachusetts: MIT Press.
Fudenberg, D., and J. Tirole. 1991. Game Theory, 579. Cambridge: MIT Press.
Osborne, M.J., and A. Rubinstein. 1994. A Course in Game Theory, 373. Cambridge, Massachusetts: The MIT Press.
Mailath, G.J., and L. Samuelson. 2006. Repeated Games and Reputations: Long-Run Relationships. New York: Oxford University Press, XVIII+645pp.
Sorin, S. 2002. A First Course on Zero-Sum Repeated Games. Berlin: Springer, XV+204pp.
Osborne, M.J. 2009. An Introduction to Game Theory, International ed, 685. Oxford: Oxford University Press.
Brams, S.J. 1994. Theory of Moves. Cambridge: Cambridge University Press, XII+248pp.
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Ungureanu, V. (2018). Strategic Form Games on Digraphs. In: Pareto-Nash-Stackelberg Game and Control Theory. Smart Innovation, Systems and Technologies, vol 89. Springer, Cham. https://doi.org/10.1007/978-3-319-75151-1_8
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