Abstract
Gerke et al. (2019) introduced Netflix Games and proved that every such game has a pure strategy Nash equilibrium. In this paper, we explore the uniqueness of pure strategy Nash equilibria in Netflix Games. Let \(G=(V,E)\) be a graph and \(\kappa :\ V\rightarrow \mathbb {Z}_{\ge 0}\) a function, and call the pair \((G, \kappa )\) a weighted graph. A spanning subgraph H of \((G, \kappa )\) is called a DP-Nash subgraph if H is bipartite with partite sets D, P called the D-set and P-set of H, respectively, such that no vertex of P is isolated and for every \(x\in D,\) \(d_H(x)=\min \{d_G(x),\kappa (x)\}.\) We prove that whether \((G,\kappa )\) has a unique DP-Nash subgraph can be decided in polynomial time. We also show that when \(\kappa (v)=k\in \mathbb {Z}_{\ge 0}\) for every \(v\in V\), the problem of deciding whether \((G,\kappa )\) has a unique D-set is polynomial time solvable for \(k=0\) and 1, and co-NP-complete for \(k\ge 2.\)
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Notes
- 1.
It is somewhat interesting that despite the characterization for D -set Uniqueness, the problem is co-NP-complete.
- 2.
Lloyd Shapley and Alvin Roth received the 2012 Nobel Memorial Prize in Economics for their work in this area. (David Gale died in 2007.)
- 3.
Vertices being constrained in the number of neighbours they may share with seems well-suited to applications. In Netflix Games sharing bestows a benefit on neighbours, but this need not be the case. Gutin et al. [8] add constrained sharing to the Susceptible-Infected-Removed (SIR) model of disease transmission of Kermack and McKendrick [11]. Gutin et al. interpret constrained sharing as ‘social distancing’ restrictions imposed on a population and document how the reach of an epidemic is curtailed when such measures are in place.
- 4.
One example from Gerke et al. was of a group of individuals who each want to attend an event and can ride-share to get to it. Every individual will be assigned as either a Driver or a Passenger, hence the labels D and P.
References
Bramoullé, Y., Kranton, R.: Public goods in networks. J. Econ. Theory 135(1), 478–494 (2007)
Corneil, D., Perl, Y.: Clustering and domination in perfect graphs. Discrete Appl. Math. 9, 27–39 (1984)
Cygan, M., Pilipczuk, M., Wojtaszczyk, J.O.: Capacitated domination faster than \(O(2^n)\). Inf. Process. Lett. 111(23–24), 1099–1103 (2011)
Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Mon. 69(1), 9–15 (1962)
Galeotti, A., Goyal, S., Jackson, M.O., Vega-Redondo, F., Yariv, L.: Network games. Rev. Econ. Stud. 77(1), 218–244 (2010)
Gerke, S., Gutin, G., Hwang, S.H., Neary, P.R.: Netflix games: local public goods with capacity constraints (2019). arXiv:1905.01693
Goddard, W., Henning, M.A.: Independent domination in graphs: a survey and recent results. Discrete Math. 313(7), 839–854 (2013)
Gutin, G., Hirano, T., Hwang, S.H., Neary, P.R., Toda, A.A.: The effect of social distancing on the reach of an epidemic in social networks (2020). arXiv:2005.03067
Gutin, G., Neary, P.R., Yeo, A.: Uniqueness of DP-Nash subgraphs and D-sets in capacitated graphs of netflix games. arXiv:2003.07106 (2020)
Kao, M., Chen, H., Lee, D.: Capacitated domination: problem complexity and approximation algorithms. Algorithmica 72(1), 1–43 (2015)
Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. London. A 115(772), 700–721 (1927). Containing Papers of a Mathematical and Physical Character. http://www.jstor.org/stable/94815
Morris, S., Shin, H.S.: Unique equilibrium in a model of self-fulfilling currency attacks. Am. Econ. Rev. 88(3), 587–597 (1998)
Nash, J.: Non-cooperative games. Ann. Math. 54(2), 286–295 (1951)
Nash, J.F.: Equilibrium points in \(n\)-person games. Proc. Natl. Acad. Sci. U.S.A. 36(1), 48–49 (1950)
Roth, A.E.: Deferred acceptance algorithms: history, theory, practice, and open questions. Int. J. Game Theory 36(3), 537–569 (2008)
Acknowledgement
Anders Yeo’s research was partially supported by grant DFF-7014-00037B of Independent Research Fund Denmark.
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Gutin, G., Neary, P.R., Yeo, A. (2020). Uniqueness of DP-Nash Subgraphs and D-sets in Weighted Graphs of Netflix Games. In: Kim, D., Uma, R., Cai, Z., Lee, D. (eds) Computing and Combinatorics. COCOON 2020. Lecture Notes in Computer Science(), vol 12273. Springer, Cham. https://doi.org/10.1007/978-3-030-58150-3_29
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