Time-Consistent Solutions for Two-Stage Network Games with Pairwise Interactions

  • Leon Petrosyan
  • Mariia Bulgakova
  • Artem SedakovEmail author


In the paper, we consider a cooperative version of a network game with pairwise interactions in which connected players play bimatrix games. For a particular type of a network, a simplified formula for the Shapley value based on a constructed characteristic function is derived. We then show the time inconsistency of classical cooperative solutions — the Shapley value and the core. The findings are applied to two important classes of bimatrix games: prisoner’s dilemma and a coordination game.


Cooperation Network formation The Shapley value The core Time consistency Prisoner’s dilemma Coordination game 



The authors thank two anonymous referees for their comments that have helped in the improvement of the paper. This research was supported by the Russian Science Foundation (grant No. 17-11-01079).


  1. 1.
    Acemoglu D, Ozdaglar A, ParandehGheibi A (2010) A Spread of (mis)information in social networks. Game Econ Behav 70(2):194–227MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aumann R, Myerson R (1988) Endogenous formation of links between players and coalitions: An application of the Shapley value. In: Roth A (ed) The Shapley value: Essays in Honor of Lloyd S. Shapley. Cambridge University Press, Cambridge, pp 175–191Google Scholar
  3. 3.
    Bala V, Goyal S (2000) A noncooperative model of network formation. Econometrica 68(5):1181–1229MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bulgakova MA, Petrosyan LA (2016) About strongly time-consistency of core in the network game with pairwise interactions. In: Proceedings of 2016 international conference “Stability and Oscillations of Nonlinear Control Systems”, pp 157–160Google Scholar
  5. 5.
    Challita U, Saad W (2017) Network formation in the sky: Unmanned aerial vehicles for multi-hop wireless backhauling. In: Proceedings of the IEEE global communications conference (GLOBECOM), Singapore, 4–8Google Scholar
  6. 6.
    Corbae D, Duffy J (2008) Experiments with network formation. Games Econ Behav 64:81–120CrossRefGoogle Scholar
  7. 7.
    Dyer M, Mohanaraj V (2011) Pairwise-interaction games. In: Aceto, L, Henzinger, M, Sgall, J (eds) Automata, Languages and Programming. ICALP 2011. Lecture Notes in Computer Science, vol 6755, pp 159-170CrossRefGoogle Scholar
  8. 8.
    Dziubiński M, Goyal S (2013) Network design and defense. Game Econ Behav 79:30–43CrossRefGoogle Scholar
  9. 9.
    Goyal S, Vega-Redondo F (2005) Network formation and social coordination. Games Econ Behav 50:178–207MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hernández P, Muñoz-Herrera M, Sánchez Á (2013) Heterogeneous network games: Conflicting preference. Game Econ Behav 79:56–66MathSciNetCrossRefGoogle Scholar
  11. 11.
    Jackson M, Watts A (2002) On the formation of interaction networks in social coordination games. Games Econ Behav 41(2):265–291MathSciNetCrossRefGoogle Scholar
  12. 12.
    König MD, Battiston S, Napoletano M, Schweitzer F (2012) The efficiency and stability of R&D networks. Game Econ Behav 75(2):694–713MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mozaffari M, Saad W, Bennis M, Debbah M (2016) Unmanned aerial vehicle with underlaid device-to-device communications: Performance and tradeoffs. IEEE Trans Wirel Commun 15(6):3949–3963CrossRefGoogle Scholar
  14. 14.
    Petrosyan LA, Danilov NN (1979) Stability of solutions of non-zero-sum game with transferable payoffs. Vestnik Leningradskogo Universiteta Ser 1 Mat Mekhanika Astron 19:52–59zbMATHGoogle Scholar
  15. 15.
    Petrosyan LA, Sedakov AA (2014) Multistage network games with perfect information. Automat Rem Contr 75(8):1532–1540MathSciNetCrossRefGoogle Scholar
  16. 16.
    Petrosyan LA, Sedakov AA (2016) The subgame-consistent shapley value for dynamic network games with shock. Dyn Games Appl 6(4):520–537MathSciNetCrossRefGoogle Scholar
  17. 17.
    Petrosyan LA, Sedakov AA, Bochkarev AO (2016) Two-stage network games. Automat Rem Contr 77(10):1855–1866MathSciNetCrossRefGoogle Scholar
  18. 18.
    Shapley L (1953) A value for n-person games. In: Kuhn H W, Tucker A W (eds) Contributions to the theory of games II. Princeton University Press, Princeton, pp 307–317Google Scholar
  19. 19.
    Von Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, PrincetonzbMATHGoogle Scholar
  20. 20.
    Saad W, Han Z, Başar T, Debbah M, Hjrungnes A (2011) Network formation games among relay stations in next generation wireless networks. IEEE Trans Commun 59(9):2528–2542CrossRefGoogle Scholar
  21. 21.
    Xie F, Cui W, Lin J (2013) Prisoners dilemma game on adaptive networks under limited foresight. Complexity 18:38–47CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Saint Petersburg State UniversitySaint PetersburgRussia

Personalised recommendations