Skip to main content
SpringerLink
Log in

Nash-2 Equilibrium: Selective Farsightedness Under Uncertain Response

  • Published:
Group Decision and Negotiation Aims and scope Submit manuscript

Abstract

This paper provides an extended analysis of an equilibrium concept for non-cooperative games with boundedly rational players: Nash-2 equilibrium. Players think one step ahead and account for all profitable responses of player-specific subsets of opponents because of both the cognitive limitations on predicting everyone’s reaction and the inability to make deeper and certain predictions. They cautiously reject improvements that might lead to worse profits after some reasonable response. For n-person games we introduce the notion of a reflection network consisting of direct competitors to express the idea of selective farsightedness. For almost every 2-person game with a complete reflection network, we prove the existence of a Nash-2 equilibrium. Nash-2 equilibrium sets are obtained in models of price and quantity competition, and in Tullock’s rent-seeking model with two players. It is shown that such farsighted behavior may provide strategic support for tacit collusion. The analyses of n-person Prisoner’s dilemma and oligopoly models with a star reflection structure demonstrate some possibilities of strategic collusion and a large variety of potentially stable outcomes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Notes

  1. All of these are the same concepts. We refer to existing results below when there is any intersection with ours.

  2. The authors formulated this result in terms of threatening-proof profile.

  3. For n-person games, our approach differs from Kilgour et al. (1987) considerably since we allow simultaneous responses of direct competitors, while in Kilgour et al. (1987) players deviate sequentially.

  4. Naturally, a more complicated method is to assign a probability proportional to the number of improving paths from s to the particular Nash-2 equilibrium. This can be done easily in the case of finite strategy space, but may face considerable technical difficulties in the case of continuous strategy intervals.

  5. For more complicated networks this is generally not true: some firms may propose less than best response prices at some Nash-2 equilibria.

References

  • Alaoui L, Penta A (2015) Endogenous depth of reasoning. Rev Econ Stud 83(4):1297–1333

    Article  Google Scholar 

  • Aumann RJ (1974) Subjectivity and correlation in randomized strategies. J Math Econ 1(1):67–96

    Article  Google Scholar 

  • Aumann RJ, Maschler M (1964) The bargaining set for cooperative games. Adv Game Theory 52:443–476

    Google Scholar 

  • Baye M, Kovenock D, De Vries C (1994) The solution to the Tullock rent-seeking game when \(R{>}2\): mixed-strategy equilibria and mean dissipation rates. Public Choice 81(3–4):363–380

    Article  Google Scholar 

  • Baye M, Morgan J, Scholten P (2006) Information, search, and price dispersion. Handb Econ Inf Syst 1:323–375

    Google Scholar 

  • Binmore K (1988) Modeling rational players: part II. Econ Philos 4(1):9–55

    Article  Google Scholar 

  • Bowley A (1924) The mathematical groundwork of economics: an introductory treatise. Oxford University Press, Oxford

    Google Scholar 

  • Boyd R, Richerson P (1988) The evolution of reciprocity in sizable groups. J Theor Biol 132(3):337–356

    Article  Google Scholar 

  • Brams S, Kilgour M (1988) Game theory and national security. Basil Blackwell, New York

    Google Scholar 

  • Bresnaham T (1981) Duopoly models with consistent conjectures. Am Econ Rev 71(5):934–945

    Google Scholar 

  • Cabral L (1995) Conjectural variations as a reduced form. Econ Lett 49(4):397–402

    Article  Google Scholar 

  • Camerer CF, Ho TH, Chong JK (2004) A cognitive hierarchy model of games. Q J Econ 119(3):861–898

    Article  Google Scholar 

  • Crawford V, Costa-Gomes M, Iriberri N (2013) Structural models of nonequilibrium strategic thinking: theory, evidence, and applications. J Econ Lit 51(1):5–62

    Article  Google Scholar 

  • d’Aspremont C, Dos Santos Ferreira R, Gerard-Varet L-A (2007) Competition for market share or for market size: oligopolistic equilibria with varying competitive toughness. Int Econ Rev 48(3):761–784

    Article  Google Scholar 

  • Dockner E (1992) A dynamic theory of conjectural variations. J Ind Econ 40(4):377–395

    Article  Google Scholar 

  • Fang L, Hipel K, Kilgour D (1989) Conflict models in graph form: solution concepts and their interrelationships. Eur J Oper Res 41(1):86–100

    Article  Google Scholar 

  • Forg F (2010) A generalization of correlated equilibrium: a new protocol. Math Soc Sci 60(3):186–190

    Article  Google Scholar 

  • Fraser N, Hipel K (1984) Conflict analysis: models and resolutions. North-Holland, New York

    Google Scholar 

  • Frey S, Goldstone R (2015) Flocking in the depths of strategic iterated reasoning. arXiv preprint arXiv:1506.05410

  • Friedman J (1977) Oligopoly and the theory of games. North-Holland, New York

    Google Scholar 

  • Gabszewicz JJ, Thisse J-F (1986) Spatial competition and the location of firms. In: Arnott R (ed) Location theory, Fundamentals of Pure and Applied Economics, vol 5. Harwood Academic, Amsterdam, pp 1–71

    Google Scholar 

  • Iskakov M, Iskakov A (2012a) Equilibrium in secure strategies. CORE discussion paper 2012/61

  • Iskakov M, Iskakov A (2012b) Equilibrium in secure strategies—intuitive formulation. Working paper WP7/2012/06. Mathematical methods for decision making in economics, business and politics

  • Iskakov M, Iskakov A (2012c) Solution of the Hotelling’s game in secure strategies. Econ Lett 117(1):115–118

    Article  Google Scholar 

  • Iskakov M, Iskakov A (2014) Equilibium contained by counter-threats and complex equilibrium in secure strategies. Large Scale Syst Control 51:130–157 (in Russian)

    Google Scholar 

  • Iskakov M, Iskakov A (2015) Asymmetric equilibria in secure strategies. Working paper WP7/2015/03. Mathematical methods for decision making in economics, business and politics

  • Iskakov M, Iskakov A, Zakharov A (2013) Tullock rent-seeking contest and its solution in secure strategies. Working paper WP7/2013/01. Mathematical methods for decision making in economics, business and politics

  • Kawagoe T, Takizawa H (2009) Equilibrium refinement vs. level-k analysis: an experimental study of cheap-talk games with private information. Games Econ Behav 66(1):238–255

    Article  Google Scholar 

  • Kilgour DM, Hipel KW, Fang L (1987) The graph model for conflicts. Automatica 23(1):41–55

    Article  Google Scholar 

  • Kilgour DM, Fang L, Hipel KW (1990) A decision support system for the graph model of conflicts. Theory Decis 28(3):289–311

    Article  Google Scholar 

  • Maskin E, Tirole J (1988) A theory of dynamic oligopoly, I: overview and quantity competition with large fixed costs. Econometrica 56(3):549–569

    Article  Google Scholar 

  • Moulin H, Vial JP (1978) Strategically zero-sum games: the class of games whose completely mixed equilibria cannot be improved upon. Int J Game Theory 7(3–4):201–221

    Article  Google Scholar 

  • Nash J (1951) Non-cooperative games. Ann Math 54(2):286–295

    Article  Google Scholar 

  • Pan X, Ratchford BT, Shankar V (2004) Price dispersion on the Internet: a review and directions for future research. J Interact Mark 18(4):116–135

    Article  Google Scholar 

  • Rezaei G, Kirley M, Pfau J (2009) Evolving cooperation in the n-player prisoner’s dilemma: a social network model. In: Artificial life: borrowing from biology. Springer, Berlin, pp 43–52

  • Sandomirskaia M (2016) A model of tacit collusion: Nash-2 equilibrium concept. In: Petrosyan LA, Mazalov VV (eds) Recent advances in game theory and applications. Static and dynamic game theory: foundations and applications. Springer, Cham, pp 243–260

    Google Scholar 

  • Stahl D (1993) Evolution of smart\( _n\) players. Games Econ Behav 5(4):604–617

    Article  Google Scholar 

  • Tullock G (1980) Efficient rent seeking. In: Buchanan JM, Tollison RD, Tullock G (eds) Toward a theory of the rent-seeking society. Texas A and M University Press, College Station, pp 97–112

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. National Research University Higher School of Economics, Moscow, Russia

    Marina Sandomirskaia

Authors
  1. Marina Sandomirskaia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marina Sandomirskaia.

Additional information

The study has been funded by the Russian Academic Excellence Project ‘5-100’.

The author is deeply indebted to Jacques-Francois Thisse, Sergey Kokovin, Fuad Aleskerov, and Nikolay Bazenkov for fruitful discussions and comprehensive support. The author thanks Ariel Rubinstein and Francis Bloch for beneficial remarks and suggestions. The author highly appreciates all discussants at the seminars in the Center for Market Studies and Spatial Economics HSE, St. Petersburg Institute for Economics and Mathematics RAS, Trapeznikov Institute of Control Sciences RAS, International College of Economics and Finance (ICEF) HSE, The Center for Research in Economics of the University Saint-Louis (Brussels), and participants of Summer School of Econometric Society in Kyoto (2016) for valuable debates. The author also thanks the anonymous reviewers whose comments and suggestions allowed to improve the paper considerably.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sandomirskaia, M. Nash-2 Equilibrium: Selective Farsightedness Under Uncertain Response. Group Decis Negot 28, 275–304 (2019). https://doi.org/10.1007/s10726-018-9602-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10726-018-9602-x

Keywords

JEL Classification

Search

Navigation