Skip to main content
SpringerLink
Log in

Two-person pairwise solvable games

  • Original Paper
  • Published:
International Journal of Game Theory Aims and scope Submit manuscript

Abstract

A game is solvable if the set of Nash equilibria is nonempty and interchangeable. A pairwise solvable game is a two-person symmetric game in which any restricted game generated by a pair of strategies is solvable. We show that the set of equilibria in a pairwise solvable game is interchangeable. Under a quasiconcavity condition, we derive a complete order-theoretic characterization and some topological sufficient conditions for the existence of equilibria, and show that if the game is finite, then an iterated elimination of weakly dominated strategies leads precisely to the set of Nash equilibria, which means that such a game is both solvable and dominance solvable. All results are applicable to symmetric contests, such as the rent-seeking game and the rank-order tournament, which are shown to be pairwise solvable. Some applications to evolutionary equilibria are also given.

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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. In fact, Nash (1951, p. 290) simply defines solvability as the interchangeability. This is because he works on the mixed extension of a finite strategic game, which is shown by himself to have at least one equilibrium. However, the formal definition of interchangeability is vacuously satisfied if there is no equilibrium. In more general settings, therefore, one may define solvability as the nonemptiness and the interchangeability of the set of all equilibria.

  2. The definition of dominance solvability varies. We adopt one in Moulin (1979, 1986), which implies neither uniqueness nor the payoff equivalence of equilibria. In Moulin (1984), dominance solvability implies uniqueness.

  3. Unless explicitly stated otherwise, we only consider equilibria in pure strategies.

  4. An equilibrium existence result for n-person symmetric weakly unilaterally competitive games is obtained in Iimura and Watanabe (2016).

  5. Accordingly, we consider the mixed extension of a finite game only when we do so explicitly.

  6. In contrast, the mixed extension of a finite two-person zero-sum game is zero-sum.

  7. The necessary and sufficient condition for G to be pairwise solvable is that in its zero-diagonal form, \(u_0(x,y)>0\) if and only if \(u_0(y,x)<0\) for all \(x,y\in S\).

  8. The linear order on the strategy set should not be confused with binary relation \(\succ \), which is also defined on the strategy set. They are distinct.

  9. Technically, we may phrase the definition as follows: A two-person symmetric game \(G=\langle S, u\rangle \) is called quasiconcave at the diagonal if there exists a linear order on S according to which (QCD) holds. As long as one linear order is fixed throughout a given context, two definitions are equivalent.

  10. Quasiconcavity at the diagonal is an adaptation of the notion of quasiconcavity at a point, defined by Mangasarian (1969, Chapter 9). Specifically, regarding u as a function of single variable \(u(\cdot , x)\), and applying his definition at the point (x, x), we obtain (QCD).

  11. Actually, one can verify that it is single-peaked.

  12. An ordered pair (L, R) of nonempty subsets of a linearly ordered set P is a Dedekind cut if \(L\cup R=P\) and \(x<y\) whenever \(x\in L\) and \(y\in R\).

  13. What if the game at hand fails to satisfy the assumptions of Lemma 5.8? If \(L_G=S\), say, then \({{\mathcal {E}}}=R_G\times R_G\). Such games should be considered on a case by case basis.

  14. This follows from the fact that any rationalizable strategy survives the iterated eliminations of strictly dominated strategies (Pearce 1984, p. 1035). Moreover, the unique equilibrium in this case is robust in the sense of Kajii and Morris (1997).

  15. This is true even when we allow the domination by mixed strategies. To see this, we only need to recall that Lemma 6.2 can be strengthened to accommodate the domination by mixed strategies.

  16. The notion of dominance solvability does not imply payoff equivalence between equilibria. Moulin (1986, p. 73) explicitly allows games like Figs. 2b and 3b to be dominance solvable.

  17. Note that Schaffer (1989) restricts his attention on symmetric evolutionary equilibria. We shall see shortly that this restriction causes no loss of generality.

  18. Milgrom and Roberts (1990) show that in a supermodular game, the iterated elimination of strictly dominated strategies leads to a minimum and a maximum Nash equilibria. Subsequently, Milgrom and Shannon (1994, p. 175) generalize this result to the class of games with ordinal strategic complementarities. These results need not apply to pairwise solvable games. Consider the game in Fig. 3a. In this game, the difference \(u(3/5,y)-u(2/5,y)\) alternates its sign twice as y increases from 0 to 1, which means that the game lacks the single crossing property (Milgrom and Shannon 1994, p. 160), or that the game is without ordinal strategic complementarities.

  19. Yasuda (2016) shows that a version of interchangeability must hold in any two-person supermodular game.

  20. Note that the game in Fig. 4a has a generalized potential in the sense of Morris and Ui (2005), which is maximized at the set of Nash equilibria, \(\{s_2,s_3\}\times \{s_2,s_3\}\).

References

  • Ania AB (2008) Evolutionary stability and Nash equilibrium in finite populations, with an application to price competition. J Econ Behav Organ 65:472–488

    Article  Google Scholar 

  • Bernheim BD (1984) Rationalizable strategic behavior. Econometrica 52:1007–1028

    Article  Google Scholar 

  • Dixit AK, Skeath S (2004) Games of strategy. W.W. Norton, New York

    Google Scholar 

  • Duersch P, Oechssler J, Schipper BC (2012a) Pure strategy equilibria in symmetric two-player zero-sum games. Int J Game Theory 41:553–564

    Article  Google Scholar 

  • Duersch P, Oechssler J, Schipper BC (2012b) Unbeatable imitation. Games Econ Behav 76:88–96

    Article  Google Scholar 

  • Friedman JW (1983) On characterizing equilibrium points in two person strictly competitive games. Int J Game Theory 12:245–247

    Article  Google Scholar 

  • Hehenkamp B, Leininger W, Possajennikov A (2004) Evolutionary equilibrium in Tullock contests: spite and overdissipation. Eur J Polit Econ 20:1045–1057

    Article  Google Scholar 

  • Hotelling H (1929) Stability in competition. Econ J 39:41–57

    Article  Google Scholar 

  • Iimura T, Watanabe T (2016) Pure strategy equilibrium in finite weakly unilaterally competitive games. Int J Game Theory 45:719–729

    Article  Google Scholar 

  • Iimura T, Maruta T, Watanabe T (2019) Equilibria in games with weak payoff externalities. Econ Theory Bull 7:245–258

    Article  Google Scholar 

  • Kajii A, Morris S (1997) The robustness of equilibria to incomplete information. Econometrica 65:1283–1309

    Article  Google Scholar 

  • Kats A, Thisse J-F (1992) Unilaterally competitive games. Int J Game Theory 21:291–299

    Article  Google Scholar 

  • Lazear EP, Rosen S (1981) Rank-order tournaments as optimum labor contracts. J Polit Econ 89:841–864

    Article  Google Scholar 

  • Mangasarian OL (1969) Nonlinear programming. McGraw-Hill, New York

    Google Scholar 

  • Milgrom P, Roberts J (1990) Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica 58:1255–1277

    Article  Google Scholar 

  • Milgrom P, Shannon C (1994) Monotone comparative statics. Econometrica 62:157–180

    Article  Google Scholar 

  • Monderer D, Shapley LS (1996) Potential games. Games Econ Behav 14:124–143

    Article  Google Scholar 

  • Morris S, Ui T (2005) Generalized potentials and robust sets of equilibria. J Econ Theory 124:45–78

    Article  Google Scholar 

  • Moulin H (1979) Dominance solvable voting schemes. Econometrica 47:1337–1351

    Article  Google Scholar 

  • Moulin H (1984) Dominance solvability and Cournot stability. Math Soc Sci 7:83–102

    Article  Google Scholar 

  • Moulin H (1986) Game theory for the social sciences. New York University Press, New York

    Google Scholar 

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

    Article  Google Scholar 

  • Pearce D (1984) Rationalizable strategic behavior and the problem of perfection. Econometrica 52:1029–1050

    Article  Google Scholar 

  • Schaffer ME (1988) Evolutionarily stable strategies for a finite population and a variable contest size. J Theor Biol 132:469–478

    Article  Google Scholar 

  • Schaffer ME (1989) Are profit-maximisers the best survivors? A Darwinian model of economic natural selection. J Econ Behav Organ 12:29–45

    Article  Google Scholar 

  • Shapley LS (1964) Some topics in two-person games. In: Dresher M, Shapley LS, Tucker AW (eds) Advances in game theory. Princeton University Press, Princeton, pp 1–28

    Google Scholar 

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

    Google Scholar 

  • von Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, Princeton

    Google Scholar 

  • Voorneveld M (2000) Best-response potential games. Econ Lett 66:289–295

    Article  Google Scholar 

  • Watson J (2013) Strategy. W.W. Norton, New York

    Google Scholar 

  • Yasuda Y (2016) Reformulations of Nash equilibrium with an application to interchangeability. Mimeo, Osaka University

Download references

Author information

Authors and Affiliations

  1. Tokyo Metropolitan University, 1-1 Minami Ohsawa, Hachioji, Tokyo, 192-0397, Japan

    Takuya Iimura & Takahiro Watanabe

  2. College of Economics, Nihon University, 1-3-2 Kanda-Misakicho, Chiyoda, Tokyo, 101-8360, Japan

    Toshimasa Maruta

Authors
  1. Takuya Iimura
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Toshimasa Maruta
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Takahiro Watanabe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Toshimasa Maruta.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Takuya Iimura and Takahiro Watanabe gratefully acknowledge financial support from KAKENHI 25380233 and 16K03553. Toshimasa Maruta gratefully acknowledges financial support from KAKENHI 17K03631. The authors are grateful to the handling editor and a referee for their comments.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Iimura, T., Maruta, T. & Watanabe, T. Two-person pairwise solvable games. Int J Game Theory 49, 385–409 (2020). https://doi.org/10.1007/s00182-020-00709-1

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00182-020-00709-1

Keywords

JEL Classification

Search

Navigation