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Uniqueness, stability and comparative statics for two-person Bayesian games with strategic substitutes

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Abstract

This paper considers a class of two-player symmetric games of incomplete information with strategic substitutes. First, we provide sufficient conditions under which there is either a unique equilibrium which is stable (in the sense of best-reply dynamics) and symmetric or a unique (up to permutations) asymmetric equilibrium that is stable (together with an unstable symmetric equilibrium). Thus, (i) there is always a unique stable equilibrium, (ii) it is either symmetric or asymmetric, and hence, (iii) a very simple local condition—stability of the symmetric equilibrium (i.e., the slope of the best-response function at the symmetric equilibrium)—identifies which case applies. Using this, we provide a very simple sufficient condition on primitives for when the unique stable equilibrium is asymmetric (and similarly for when it is symmetric). Finally, we show that the conditions guaranteeing the uniqueness described above also yield novel comparative statics results for this class of games.

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Notes

  1. By unique up to permutations we mean that, as the game is symmetric, if \(\left( x,y\right) \) is an equilibrium so is \(\left( y,x\right) \). Henceforth, we refer to this as unique and drop the clause “up to permutations.”

  2. It would be interesting, but beyond the scope of this paper, to extend the results to more players and actions, and also to extend those results that would apply, such as the comparative-statics results, to asymmetric environments.

  3. Obviously, the cost could be a benefit, and one could have both; we focus wlog on the case of costs as it is more natural in some of the examples we consider and simplifies the writing.

  4. Most commonly studied distributions have log-concave densities, see Bagnoli and Bergstrom (2005).

  5. Strategic substitutes (\(U^{\mathrm{H}}<U^{\mathrm{L}}\)) together with the assumptions on f imply that \(f\left( U^{\mathrm{H}}\right) >f\left( U^{\mathrm{L}}\right) \), which is why these conditions are sufficient but not necessary.

  6. For games with strategic complementarities (supermodular games), there is a significant body of work on the structure of equilibria and their stability and comparative statics (e.g., Milgrom and Roberts 1990; Vives 1990). However, as noted, those results do not apply in our strategic substitutes (submodular) context. It is true that a two-player game of strategic substitutes can be transformed into one with strategic complements by permuting the actions of one player (specifically by reversing the order). However, the symmetry of the game is not preserved in this permutation, so the results on symmetric submodular games do not apply in our case.

  7. Zimper (2007) and Roy and Sabarwal (2012) provide conditions on the best-reply function that guarantee dominance solvability in lattice games with strategic substitutes. Roy and Saberwal in particular relate this to global stability.

  8. It remains an open question to what extent the results herein can be extended to general lattice games, and not only those with a differentiable structure as we assume.

References

  • Acemoglu, D., Jensen, M.K.: Aggregate comparative statics. Games Econ. Behav. 81, 27–49 (2013)

    Article  Google Scholar 

  • Amir, R., Garcia, F., Knauff, M.: Symmetry-breaking in two-player games via strategic substitutes and diagonal nonconcavity: a synthesis. J. Econ. Theory 145, 1968–1986 (2010)

    Article  Google Scholar 

  • Amir, R.: R&D returns, market structure and research joint ventures. J. Inst. Theor. Econ. 156, 583–598 26(2), 445–469 (2000)

    Google Scholar 

  • Bagnoli, M., Bergstrom, T.: Log-concave probability and its applications. Econ. Theory 26(2), 445–469 (2005). https://doi.org/10.1007/s00199-004-0514-4

    Article  Google Scholar 

  • Becker, G.S.: Division of labor in households and families. In: Becker, G.S. (ed.) A Treatise on the Family, Chapter 2. Harvard University Press, Cambridge (1993)

    Google Scholar 

  • Bulow, J.I., Geanakoplos, J.D., Klemperer, P.D.: Multimarket oligopoly: strategic substitutes and complements. J. Polit. Econ. 93(3), 488–511 (1985)

    Article  Google Scholar 

  • Cournot, A.: Researches into the Mathematical Principles of the Theory of Wealth. McMillan, New York (1897)

    Google Scholar 

  • de Frutos, M.-A., Fabra, N.: Endogenous capacities and price competition: the role of demand uncertainty. CEPR D. P. 6096 (2007)

  • Hadfield, G.K.: A coordination model of the sexual division of labor. J. Econ. Behav. Organ. 40, 125–153 (1999)

    Article  Google Scholar 

  • Hefti, A.: On the relationship between uniqueness and stability in sum-aggregative, symmetric and general differentiable games. Math. Soc. Sci. 80, 83–96 (2016a)

    Article  Google Scholar 

  • Hefti, A.: Symmetric stability in symmetric games. Theor. Econ. Lett. 6, 488–493 (2016b)

    Article  Google Scholar 

  • Hefti, A.: Equilibria in symmetric games: theory and applications. Theor. Econ. 12, 979–1002 (2017)

    Article  Google Scholar 

  • Matsuyama, K.: Symmetry-breaking. In: Blume, L., Durlauf, S. (eds.) The New Palgrave Dictionary of Economics, 2nd edn. Palgrave Macmillan, Basingstoke (2008)

    Google Scholar 

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

    Article  Google Scholar 

  • Roy, S., Sabarwal, T.: Monotone comparative statics for games with strategic substitutes. J. Math. Econ. 46(5), 793–806 (2010)

    Article  Google Scholar 

  • Roy, S., Sabarwal, T.: Characterizing stability properties in games with strategic substitutes. Games Econ. Behav. 75, 337–353 (2012)

    Article  Google Scholar 

  • Vives, X.: Nash equilibrium with strategic complementarities. J. Math. Econ. 19, 305–321 (1990)

    Article  Google Scholar 

  • Zimper, A.: A fixed-point characterization of the dominance-solvability of lattice games with strategic substitutes. Int. J. Game Theory 36(1), 107–117 (2007)

    Article  Google Scholar 

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Authors and Affiliations

  1. Northwestern University, 2211 Campus Drive, Evanston, IL, USA

    Eddie Dekel

  2. Tel Aviv University, Ramat Aviv, 69978, Tel Aviv, Israel

    Eddie Dekel

  3. The Eitan Berglas School of Economics, Tel Aviv University, Ramat Aviv, 69978, Tel Aviv, Israel

    Ady Pauzner

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  1. Eddie Dekel
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  2. Ady Pauzner
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Correspondence to Eddie Dekel.

Additional information

We acknowledge helpful comments from the referees and the issue editor, Rabah Amir, and financial support from the Pinhas Sapir Center, and the National Science Foundation Grant SES-1227434 for financial support.

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Dekel, E., Pauzner, A. Uniqueness, stability and comparative statics for two-person Bayesian games with strategic substitutes. Econ Theory 66, 747–761 (2018). https://doi.org/10.1007/s00199-017-1083-7

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