The nonatomic supermodular game. (English) Zbl 1283.91010
Summary: We introduce the nonatomic supermodular game, where no player’s action has any discernible impact on other players’ payoffs and yet strategic complementarities prevail among all players’ types and actions. For both semi-anonymous and anonymous games, we show that monotone equilibria form nonempty complete lattices and among these equilibria, the largest and smallest members vary in monotone fashions with respect to certain game-changing parameters. Results here complement existing nonatomic-game works, which focused more on pure equilibria of anonymous games where opponents’ types are not influential. They are also applicable to price competition involving diverse cost/quality parameters, as well as a slew of other situations.
Keywords:
nonatomic game; supermodular game; strong set order; stochastic dominance order; complete latticeReferences:
| [1] | Akerlof, G., The economics of caste and of the rat race and other woeful tales, Quart. J. Econ., 90, 599-617 (1976) |
| [2] | Balder, E. J., A unifying pair of Cournot-Nash equilibrium existence results, J. Econ. Theory, 102, 437-470 (2002) · Zbl 1040.91016 |
| [3] | Diamond, P. A., Aggregate demand management in search equilibrium, J. Polit. Economy, 91, 881-894 (1982) |
| [4] | Khan, M. A.; Rath, K. P.; Sun, Y., On the existence of pure strategy equilibrium in games with a continuum of players, J. Econ. Theory, 76, 13-46 (1997) · Zbl 0883.90137 |
| [5] | Mas-Colell, A., On a theorem of Schmeidler, J. Math. Econ., 13, 201-206 (1984) · Zbl 0563.90106 |
| [6] | Milgrom, P. R.; Roberts, J., Rationalizability, learning, and equilibrium in games with strategic complementarities, Econometrica, 58, 1255-1277 (1990) · Zbl 0728.90098 |
| [7] | Milgrom, P. R.; Shannon, C., Monotone comparative statics, Econometrica, 62, 157-180 (1994) · Zbl 0789.90010 |
| [8] | Radner, R.; Rosenthal, R., Private information and pure-strategy equilibria, Math. Oper. Res., 7, 401-409 (1982) · Zbl 0512.90096 |
| [9] | Rashid, S., Equilibrium points of non-atomic games: asymptotic results, Econ. Letters, 12, 7-10 (1983) · Zbl 1273.91040 |
| [10] | Schmeidler, D., Equilibrium points of nonatomic games, J. Stat. Phys., 7, 295-300 (1973) · Zbl 1255.91031 |
| [11] | Shaked, M.; Shanthikumar, J. G., Stochastic Orders (2007), Springer: Springer New York · Zbl 1111.62016 |
| [12] | Tarski, A., A lattice-theoretical fixpoint theorem and its applications, Pac. J. Math., 5, 285-309 (1955) · Zbl 0064.26004 |
| [13] | Topkis, D. M., Minimizing a submodular function on a lattice, Oper. Res., 26, 305-321 (1978) · Zbl 0379.90089 |
| [14] | Topkis, D. M., Equilibrium points in nonzero-sum \(n\)-person submodular games, SIAM J. Control Optim., 17, 773-787 (1979) · Zbl 0433.90091 |
| [15] | Topkis, D. M., Supermodularity and Complementarity (1998), Princeton University Press: Princeton University Press Princeton, New Jersey |
| [16] | Van Zandt, T.; Vives, X., Monotone equilibrium in Bayesian games of strategic complementarities, J. Econ. Theory, 134, 339-360 (2007) · Zbl 1156.91312 |
| [18] | Vives, X., Nash equilibrium with strategic complementarities, J. Math. Econ., 19, 305-321 (1990) · Zbl 0708.90094 |
| [19] | Vives, X., Oligopoly Pricing (1999), MIT Press: MIT Press Cambridge, Massachusetts |
| [20] | Vives, X., Complementarities and games: new developments, J. Econ. Lit., 43, 437-479 (2005) |
| [21] | Zhou, L., The set of Nash equilibria of a supermodular game is a complete lattice, Games Econ. Behav., 7, 295-300 (1994) · Zbl 0809.90138 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
