$$p$$-best response set.(English)Zbl 1142.91345

Summary: This paper introduces a notion of $$p$$-best response set ($$p$$-BR). We build on this notion in order to provide a new set-valued concept: the minimal $$p$$-best response set ($$p$$-MBR). After proving general existence results of the $$p$$-MBR, we show that it characterizes set-valued stability concepts in a dynamic with Poisson revision opportunities borrowed from A. Matsui and K. Matsuyama [J. Econ. Theory 65, 415–434 (1995; Zbl 0835.90121)]. Then, we study equilibrium selection. In particular, using our notion of $$p$$-BR, we generalize S. Morris et al. [ Econometrica 63, No. 1, 145–157 (1995; Zbl 0827.90138 )] that aimed to provide sufficient conditions under which a unique equilibrium is selected in the presence of higher order uncertainty.

MSC:

 91A22 Evolutionary games 91A10 Noncooperative games 91A26 Rationality and learning in game theory

Citations:

Zbl 0835.90121; Zbl 0827.90138
Full Text:

References:

 [1] Aumann, R.J.; Brandeburger, A., Epistemic conditions for Nash equilibrium, Econometrica, 63, 1161-1180, (1995) · Zbl 0841.90125 [2] Basu, K.; Weibull, J.W., Strategy subsets closed under rational behavior, Econ. letters, 36, 141-146, (1991) · Zbl 0741.90097 [3] Battigalli, P.; Siniscalchi, M., Rationalizable bidding in first price auctions, Games econ. behav., 45, 38-72, (2003) · Zbl 1063.91018 [4] Bernheim, D., Rationalizable strategic behavior, Econometrica, 52, 1007-1028, (1984) · Zbl 0552.90098 [5] J. Durieu, P. Solal, O. Tercieux, Adaptive learning and curb set selection, (2003) Mimeo. · Zbl 1233.91035 [6] Ellison, G., Basins of attraction, long-run stochastic stability, and the speed of step-by-step evolution, Rev. econ. stud., 67, 17-45, (2000) · Zbl 0956.91027 [7] Glicksberg, I.L., A further generalization of the Kakutani fixed point theorem with application to Nash equilibrium points, Proc. amer. math. soc., 38, 170-174, (1952) · Zbl 0046.12103 [8] Harsanyi, J.; Selten, R., A general theory of equilibrium in games, (1988), Cambridge MIT Press Cambridge [9] Hurkens, S., Learning by forgetful players, Games econ. behav., 11, 304-329, (1995) · Zbl 0841.90126 [10] Kajii, A.; Morris, S., The robustness of equilibria to incomplete information, Econometrica, 65, 1283-1309, (1997) · Zbl 0887.90186 [11] Kalai, E.; Samet, D., Persistent equilibria in strategic games, Int. J. game theory, 13, 129-144, (1984) · Zbl 0541.90097 [12] F. Kojima, Risk-dominance and perfect foresight dynamics in $$N$$-player games, J. Econ. Theory, forthcoming. · Zbl 1153.91313 [13] Maruta, T., On the relationship between risk-dominance and stochastic stability, Games econ. behav., 19, 221-234, (1997) [14] Matsui, A.; Matsuyama, K., An approach to equilibrium selection, J. econ. theory, 65, 415-434, (1995) · Zbl 0835.90121 [15] A. Matsui, D. Oyama, Rationalizable foresight dynamics, Games Econ. Behav., (2002), forthcoming. · Zbl 1177.91030 [16] Monderer, D.; Samet, D., Approximating common knowledge with common beliefs, Games econ. behav., 1, 170-190, (1989) · Zbl 0755.90110 [17] Morris, S.; Rob, R.; Shin, H.S., $$p$$-dominance and belief potential, Econometrica, 63, 145-157, (1995) · Zbl 0827.90138 [18] Morris, S.; Ui, T., Generalized potentials and robust sets of equilibria, J. econ. theory, 124, 34-78, (2005) · Zbl 1100.91004 [19] Nash, J., Equilibrium points in $$n$$-person games, Proc. natl. acad. sci., 36, 48-49, (1950) · Zbl 0036.01104 [20] Oyama, D., $$p$$-dominance and equilibrium selection under perfect foresight dynamics, J. econ. theory, 107, 288-310, (2002) · Zbl 1033.91002 [21] D. Oyama, S. Takahashi, J. Hofbauer, Monotone methods for equilibrium selection under perfect foresight dynamics, (2003) Mimeo. [22] Pearce, D., Rationalizable strategic behavior and the problem of perfection, Econometrica, 52, 1029-1050, (1984) · Zbl 0552.90097 [23] Reny, P., On the existence of pure and mixed Nash equilibria in discontinuous games, Econometrica, 67, 1029-1056, (1999) · Zbl 1023.91501 [24] Rubinstein, A., The electronic mail game: strategic behavior under almost common knowledge, Amer. econ. rev., 79, 385-391, (1989) [25] O. Tercieux, $$p$$-Best response set and the robustness of equilibria to incomplete information, Games Econ. Behav., forthcoming. · Zbl 1177.91019 [26] Young, P., The evolution of conventions, Econometrica, 61, 57-84, (1993) · Zbl 0773.90101
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.