Trembling hand perfection for mixed quantal/best response equilibria

The quantal response equilibrium (QRE) is a powerful alternative to full rationality equilibrium concepts. At a QRE, all joint moves have non-zero probability. However in “mixed scenarios”, where some players use quantal response and some use best response, equilibrium strategy profiles can have joint moves with zero probability. This raises the question of applying the trembling hand refinement to such mixed scenarios. To address this I first show how to reformulate the QRE as a “best response” equilibrium where expected utilities are replaced by more general objective functions. I then show that under this reformulation the two popular types of trembling hand perfection can differ when some players use quantal response and some use best response. I end by showing that one of those types of trembling hand perfection cannot be used to remove certain troubling kinds of equilibrium in such mixed scenarios, while the other type can. The conclusion is that only the one type of trembling hand perfection should be applied when we allow some players to be quantal response and some to be best response.

[1]  E. Maskin,et al.  The Existence of Equilibrium in Discontinuous Economic Games, I: Theory , 1986 .

[2]  Won Kyu Kim,et al.  Nash equilibrium and minimax theorem with C-concavity , 2007 .

[3]  E.E.C. van Damme,et al.  Games with imperfectly observable commitment , 1997 .

[4]  Haishu Lu,et al.  On the Existence of Pure Strategy Nash Equilibrium for Non-cooperative Games in L-convex Spaces , 2007, 2009 International Conference on Intelligent Human-Machine Systems and Cybernetics.

[5]  H. Nikaidô,et al.  Note on non-cooperative convex game , 1955 .

[6]  E. Hopkins.  Two Competing Models of How People Learn in Games (first version) , 1999 .

[7]  Jacob K. Goeree,et al.  Minimum-Effort Coordination Games: Stochastic Potential and Logit Equilibrium , 2001, Games Econ. Behav..

[8]  R. McKelvey,et al.  Quantal Response Equilibria for Normal Form Games , 1995 .

[9]  Jeff S. Shamma,et al.  Dynamic fictitious play, dynamic gradient play, and distributed convergence to Nash equilibria , 2005, IEEE Transactions on Automatic Control.

[10]  T. Başar,et al.  Dynamic Noncooperative Game Theory, 2nd Edition , 1998 .

[11]  Roger B. Myerson,et al.  Game theory - Analysis of Conflict , 1991 .

[12]  D. Fudenberg,et al.  Steady state learning and Nash equilibrium , 1993 .

[13]  R. Selten.  Reexamination of the perfectness concept for equilibrium points in extensive games , 1975, Classics in Game Theory.

[14]  T. Başar,et al.  Dynamic Noncooperative Game Theory , 1982 .

[15]  David M. Kreps,et al.  Learning Mixed Equilibria , 1993 .

[16]  J K Goeree,et al.  Stochastic game theory: for playing games, not just for doing theory. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[17]  R. McKelvey,et al.  Quantal Response Equilibria for Extensive Form Games , 1998 .

[18]  Jacob K. Goeree,et al.  Quantal Response Equilibrium and Overbidding in Private-Value Auctions , 2002, J. Econ. Theory.