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5.3 Fictitious-Play-Based Algorithms > 5.3.5 Is Fictitious Play Applicable in W... - Pg. 137

5.3 Fictitious-Play-Based Algorithms 137 The stochastic process {(x j,t , y j,t )} t satisfies the conditions in Bena¨ m (1999): i these conditions are reported and explained in Appendix A.2. We then use it and approximate the stochastic process by the following system of ordinary differential equations: x j (t) = 2 1 K y j (t) 1 K-1 2 K i=j i (x i (t)) - x j (t) y j (t) = - y j (t) A rest point x is linearly unstable with respect to a dynamic process x = (x) if its linearization D(x) evaluated at x has at least one eigenvalue with a positive real part. ·························································································· · · · · · · · If x is a linearly unstable equilibrium for the system x = (x) - x then: · · · · lim x 1,t = . . . = lim x K,t = x (5.11) · · t - - t - - · · · · with probability zero. · Theorem 149: Fudenberg and Takahashi (2008) and Pemantle (1990)