CONES OF COOPERATION, PERRON-FROBENIUS THEORY AND THE INDEFINITELY REPEATED PRISONERS-DILEMMA

The continuation probability and discount parameter of indefinitely re peated games define an infinite matrix with nonnegative entries. The s olutions of a matrix inequality are subgame perfect equilibria for the repeated Prisoners' Dilemma game and form a 'cone of cooperation'. Th e cone's geometry quantifies the intuition that more cooperation is po ssible as the probability of continuation increases. The spectral radi us acts as a bifurcation point; comparing a parameter of the stage gam e to the spectral radius indicates whether a cooperative equilibrium ( eigenvector) exists. The structure of the matrix yields the spectral r adius and eigenvectors; surprisingly, Perron-Frobenius theory is fruit less. (C) 1998 Elsevier Science S.A. All rights reserved.