Cones of cooperation, Perron-Frobenius theory and the indefinitely repeated Prisoners' Dilemma

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