Gradualism and Irreversibility

This paper considers a class of two-player dynamic games in which each player controls a one-dimensional variable which we interpret as a level of cooperation. In the base model, there is an irreversibility constraint stating that this variable can never be reduced, only increased. It otherwise satis…es the usual discounted repeated game assumptions. Under certain restrictions on the payo¤ function, which make the stage game resemble a continuous version of the Prisoners’ Dilemma, we characterize e¢cient symmetric equilibria, and show that cooperation levels exhibit gradualism and converge, when payo¤s are smooth, to a level strictly below the one-shot ef- …cient level: the irreversibility induces a steady-state as well as a dynamic ine¢ciency. As players become very patient, however, payo¤s converge to (though never attain) the e¢cient level. We also show that a related model in which an irreversibility arises through players choosing an incremental variable, such as investment, can be transformed into the base model with similar results. Applications to a public goods sequential contribution model and a model of capacity reduction in a declining industry are discussed. The analysis is extended to incorporate partial reversibility, asymmetric equilibria, and sequential moves.