The grim-trigger strategy introduced by Friedman is often used in economic models, mainly because of its simplicity, to show that collusion can be sustained by means of a subgame perfect equilibrium. In this work we show that, under simple conditions, it is possible to improve on the grim-trigger strategy while retaining subgame perfection and in some cases adding weak renegotiation proofness (in the sense of Farrell and Maskin). The basic idea is that, following a deviation, the cheater, instead of continuing in the Nash equilibrium forever, chooses an autopenalty which signals, in a very strong way, that he/she would not deviate any more if cooperation were reestablished. We check the nice working of this strategy in the Cournot's oligopoly model.
Subgame Perfection for Equilibria in Quantitative Reachability Games