This paper studies the dynamic and equilibrium-selecting behavior of a multi-agent system consisting of adaptive bargaining agents. We model an adaptive agent as a collection of strategies which is optimized by an evolutionary algorithm (EA). EAs are stochastic search methods based upon the principles of natural genetic systems. These algorithms have been used in the past, with considerable success, to solve difficult optimization problems. Examples include problems with huge search spaces, multiple local optima, discontinuities, and noise. Adaptive agents learn in three different ways in an evolutionary setting: (i) by selection and reproduction of successful strategies, (ii) by recombining or ``crossing over'' previously-tested strategies, and (iii) by random experimentation (by ``mutating'' existing strategies). Such agents are boundedly rational because they only experience the profit of their interactions with other agents and learn by trial-and-error instead of abstract reasoning. Their equilibrium-selecting behavior is interpreted in this paper by comparison with game-theoretic (subgame-perfect equilibrium) predictions for fully rational agents. This paper shows that game-theoretic approaches are very useful to interpret equilibrium-selecting behavior in evolutionary systems of adaptive bargaining agents. The adaptive agents are boundedly rational because they only experience the profit of their interactions with other agents. Nevertheless, they display behavior that is surprisingly "rational" and fully informed in many instances. Agreement between theory and experiment is especially good when the agents experience an intermediate time pressure. In extreme situations (i.e., when time pressure becomes either strong or weak) significant deviations from game-theoretic predictions can occur, however. A good example is the case of extreme time pressure. In this case, highly nonlinear transients can occur if the deal reached by the adaptive agents approaches the extreme outcome predicted by game theory. Two other experimental observations should also be mentioned here. First, the finite horizon of the negotiations is not always fully exploited by the last agent in turn (even if time pressure is rather weak). In fact, the boundedly-rational agents often act as if the length of the game is actually much longer. This lends more support to the "infinite-horizon" assumption frequently employed in game-theoretic work. This approximation may yield surprisingly accurate results when the agents do not perceive the deadline of the negotiations. Second, we observe (and explain) discrepancies between theory and experiment if the agents experience an unequal time pressure. More in general, this work presents a systematic validation of evolutionary and computational techniques in the field of bargaining. Our model has also served as a starting point for further explorations. Several important topics have been addressed in these works: complex multi-issue and multi-opponent bargaining problems, economic modelling issues, learning by co-evolution, the development of powerful bargaining strategies, etc. We hope that these different lines of research will be extended further in future works.
