N-Person Bargaining and Strategic Complexity¤

We investigate the e®ect of introducing costs of complexity in the n -person unanimity bargaining game. In particular, the paper provides a justi ̄cation for stationary equilibrium strategies in the class of games where complexity costs matter. As is well-known, in this game every individually rational allocation is sustainable as a Nash equilibrium (also as a subgame perfect equilibrium if players are su±ciently patient and if n > 2). Moreover, delays in agreement are also possible in such equilibria. By limiting ourselves to strategies that can be implemented by a machine (automaton) and by suitably modifying the de ̄nition of complexity for the purpose of analysing a single extensive form, we ̄nd that complexity costs do not reduce the range of possible allocations but they do limit the amount of delay that can occur in any agreement. In particular, we show that in any n-player game, for any allocation z; an agreement on z at any period t can be sustained as a Nash equilibrium of the machine game with complexity costs if and only if t · n: We use the limit on delay result to establish that, in equilibrium, the machines implement stationary strategies. ¤The ̄rst version of this paper was written in 1994. Chatterjee wishes to acknowledge the hospitality of St.John's College, Cambridge during the initial period of work on this paper. Another substantial revision was done when Chatterjee was a guest at CRIEFF, University of St. Andrews, and we are grateful also for the stimulating environment there. We also thank Luca Anderlini, Ani Dasgupta, Vijay Krishna and Ariel Rubinstein for comments on earlier versions. A shorter paper with a related but di®erent de ̄nition of complexity, "Multiperson Bargaining and Strategic Complexity", is forthcoming in Econometrica. Three anonymous referees and a Co-Editor made comments that helped us greatly to refocus the contents of the two papers. yThe current, March 1999, version is based primarily on one written in April 1998, and presents our results in some detail.