Identication of Stochastic Sequential Bargaining Models1

Stochastic sequential bargaining games (Merlo and Wilson (1995, 1998)) have found wide applications in various …elds including political economy and macroeconomics due to their ‡exibility in explaining delays in reaching an agreement. In this paper, we present new results in nonparametric identi…cation of such models under di¤erent scenarios of data availability. First, we give conditions for an observed distribution of players’decisions and agreed allocations of the surplus, or the "cake", to be rationalized by a sequential bargaining model. We show the common discount rate is identi…ed, provided the surplus is monotonic in unobservable states (USV) given observed ones (OSV). Then the mapping from states to surplus, or the "cake function", is also recovered under appropriate normalizations. Second, when the cake is only observed under agreements, the discount rate and the impact of observable states on the cake can be identi…ed, if the distribution of USV satis…es some exclusion restrictions and the cake is additively separable in OSV and USV. Third, if data only report when an agreement is reached but never report the size of the cake, we propose a simple algorithm that exploits shape restrictions on the cake function and the independence of USV to recover all rationalizable probabilities for agreements under counterfactual state transitions. Numerical examples show the set of rationalizable counterfactual outcomes so recovered can be informative. Key words: Nonparametric identi…cation, non-cooperative bargaining, stochastic sequential bargaining, rationalizable counterfactual outcomes JEL codes: C14, C35, C73, C78 1We thank Steven Berry, Herman Bierens, Hanming Fang, Philip Haile, Isabelle Perrigne, Katja Seim, Elie Tamer, Petra Todd, Quang Vuong, Ken Wolpin and seminar participants at Penn State and Wharton School at U Penn for comments and discussions. The usual disclaimer applies.