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Solution uniqueness is an important property for a bargaining model. Rubinstein's (1982) seminal 2-person alternating-offer bargaining game has a unique Subgame Perfect Equilibrium outcome. Is it possible to obtain uniqueness results in the much enlarged setting of multilateral bargaining with a characteristic function? This paper investigates a random-proposer model first studied in Okada (1993) in which each period players have equal probabilities of being selected to make a proposal and bargaining ends after one coalition forms. Focusing on transferable utility environments and Stationary Subgame Perfect Equilibria (SSPE), we find ex ante SSPE payoff uniqueness for symmetric and convex characteristic functions, considerably expanding the conditions under which this model is known to exhibit SSPE payoff uniqueness. Our model includes as a special case a variant of the legislative bargaining model in Baron and Ferejohn (1989), and our results imply (unrestricted) SSPE payoff uniqueness in this case.

A longstanding criticism of the core is that it is too sensitive to small changes in player numbers, as in a well known example where one extra seller (resp. buyer) causes the entire surplus to go to the buyer's (seller's) side. We test this example in the lab, using several different trading institutions. We find that successful collusion is relatively infrequent and decreasing over time even with institutions that facilitate collusion and, consistent with core theory, a disproportionate share of the surplus typically goes to the less numerous side. Our study also illuminates the boundaries of competitive equilibrium.