Discounting and efficiency in coalitional bargaining with random proposers

This paper analyzes a random-proposer coalitional bargaining game with different discount factors, which is a generalized version of Okada's (1996) model. We consider limit subgame efficiency which means that when the discount factors are sufficiently close to unity, the full coalition is formed in each subgame. In this paper, a negative result is shown: The limit subgame efficiency is attained if and only if values of the characteristic function are zero for all coalitions but the grand coalition. This result implies that under different discount factors, even under a naturally generalized condition of Okada''s necessary and sufficient condition for the limit subgame efficiency, the limit subgame efficiency is not necessarily achieved. On the other hand, it is shown that under a condition on the region of players'' discount factors, the generalized condition of Okada''s condition is almost necessary and sufficient for the limit subgame efficiency.