Huang, Chen-Ying: Multilateral Bargaining: Conditional and Unconditional Offers
World Conference Econometric Society, 2000, Seattle

Chen-Ying Huang, National Taiwan University
Multilateral Bargaining: Conditional and Unconditional Offers
Session: C-5-15  Sunday 13 August 2000  by Huang, Chen-Ying
We present a multilateral bargaining game in which n persons split a cake. A novel aspect of the model is the distinction between conditional and unconditional offers. Since in the pure bargaining situation only the grand coalition has non-zero value, we interpret an offer in terms of cake as a conditional offer. This only takes effect conditional on other responders' consent (for unanimous agreement is naturally required to cut the cake). On the other hand, an offer in terms of the proposer's outside resource is called an unconditional offer. As long as the proposed responder agrees, an unconditional offer binds because the proposer has complete control over his outside resource. This distinction is not void. It sheds light on the contrasting results of the previous literature on multilateral bargaining. For instance, Shaked's (reported by Sutton [2]) unanimity game admits multiplicity of equilibria because offers are conditional while Krishna and Serrano [1] get a unique equilibrium outcome with only unconditional offers.
We find that when both conditional and unconditional offers are allowed, we obtain uniqueness again. We further suggest the intuition for this result and indicate the importance of unconditional offers (and the irrelevance of conditional offers) in pinning down the equilibrium. Although Krishna and Serrano [1] also get a unique equilibrium outcome, we argue that our game is a more natural setup for several important reasons. Moreover, the uniqueness result is quite strong that so long as players propose according to some fixed periodic protocol (for instance, when there are 3 players, a protocol specifying 1, 2, 1, 3 is of periodicity 4), our game makes a unique prediction. In fact, when the discount factor is large enough, a player's equilibrium payoff exactly reflects his bargaining position so that the more times a player can propose in a cycle, the more he ends up with in equilibrium (for instance, when the protocol is 1, 2, 1, 3, player 1 gets about a half of the cake while 2 and 3 both get a quarter). No other paper in the literature of multilateral bargaining makes this prediction.
The proof for uniqueness is simple. We argue by contradiction that if uniqueness does not hold, then we can create a series of deviations by a single player such that when a proposing cycle is completed, this player's payoff is even lower after these deviations. If we iterate this argument (i.e., to run through the proposing cycle) for enough times, eventually this deviating player is getting something negative, which is impossible. Thus by exploiting the stationary structure of the game, we obtain the unique equilibrium outcome. The proof is far simpler than that of Krishna and Serrano [1].
Finally, to further study robustness of the uniqueness result, we discuss two extensions. The first extension deals with the situation where instead of a fixed protocol, players are randomly selected as the proposer with some fixed probability. The second extension deals with coalitional bargaining. We find that uniqueness still obtains in both cases and the spirit of the proofs is very similar. In particular, we rely heavily on the use of unconditional offers. We further comment on the unique equilibrium outcome of the coalitional bargaining game.

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