Our concern is the extension of the theory of the Shapley value to problems involving externalities. Using the standard axiom systems behind the Shapley value leads to the identification of bounds on players' payoffs around an “externality-free” value. The approach determines the direction and maximum size of Pigouvian-like transfers among players, transfers based on the specific nature of externalities that are compatible with basic normative principles. Examples are provided to illustrate the approach and to draw comparisons with previous literature.

Non-equilibrium models of choice (e.g. level-k reasoning) have significantly different, sometimes more accurate, predictions in games than does Nash equilibrium. When it comes to the maximal set of functions that are implementable in mechanism design, however, they turn out to have similar implications. Focusing on single-valued rules, we discuss the role and implications of different behavioural anchors (arbitrary level-0 play), and prove a level-k revelation principle. If a function is level-k implementable given any level-0 play, it must obey a slight weakening of standard strict incentive constraints. Further, the same condition is also sufficient for level-k implementability, although the role of specific level-0 anchors is more controversial for the sufficiency argument. Nonetheless, our results provide tight characterizations of level-k implementable functions under a variety of level-0 play, including truthful, uniform, and atomless anchors.

The authors study a multilateral bargaining procedure that extends A. Rubinstein's (1982) alternating offer game to the case of n players. The procedure captures the notion of consistency in the sense familiar in cooperative game theory and they use it to establish links to the axiomatic theory of bargaining.