Four axioms, based on Shapley's original development of the Shapley value, are proposed for values of cooperative games with quarrelling. The collection of all values satisfying these axioms is shown to be the collection of values determined by a construction utilizing the proper choices of several parameters. This facilitates the proof of several common properties of all these values which are analogous to properties of the usual Shapley value. Finally, the parameters corresponding to the two specific values for games with quarrelling proposed in [ 2 ], are presented.
The Shapley value of a cooperative game has proven to be one of the most successful concepts of game theory. In his original development [4], Shapley showed that three simple axioms uniquely determine the value. He then devised a simple model of play for a cooperative game, 1 under which each player's expected reward is exactly his Shapley value. Recently, Aumann and Shapley 2 have commented that "this 'constructive' approach ... [which] complements ... the 'axiomatic' approach ... has run like a countermotif through the development of value theory from its beginnings." The present work concerns the interrelationship of these two approaches to the Shapley value for games with quarrelling.
In a cooperative game with quarrelling, certain players (the quarrellers) refuse to join any coalition already containing another quarreller. By assigning a Shapley value to such games, one gains information on the worth of cooperation within certain coalitions. For example, the difference between a player's Shapley values with and without quarrelling measures the worth, to him, of cooperation among the quarrellers (if it were possible). In [2], two definitions of a Shapley value for games with quarrelling were proposed. Each definition was obtained using the 'constructive' approach; the two models of play, which can be called 'prior selection' and 'posterior selection '3 , led to two values which are, in general, distinct. In what follows, the 'axiomatic' approach to the Shapley value for games with quarrelling will be adopted. It will be shown that four axioms, closely related to Shapley's original axioms in [4], determine an infinitely large family of 'Shapley values' for games with quarrelling, including both values defined previously. In addition, some further properties shared by all members of this family will be deduced.