Equilibrium points in games with vector payoffs
Authors: Shapley L. S. , Rigby Fred D. 
Published: May 2007

Historical. The topic of games with vector payoffs is one which could be expected to attract attention on the basis of its intrinsic interest. However, the history of the particular problem treated in Dr. Shapley's paper was not of this kind and m a y have some interest of its own. During an interval when the writer of this note was engaged in operations research his group was asked t o analyze a combat situation in which movement of f o r c e s and the inhibition of such movement were critical It turned out to be feasible to represent essential aspects of this situation by a game‐like model with reasonably well defined courses of action corresponding to pure strategies for each side. However, each pair of these strategies, one for each player, generated both a time delay in the movement and losses to the moving forces not balanced by losses to the inhibiting player. That is the model was a game with vector pay off in the sense of this paper. Since the values of these delays and losses would be realized in a subsequent battle in the area approached by the movement in question, it was clear that some s o r t of exchange ratio between attrition and delay must exist. Efforts to obtain estimates of such a ratio failed completely. In fact, several rather arbitrary weightings of delay and attrition were assumed and the resulting numerical games solved for the sake of the insight they could provide, but this was a rather unsatisfactory expedient. Had the theory presented in this paper been available, it would have been possible at least to sharpen the questions asked in pursuit of the „exchange ratio,”︁ and probably to eliminate rationally considerable sets of strategy pairs a s unsuitable or a s representing extreme cases in the weighting.

In any event, the theory was not a t hand and the study could not wait on its development. The problem was, however, brought to the attention of Dr. Shapley, with this paper a s a consequence.

Nature of the Results. Perhaps a comment on the kind of results which are presented is in order since there is a sort of fitness to them. As was noted above, the occurrence of vector payoff represents a failure to resolve some of the questions whose answers are needed in order to construct a game model. This being the case, one should not be disappointed that the theory does not produce a clear cut, well defined solution concept. What it does produce, if the payoff vectors have K + 1 components, is a ZK parameter set of equilibrium points, in comparison with a unique solution for a numerical game similarly specified. (The fact that numerical game solutions need not be unique has its counterpart among vector game equilibrium points also, of course.) Such an „equilibrium set,”︁ however, need not cover a major part of the joint mixed strategy space, even in games with only a few pure strategies for each player. Accordingly, although finding these sets may be extremely tedious for a particular game, they may well turn out to be worth having in the sense that they limit the domain over which weighting comparisons of the payoff vector components need be made. Since Dr. Shapley's results concern both „strong”︁ and „weak”︁ equilibrium points it is pertinent to note that this distinction appears „usually”︁ to concern the boundaries of the „equilibrium sets.”︁ That is one might expect that the set of strong equilibrium points omits boundary points which are included in the weak case. It should be remarked, however, that this characterization of the distinction is not precise or universal, and instances of its failure can be produced.