Tournament Solutions Extensions of Maximality and their Applications to Decision-Making

Now the notion of domination on which we rely is, indeed, not transitive. [.. . ] This lack of transitivity, especially in the above formalistic presentation, may appear to be an annoying complication and it may even seem desirable to make an effort to rid the theory of it. Yet the reader who takes another look at the last paragraph will notice that it really contains only a circumlocution of a most typical phenomenon in all social organizations. The domination relationships between various imputations x, y, z,. .. — i.e. between various states of society—correspond to the various ways in which these can unstabilize—i.e. upset—each other. That various groups of participants acting as effective sets in various relations of this kind may bring about cyclical dominations—e.g., y over x, z over y, and x over z—is indeed one of the most characteristic difficulties which a theory of these phenomena must face. Thus our task is to replace the notion of the optimum—i.e. of the first element—by something which can take over its functions in a static equilibrium. This becomes necessary because the original concept has become untenable. We first observed its breakdown in the specific instance of a certain three-person game [.. . ] But now we have acquired a deeper insight into the cause of its failure: it is the nature of our concept of domination, and specifically its intransitivity. This type of relationship is not at all peculiar to our problem. Other instances of it are well known in many fields and it is regretted that they have never received a generic mathematical treatment. We mean all those concepts which are in the general nature of a comparison of preference or superiority, or of order, but lack transitivity: e.g., the strength of chess players in a tournament, the paper form in sports and races, etc.