Ordinal Game Theory and Applications – A New Framework For Games Without Payoff Functions

Decision-making problems in engineering, business, management, and economics that involve two or more decision-makers with competing objectives are often optimized using the theory of games. This theory, initially developed by Von Neumann and Morgenstern, and later by Nash, requires that each point in the decision space be mapped, through a payoff function, into a real number representing the value of the collective set of decisions to each decision maker. This theory, which is cardinal in nature, requires that each decision-maker determine its decision by maximizing its payoff function taking into account the choice of decisions by all other decision-makers. While this theory has been very useful in addressing some aspects of quantitative decision-making in engineering and economics, it has not been able to adequately address qualitative problems in fields such as social and political sciences, as well as a large segment of complex problems in engineering, business and management imbedded in a competitive environment. The main reason for this is the inherent difficulty in defining an adequate payoff function for each decision maker in these types of problems. In this paper, we present a theory where, instead of a payoff function, the decision-makers are able to rank order their decisions against decision choices by the other decision-makers. Such a rank ordering could be the result of personal, subjective, preferences derived from qualitative analysis, as is the case in many social sciences problems. In such problems a heuristic, knowledge-based, rank ordering of decision choices in a finite decision space can be viewed as a first step in the process of modeling complex problems for which a mathematical description is usually extremely difficult, if not impossible, to obtain. In order to distinguish between these two types of games, we will refer to traditional payoffbased games as “Cardinal Games” and to these new types of rank ordering-based games as “Ordinal Games”. In this paper, we review the theory of ordinal games and discuss associated solution concepts such as the Nash equilibrium. We will also show that these solutions are general in nature and can be characterized, in terms of existence and uniqueness, with conditions that are more intuitive and much less restrictive than those of the traditional cardinal games. We will illustrate these concepts with several examples of deterministic matrix games, including an example of team composition and task assignment by a top commander in a military operation where payoff functions are not readily available. We feel that this new theory of ordinal games will be very useful when dealing with complex decisionmaking problems that involve more than one decision makers in a competitive environment.