Ordinal Games and Generalized Nash and Stackelberg Solutions

The traditional theory of cardinal games deals with problems where the players are able to assess the relative performance of their decisions (or controls) by evaluating a payoff (or utility function) that maps the decision space into the set of real numbers. In that theory, the objective of each player is to determine a decision that minimizes its payoff function taking into account the decisions of all other players. While that theory has been very useful in modeling simple problems in economics and engineering, it has not been able to address adequately problems in fields such as social and political sciences as well as a large segment of complex problems in economics and engineering. The main reason for this is the difficulty inherent in defining an adequate payoff function for each player in these types of problems. In this paper, we develop a theory of games where, instead of a payoff function, the players are able to rank-order their decision choices against choices by the other players. Such a rank-ordering could be the result of personal subjective preferences derived from qualitative analysis, as is the case in many social or political science problems. In many complex engineering problems, a heuristic knowledge-based rank ordering of control choices in a finite control space can be viewed as a first step in the process of modeling large complex enterprises 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 payoff-based games as cardinal games and to these new types of rank ordering-based games as ordinal games. In the theory of ordinal games, rather than minimizing a payoff function, the objective of each player is to select a decision that has a certain rank (or degree of preference) taking into account the choices of all other players. In this paper, we will formulate a theory for ordinal games and develop solution concepts such as Nash and Stackelberg for these types of games. We 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 numerous examples of deterministic matrix games. We feel that this new theory of ordinal games will be very useful to social and political scientists, economists, and engineers who deal with large complex systems that involve many human decision makers with often conflicting objectives.