Ordinal Game Theory and Applications – a New Framework for Games Without Payoff Functions
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Ordinal Game Theory and Applications – A New Framework For Games Without Payoff Functions Jose B. Cruz*, Jr. and Marwan A. Simaan# *Department of Electrical Engineering The Ohio State University, Columbus, OH 43210 Email: [email protected] #Department of Electrical Engineering University of Pittsburgh, Pittsburgh, PA 15261 Email: [email protected] Keywords: Games Theory, Decision-Making, ordering could be the result of personal, Payoff Functions, Nash Solutions, Rank Ordering. subjective, preferences derived from qualitative analysis, as is the case in many social sciences ABSTRACT problems. In such problems a heuristic, knowledge-based, rank ordering of decision Decision-making problems in engineering, choices in a finite decision space can be viewed as business, management, and economics that a first step in the process of modeling complex involve two or more decision-makers with problems for which a mathematical description is competing objectives are often optimized using usually extremely difficult, if not impossible, to the theory of games. This theory, initially obtain. In order to distinguish between these two developed by Von Neumann and Morgenstern, types of games, we will refer to traditional payoff- and later by Nash, requires that each point in the based games as “Cardinal Games” and to these decision space be mapped, through a payoff new types of rank ordering-based games as function, into a real number representing the value “Ordinal Games”. In this paper, we review the of the collective set of decisions to each decision theory of ordinal games and discuss associated maker. This theory, which is cardinal in nature, solution concepts such as the Nash equilibrium. requires that each decision-maker determine its We will also show that these solutions are general decision by maximizing its payoff function taking in nature and can be characterized, in terms of into account the choice of decisions by all other existence and uniqueness, with conditions that are decision-makers. While this theory has been very more intuitive and much less restrictive than those useful in addressing some aspects of quantitative of the traditional cardinal games. We will decision-making in engineering and economics, it illustrate these concepts with several examples of has not been able to adequately address qualitative deterministic matrix games, including an example problems in fields such as social and political of team composition and task assignment by a top sciences, as well as a large segment of complex commander in a military operation where payoff problems in engineering, business and functions are not readily available. We feel that management imbedded in a competitive this new theory of ordinal games will be very environment. The main reason for this is the useful when dealing with complex decision- inherent difficulty in defining an adequate payoff making problems that involve more than one function for each decision maker in these types of decision makers in a competitive environment. problems. 1. Introduction In this paper, we present a theory where, instead of a payoff function, the decision-makers are able Systems that involve more than one decision- to rank order their decisions against decision maker in competition are often optimized using choices by the other decision-makers. Such a rank the theory of games. This theory, initially developed by Von Neumann and Morgenstern [1], Clearly, the usefulness of game theory as in the and later by Nash [2], requires that the collective example described above is limited to problems decisions of the decision-makers be mapped where the payoff functions (costs) can be through payoff functions1 into a set of real expressed, and mathematically defined, for all numbers representing the values of these decisions decision-makers in the game. It is not difficult, to each of the decision-makers. In order to however, to imagine many real life decision- illustrate this concept, let us consider the simple making problems where the payoff functions matrix game illustrated in Figure 1. cannot be easily determined. For example, problems that most individuals face in buying a house, planning a vacation, or resolving a conflict, cannot be easily expressed in terms of a payoff (DM2) function. Even, in such fields as military decision-making applications where the theory of y games has traditionally received considerable y1 y2 3 attention, planning and managing an air operation $5,100 $8,000 $5,100 x in the presence of an intelligent adversary could 1 $3,200 $3,200 $2,900 be extremely difficult, if not impossible, to (DM1) x $6,000 $5,100 $1,900 2 $9,000 $8,300 $6,300 formulate using this theory of games. In these $8,000 $4,200 $1,100 problems, the main difficulty lies in the inability x 3 $7,300 $4,400 $7,300 to formulate appropriate payoff functions for all decision-makers involved. On the other hand, in Figure 1: A Simple Matrix Game the majority of such problems, instead of payoff functions, the decision-makers may have certain preferences that can be expressed easily as a rank DM1 (Decision-Maker 1) must choose from three ordering of the various options available to them. Furthermore, these preferences may be completely options in his decision space Xxxx= {, , } and 123 subjective in nature reflecting certain biases or DM2 has also three options Yyyy= {,123 , }to experiences that they may have. A new theory of choose from. For each pair of choices {,xijy }the games that deals with rank ordering of the options, entries P (,xy )and P (,xy ) in the matrix of rather than payoff functions as a basis for 1 ij 2 ij optimization has recently been developed by the Figure 1 represent costs associated with these authors [3]. For the sake of completeness, we will choices that each DM will incur. The objective of review below the aspects of this theory that are ∈ DM1 is to choose an option xi X to minimize relevant to business decision making. his costs C1, and the objective of DM2 is to ∈ choose an option yYj so as to minimize his As a simple example, to illustrate the main idea of rank ordering in games, consider the situation of costs C2. Obviously, in this example, both decision-makers are interdependent in that the two friends, John and Mary, planning a weekend costs incurred by each is determined not only by vacation together [3]. John likes to go the its own choice, but also by the choice made by the mountains and Mary prefers the beach. They both other DM. Hence, in making a choice each DM like to be together, if possible, but they have has to take into account the choice made by the strong preferences for the various options other DM. available to them. Let us assume that of the nine possible options, John ranks his preferences as illustrated in the table (or matrix) of Figure 2. 1 Other common terms used instead of payoff function include utility function, objective function, cost function, Mary may have completely different preferences. loss function, performance function, profit function, etc. A Let us suppose that hers are ranked as shown in best selection of decision will involve either maximizing or minimizing this function depending on its definition. Figure 3. Superimposing both preferences2 we to the mountains is John’s 1st preference but get the matrix game shown in Figure 4. Mary’s 7th preference, and both going to the beach is Mary’s 1st preference but John’s 4th preference; however, John staying at home while Mary going to the beach is his 2nd preference but her 6th and so Mary (P2) on. These are purely subjective preferences that may have no associated payoff or value. Mountain Beach Home Mountain 1 6 5 John Mary (P2) Beach 8 4 9 (P1) Mountain Beach Home Home 7 2 3 Mountain 1 , 7 6 , 8 5 , 9 John Figure 2: John’s Preferences Beach 8 , 2 4 , 1 9 , 3 (P1) Home 7 , 4 2 , 6 3 , 5 Mary (P2) Figure 4: John and Mary’s Ordinal Game Mountain Beach Home Mountain 7 8 9 In many ways, ordinal games can be viewed as John extensions of ordinal optimization problems in the Beach 2 1 3 (P1) same manner as cardinal games are extensions of cardinal optimization problems. Ordinal Home 4 6 5 optimization is a means of finding good, better, or best designs from a set of ordered options rather than using a formal cardinal process of calculating Figure 3: Mary’s Preferences the payoff cost or value of each option. It is a simple and yet very effective method of Note that in contrast to the game of Figure 1, the optimizing systems as demonstrated by Ho et al game of Figure 4 is formulated using preferential [4] in optimizing Discrete Event Dynamic rankings instead of payoff functions. The entries Systems (DEDS). in this game are positive integers representing the order (or ranking) of the various options for each The use of ordinal methods instead of payoff player. In order to differentiate such games from functions in decision-making problems is a the traditional, cardinal games, we will refer to concept that has received considerable attention in them as Ordinal Games [3]. Note that in the the past 20 years or so. The Analytic Hierarchy ordinal game of Figure 4, all nine options are rank Process (AHP) developed by Saaty in 1980 [5-6] ordered for each player. For example, both going is a very effective tool for optimizing complex 2 In this table, we use the number 1 to indicate 1st preference, multi-criteria decision-making problems. The 2 for 2nd preference and so on. Since there are 9 options, the AHP requires that the decision-maker consider number 9 will correspond to the 9th (last) preference.