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Top1. Introduction
Human judgments, including expressing preferences over a set of feasible outcomes or states in a conflict, are usually imprecise. Situations characterized by vagueness, impreciseness, incompleteness and ambiguity, are often reflected in a decision maker’s preferences. Whether the preferences are cardinal or ordinal, fuzziness in the preferences may influence the equilibria of the game under study (Kilgour, Fang, & Hipel, 1995).
In order to better comprehend, model and analyze conflicts, a number of methodologies have been proposed. Among, but not limited to, those methodologies, are the graph model for conflict resolution (Fang, Hipel, & Kilgour, 1993), conflict analysis (Fraser & Hipel, 1984), theory of moves (Brams, 1994), theory of fuzzy moves (Kandel & Zhang, 1998; Li, Karray, Hipel, & Kilgour, 2001), drama theory (Bennett, 1998; Howard, 1999, 1994), and metagame analysis (Howard, 1971). The common dominator among all of them is that they are based on game theoretic approaches. The recent history of game theory dates back to the year 1944 when von Neumann and Morgenstern published their seminal book entitled The Theory of Games and Economic Behaviour (Neumann & Morgenstern, 1953). John Nash (Nash, 1950, 1951) introduced the concept of a Nash Equilibrium into classical game theory in 1950 and 1951. In fact, game theory is a widely accepted procedure for addressing problems arising in engineering, economics, and politics (Bennett, 1995; Binmore, 1992; Brams, 1994; Fang, Hipel, & Kilgour, 1993; Osborne, 2003).
When modeling a game, it is assumed that one is cognizant of the decision makers or players, the options or courses of actions available to each, and their preferences over the possible feasible states that can take place. Among the well-known and widely applied solution concepts are Nash stability (Nash, 1950, 1951), general metarationality (Howard, 1971), symmetric metarationality (Howard, 1971), and sequential stability (Fraser & Hipel, 1984). Fang et al. (Fang, Hipel, & Kilgour, 1993) define these and other solution concepts within the framework of the graph model for conflict resolution.
For the cardinal case, the payoffs of the states are known in terms of numerical values, where the difference between any two numbers does have a meaning. On the other hand, for the ordinal situation, the exact cardinal payoffs are not known and the decision maker can only express his or her preferences in terms of an ordering of the states from most to least preferred, where ties are allowed. In both cases, preferences might involve some fuzziness about the values (cardinal) or about the ordering (ordinal). Yet, one needs to express his or her preferences among feasible states hoping for a better or more rewarding strategic result.
This research is an enhancement and expansion of the earlier work of Al-Mutairi et al. (Almutairi, Hipel, & Kamel, 2006, 2008). Following a general discussion of preferences in the next section, a fuzzy preference structure is developed for use with modified versions of the solution concepts of Nash and sequential stability within the paradigm of the graph model for conflict resolution. Subsequently, this new approach to uncertain preferences, along with associated solution concepts, are applied to the game of Prisoner’s Dilemma.