Fuzzy Optimal Approaches to 2-P Cooperative Games

Fuzzy Optimal Approaches to 2-P Cooperative Games

Mubarak S. Al-Mutairi (Hafr Al-Batin Community College, King Fahd University of Petroleum and Minerals (KFUPM), Hafr Al-Batin, Saudi Arabia)
Copyright: © 2016 |Pages: 14
DOI: 10.4018/IJAIE.2016070102


In game theory, two or more parties need to make decisions with fully or partially conflicting objectives. In situations where reaching a more favourable outcome depends upon cooperation between the two conflicting parties, some of the mental and subjective attitudes of the decision makers must be considered. While the decision to cooperate with others bears some risks due to uncertainty and loss of control, not cooperating means giving up potential benefits. In practice, decisions must be made under risk, uncertainty, and incomplete or fuzzy information. Because it is able to work well with vague, ambiguous, imprecise, noisy or missing information, the fuzzy approach is effective for modeling such multicriteria conflicting situations. The well-known game of Prisoner's Dilemma, which reflects a basic situation in which one must decide whether to cooperate or not with a competitor, is systematically solved using a fuzzy approach. The fuzzy procedure is used to incorporate some of the subjective attitudes of the decision makers that are difficult to model using classical game theory. Furthermore, it permits researchers to consider the subjective attitudes of the decision makers and make better decisions in subjective, uncertain, and risky situations.
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1. Introduction

Game theory is the most accepted and applicable procedure for problems arising in engineering, economics, and politics (Binmore, 1992; Brams, 1994; Osborne, 1994; Fang, Hipel, & Kilgour, 1993; & Fraser, & Hipel, 1994). The history of game theory dates back to the year 1944 when von Neumann and Morgenstern published their book The Theory of Games and Economic Behaviour (Neumann, 1944). John Nash (Nash, 1951; & Binmore, 1992), introduced the concept of Nash Equilibrium in classical game theory.

When a group of people or agents (referred to them as players in game theory) interact with each other, they are mostly faced with conflicting objectives. While they are trying to maximize their profit or minimize their costs, it might be in their interest to help others achieve their goals. This could as a result of expecting to be reciprocated in future interaction or to achieve a better situation that seems unachievable without the others’ help.

Cooperation is the situation where people (or agents) work together to achieve a goal that without working together wouldn’t be achieved or achieved but with less rewards. When engaging in a cooperative action, the chances of success are dependent on the actions and the number of partners involved. This increases the chances for risks and uncertainties. Thus, it could be claimed that trust is a necessary condition for cooperation and that it is a product of a successful cooperative action (Sztompka, 1999). Trust is a good way of motivating cooperative actions but might not be sufficient alone. Besides trust there should be some sort of common goals, shared values, or even a kind of reciprocation. The role of trust in the case of cooperation is mainly in the elimination of fear of being betrayed or not being reciprocated (Lubell, 2001). The two main conditions to be met beside trust to facilitate cooperative actions are:

  • 1.

    Having a common goal or sharing some values.

  • 2.

    Expecting the others to cooperate.

If any of the above two conditions is not met, the chances for cooperation are relatively low. In general, we could say that cooperation happens when there is a non-mutually exclusive goal in which everyone wants to reach a win-win situation. On the other hand, the presence of distrust could eliminate any chances for cooperation. (Gambetta, 1988) in page 219 stated that “if distrust is complete, cooperation will fail among free agents”.

Trying to maximize his or her expected payoff, he or she investigates the possibility of cooperating or joining a coalition that promises the best individual expectations. In such highly subjective situations, nothing is guaranteed. It is subjected to many unpredictable and hard to evaluate factors. Not being able to predict the commitment of the game players, the surrounding environment influences, and the game roles, it is unlikely to be able to predict the outcomes. Such uncertainty, imprecision, vagueness is best modeled through the use of fuzzy logic.

Fuzzy sets theory (Zadeh, 1973; Mares, 2001; & Mohanty, 1994) is good when dealing with vague, imprecise, noisy or missing information. Instead of using the Boolean {0,1} values, fuzzy sets tries to map the degree an element belongs to a certain group in a continuous scale between [0,1].

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