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**Fuzzy and multiobjective games for conflict resolution.**
*(English)*
Zbl 0973.91001

Studies in Fuzziness and Soft Computing. 64. Heidelberg: Physica-Verlag. xiii, 258 p. DM 128.29; sFr 110.75; £44.00; $ 78.00 (2001).

In some real-life situations, the outcome \(f(x)\) of a person’s action \(x\) depends only on this action; in such situations, we must select an action \(x\) for which the outcome \(f(x)\) is the most preferable. These situations are naturally formalized as optimization problems.

In most real-life situations, however, the outcome of a person’s action depends not only on this action, but also on the actions of other people whose preferences may be different. In such conflict situations, we must help each of the participants \(i\) select an action \(x_i\) which is – in some sense – reasonable. Techniques for resolving such conflict situations are known as game theory.

Traditionally, in game theory, it is assumed that we know the exact consequence \(f(x)\) of each action selection \(x=(x_1,\ldots,x_n)\), and that each participant has a clear preference between different consequences. In real life, we often have only a vague (“fuzzy”) prediction about the consequences of each action selection \(x\), and only a fuzzy preference between different consequences. The fuzzy character of a preference is often described in the following multicriteria terms: for each \(x\), we know the resulting characteristics \(f_1(x),\ldots,f_k(x)\), but we are not sure how to compare the \(k\)-dimensional vectors \((f_1(x),\ldots,f_k(x))\) and \((f_1(x'),\ldots,f_k(x'))\) corresponding to different action selections \(x\) and \(x'\).

Both uncertainties – fuzzy and multicriteria – can be naturally described in terms of fuzzy logic. The book under review extends all three major types of games – zero-sum two-person games, general two-person games, and cooperative \(n\)-person games – to such fuzzy and multicriteria situations. The authors overview different formalizations of thus generalized games, present theorems about these generalized games, and present algorithms for solving these games. A large portion of the results comes from the authors themselves, including some results that has never been published before.

Overall, algorithm-wise, the picture is very reassuring. For example, for a zero-sum game, for which, in crisp case, linear programming is the main tool, this same tool also works in important fuzzy cases – e.g., when the membership functions are piece-wise linear. Similarly, fuzzy non-zero-sum two-person games can be resolved by repeatedly using quadratic programming – a technique that is appropriate for crisp games of this type.

For cooperative games, an additional complexity comes from the fact that, in additional to crisp coalitions, it is natural to consider fuzzy ones. In spite of this complexity, it turns out to be possible to reduce the problem to several almost crisp (interval-valued) ones, with intervals corresponding to \(\alpha\)-cuts of the corresponding fuzzy numbers. In all three cases, reductions are far from trivial, often requiring mathematical ingenuity.

An interesting open question is how efficient are the authors’ algorithms. In a somewhat old-fashioned approach to numerical algorithms, the authors stop after proving that an algorithm converges without providing any estimates for the number of computational steps. The authors’ examples seem to indicate that this number of steps is not large, but a detailed analysis is definitely in order.

In most real-life situations, however, the outcome of a person’s action depends not only on this action, but also on the actions of other people whose preferences may be different. In such conflict situations, we must help each of the participants \(i\) select an action \(x_i\) which is – in some sense – reasonable. Techniques for resolving such conflict situations are known as game theory.

Traditionally, in game theory, it is assumed that we know the exact consequence \(f(x)\) of each action selection \(x=(x_1,\ldots,x_n)\), and that each participant has a clear preference between different consequences. In real life, we often have only a vague (“fuzzy”) prediction about the consequences of each action selection \(x\), and only a fuzzy preference between different consequences. The fuzzy character of a preference is often described in the following multicriteria terms: for each \(x\), we know the resulting characteristics \(f_1(x),\ldots,f_k(x)\), but we are not sure how to compare the \(k\)-dimensional vectors \((f_1(x),\ldots,f_k(x))\) and \((f_1(x'),\ldots,f_k(x'))\) corresponding to different action selections \(x\) and \(x'\).

Both uncertainties – fuzzy and multicriteria – can be naturally described in terms of fuzzy logic. The book under review extends all three major types of games – zero-sum two-person games, general two-person games, and cooperative \(n\)-person games – to such fuzzy and multicriteria situations. The authors overview different formalizations of thus generalized games, present theorems about these generalized games, and present algorithms for solving these games. A large portion of the results comes from the authors themselves, including some results that has never been published before.

Overall, algorithm-wise, the picture is very reassuring. For example, for a zero-sum game, for which, in crisp case, linear programming is the main tool, this same tool also works in important fuzzy cases – e.g., when the membership functions are piece-wise linear. Similarly, fuzzy non-zero-sum two-person games can be resolved by repeatedly using quadratic programming – a technique that is appropriate for crisp games of this type.

For cooperative games, an additional complexity comes from the fact that, in additional to crisp coalitions, it is natural to consider fuzzy ones. In spite of this complexity, it turns out to be possible to reduce the problem to several almost crisp (interval-valued) ones, with intervals corresponding to \(\alpha\)-cuts of the corresponding fuzzy numbers. In all three cases, reductions are far from trivial, often requiring mathematical ingenuity.

An interesting open question is how efficient are the authors’ algorithms. In a somewhat old-fashioned approach to numerical algorithms, the authors stop after proving that an algorithm converges without providing any estimates for the number of computational steps. The authors’ examples seem to indicate that this number of steps is not large, but a detailed analysis is definitely in order.

Reviewer: O.M.Kosheleva (El Paso)