Grey Game Theory and Its Applications in Economic Decision‐Making

Citation

Andrew, A.M. (2010), "Grey Game Theory and Its Applications in Economic Decision‐Making", Kybernetes, Vol. 39 No. 7, pp. 1205-1207. https://doi.org/10.1108/03684921011062782

The need to accept and deal with uncertainty has been epitomized by Robert Vallée by observing that we live in a grey world. Uncertainty can arise due to imprecision of measurement and/or memory and/or transmission, and more seriously as the result of the “small causes, large effects” feature of natural systems, with the weather a supreme example. We also know from the work of Heisenberg and Schrödinger that physical phenomena are unpredictable at a fundamental level. The various means of dealing with uncertainty have been the subject of much recent online discussion, with Chinese complexity studies, particularly those under the heading of grey systems, receiving particular attention. This book presents impressive mathematical theory, with references to applications, but does not try to demonstrate the merits of its approach relative to other means of acknowledging and coping with uncertainty.

Also in a number of other ways, the treatment in the book leaves something to be desired. One is that it uses special mathematical formalisms without first explaining them. This presents no problem if the introductory article of Lin et al. (2004) is at hand, and that article also has discussion of the value of the “grey” approach, at least to the extent of comparing it with the alternative offered by fuzzy set theory.

The approach uses “interval numbers”, which are also a feature of other initiatives under the heading of “interval arithmetic” or “interval analysis”. Of course, a certain amount of overlap with other work does not deny the value of the approach since the basic ideas can be developed in widely different ways. The idea of a “confidence interval” is also part of standard statistical theory, based on standard probability theory, but in this case the interval boundaries are produced as an output, rather than an input, of analysis that takes account of sample sizes and variances and a significance level.

There is widespread agreement, as discussed by Klir (1989) and in many later publications, that uncertainty is usefully treated in ways other than those of standard probability theory. However, I find the comparison of methods by Lin et al. (2004) unsatisfactory. They give as an example of a situation lending itself to “grey” analysis a prediction by the Chinese Government that at a certain date the population of the country would be between certain stated limits. The theory must be of limited value if it is restricted to situations that are as tailor‐made as this for interval analysis, with apparently no doubt that the outcome would conform. Most readers will feel more familiar with government forecasts regarded as goals that might or might not be met, and that might not have been sincerely meant when first advanced. The impression these authors give of fuzzy methodology is also misleading since it is presented as rather vaguely trying to represent human impressions, whereas in fact applications of fuzzy theory involve construction of precise envelopes of degrees of set membership.

The comparison of alternative analysis strategies is obviously a complicated issue, and a final choice must be on a “proof of the pudding” basis, depending on what has been found to work and on what is mathematically tractable. The authors of the present book assure readers that the “grey” methodology has proved to be highly effective over 20 years of development, which certainly makes it deserving of attention, even though fuzzy methodology (Zadeh, 1965) goes farther back. It is impossible not to feel, though, that a method that treats all values within an interval equally (as they would be in a fuzzy scheme restricted to rectangular membership functions) cannot be using available information to the full in all situations. It might be argued that values are not treated equally when some rule for “whitenisation” is applied to a grey result, but little is said about whitenisation in the new book.

Some of the choices of English‐language terminology in the book do not seem ideal, one example being “expanding matrix” to refer to what would be indicated better by “expanded matrix”. The terms “overrated” and “underrated” risk are also used in a way that I did not find to be intuitive.

Quite a lot of attention is given to the “Centipede Game” with treatment that is claimed to resolve its paradox by splitting it into sub‐games. The game is not described in detail and readers are referred to a paper by Rosenthal from 1981. The paper, however, does not refer to the game by that name, which seems to have arisen in later discussions (apparently given just because a large number of repetitions of the game move are required, arbitrarily set at 100). A full and useful treatment can be found in Wikipedia. The game has the property of the “Prisoners' dilemma” situation, in that there is a collaborative strategy that brings a large reward, but the reward is small when each player behaves selfishly. There is a paradoxical discrepancy between what is computed as optimal when looking back from some way into the game and what appears optimal at the start.

The critical comments made above, some of them referring to matters purely of presentation, should not obscure the fact that there is important material here. The use of interval numbers is nicely reconciled with matrix algebra including inversion, and thereby with linear programming and games theory following the lead of von Neumann and Morgenstern, treated more digestibly in many works such as that of Vajda (1960). It may be noted that application of fuzzy theory to linear programming was discussed by Zimmermann (1976) though probably with little overlap with the present study.

The most challenging problems of games theory arise when there is no clear rational policy that determines the opponent's moves. The opponent may act irrationally or may switch unpredictably between alternative strategies. Biological evolution was studied by Maynard‐Smith as a game against the incompletely‐known opponent nature, and the new book not only offers a fresh perspective on this, but also extends the principles to examination of the development of business agglomerations by mechanisms involving learning. The “duopoly” situation with two players competing for a market is treated with reference to a model proposed by the French Mathematician Cournot as early as 1838. A sealed‐bid auction situation is also treated.

All the topics are treated in a fresh and stimulating way and readers are assured of their effectiveness in practice.