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Mathematical models as a tool for the social sciences
Mathematical Social Sciences 2 (1982) 421-425 North-Holland
421
Publishing Company
BOOK REVIEWS
Bruce 4. West, ed., Mathematiwl ModeLs QFa Tool for the So&d &mxs.
New
York: Gordon and Breach, 1980, 120 pp., US $26.50.
Presumably this book is intended as an advanced sourcebook in the mathematical social sciences. It arose from a setr.i;;~ SKImathematical modelling. It consists of 8 papers on models in diverse: areas, all quite interesting, and written in instructional rather than journal :tyle. The first paper is 2%r~rv!ey of the controversial subject cliometrics, or historical econometrics. The author argues that cliometrics is the most important trend in economics history, and discusses in particular the argument that contrary to earlier work, slavery was very profitable in the South. Next A.O. Dick gives a paper on serial memory which explains the fact that the hardest items on a list to remember are slightly past the middle of the list. The mathematics in these essays is quite simple. Next Julian Keilson gives an algorithm for contract acceptance. Opportunities for contract of various rewards Si and lenghts of time Ti occur as independent Poisson processes at rates li. The contractor chooses a set A of types he will accept, to maximize his long run profit. The author gives a simple algorithm for determining A to give this maximum. The mathematics here requires the Poisson distribution and inequalities. Aaron Budgor and the editor present an example of the statistics of extreme events, the maximum and minimum river heights for the Nile over 1300 years. The mathematics, stochastic differential equations, is quite advanced and important. A first model gives the Gumbel distribution. A second model, taking into account the tributaries of the Nile, is not fully worked out. J.M. Kemperson discusses those proportions of genetic types which can occur as equilibrium states. This involves linear algebra. William Riker reviews cooperative game theory and argues that minimal winning coalitions, which have the highest payoff per member, tend to emerge, and that because of this international organizations such as the U.N. will not be effective in eliminating international conflict. The mathematics is elementary. Next the editor presents a model of the stock market using stochastic differential equations. In the last essay Wade W. Badger presents a model for the size distribution of incomes. Two well-known models are the Pareto distribution Ax- ’ and the lognormal distribution. The former fails for low incomes and the latter for high incomes. The author adopts a different approach. He assumes that the entropy of the distribution of utilities is a maximum subject to a constraint Of constant 0165-4896/82/0000-0000/$02.75
0 1982 North-Holland
422
Book Reviews
variance, which gives a normal distribution of utilities. He then conjectures a C’ula for the utility of money u(x) =exp( -6/s). This gives a good fit with 0.58, b= 31. F. ut. Roush Mathematics Research Group Alabama State University Montgomery, AL 36101, U.S.A.
WeW Moulin, TakeS&~fw o/socibl Choti. Paris: Cahiers de Mathbmatiquesde n, Cenl‘re de Recherche de MathOmatiques de la DCcision, UniversitC I, P#*l%, 1981. ‘IBe author in the preface suggests this book as a first or second year graduate text for a course in social choice theory, public choice or game theory. It is also an excellentsummary of much recent work for the researcher. Although, as the author mentions, it is self-contained and uses only elementary mathematical techniques, it higYy mathematical. A number of quite interesting and somewhat difficult problems are provided. The graduate student will be taken to the frontiers of current researchby this book. There are two approaches to social choice theory, the very normative one ‘what is best for a group’, subject to rationality conditions, represented by the original tatanent of Arrow’s theorem and work of Harsanyi, Deschamps, Gevers, and others on utilities, and the more positive question of ‘what group choice methods can be implemented suitably’, represented by strategyproofness results. That is, suppose voters will misrepresenttheir true preferences to gain advantages, i.e., the real world. For what social choice methods will voters have unique natural strategies leading to social choice satisfying the norms of anonymity (individuals play e@valent roles), neutrality (alternatives play equivalent roles), and Pareto effi&ncy ithere is no other choice which everyone likes better)? This book is concerned ‘Iviththe latter questior, which represents most present work in the field. One simple answer to the above question is ‘none’, provided by Gibbard and Satterthwaite, in jthe case of dominant strategies. However, reflection shows that several possibilities might suffice to make a social choice function effectively immune to strategy: in particular domain restrictions and subtler solution concepts in game theory than the dominal,ed strategy equilibrium, in which each player’s strategy is best regardless of the actions of the others. E. Maskin in particular d-e&@ the latter idea of a social choice correspondence which could be realized for *me game using a specifiedgame-theory solution concept. This is called implelion and it is, generally speaking, the main topic of this book. The book also up d:Jmain restriction (not many positive results have been achieved in and in mrticular proves the result that if a domain permits a neutral,
421
Publishing Company
BOOK REVIEWS
Bruce 4. West, ed., Mathematiwl ModeLs QFa Tool for the So&d &mxs.
New
York: Gordon and Breach, 1980, 120 pp., US $26.50.
Presumably this book is intended as an advanced sourcebook in the mathematical social sciences. It arose from a setr.i;;~ SKImathematical modelling. It consists of 8 papers on models in diverse: areas, all quite interesting, and written in instructional rather than journal :tyle. The first paper is 2%r~rv!ey of the controversial subject cliometrics, or historical econometrics. The author argues that cliometrics is the most important trend in economics history, and discusses in particular the argument that contrary to earlier work, slavery was very profitable in the South. Next A.O. Dick gives a paper on serial memory which explains the fact that the hardest items on a list to remember are slightly past the middle of the list. The mathematics in these essays is quite simple. Next Julian Keilson gives an algorithm for contract acceptance. Opportunities for contract of various rewards Si and lenghts of time Ti occur as independent Poisson processes at rates li. The contractor chooses a set A of types he will accept, to maximize his long run profit. The author gives a simple algorithm for determining A to give this maximum. The mathematics here requires the Poisson distribution and inequalities. Aaron Budgor and the editor present an example of the statistics of extreme events, the maximum and minimum river heights for the Nile over 1300 years. The mathematics, stochastic differential equations, is quite advanced and important. A first model gives the Gumbel distribution. A second model, taking into account the tributaries of the Nile, is not fully worked out. J.M. Kemperson discusses those proportions of genetic types which can occur as equilibrium states. This involves linear algebra. William Riker reviews cooperative game theory and argues that minimal winning coalitions, which have the highest payoff per member, tend to emerge, and that because of this international organizations such as the U.N. will not be effective in eliminating international conflict. The mathematics is elementary. Next the editor presents a model of the stock market using stochastic differential equations. In the last essay Wade W. Badger presents a model for the size distribution of incomes. Two well-known models are the Pareto distribution Ax- ’ and the lognormal distribution. The former fails for low incomes and the latter for high incomes. The author adopts a different approach. He assumes that the entropy of the distribution of utilities is a maximum subject to a constraint Of constant 0165-4896/82/0000-0000/$02.75
0 1982 North-Holland
422
Book Reviews
variance, which gives a normal distribution of utilities. He then conjectures a C’ula for the utility of money u(x) =exp( -6/s). This gives a good fit with 0.58, b= 31. F. ut. Roush Mathematics Research Group Alabama State University Montgomery, AL 36101, U.S.A.
WeW Moulin, TakeS&~fw o/socibl Choti. Paris: Cahiers de Mathbmatiquesde n, Cenl‘re de Recherche de MathOmatiques de la DCcision, UniversitC I, P#*l%, 1981. ‘IBe author in the preface suggests this book as a first or second year graduate text for a course in social choice theory, public choice or game theory. It is also an excellentsummary of much recent work for the researcher. Although, as the author mentions, it is self-contained and uses only elementary mathematical techniques, it higYy mathematical. A number of quite interesting and somewhat difficult problems are provided. The graduate student will be taken to the frontiers of current researchby this book. There are two approaches to social choice theory, the very normative one ‘what is best for a group’, subject to rationality conditions, represented by the original tatanent of Arrow’s theorem and work of Harsanyi, Deschamps, Gevers, and others on utilities, and the more positive question of ‘what group choice methods can be implemented suitably’, represented by strategyproofness results. That is, suppose voters will misrepresenttheir true preferences to gain advantages, i.e., the real world. For what social choice methods will voters have unique natural strategies leading to social choice satisfying the norms of anonymity (individuals play e@valent roles), neutrality (alternatives play equivalent roles), and Pareto effi&ncy ithere is no other choice which everyone likes better)? This book is concerned ‘Iviththe latter questior, which represents most present work in the field. One simple answer to the above question is ‘none’, provided by Gibbard and Satterthwaite, in jthe case of dominant strategies. However, reflection shows that several possibilities might suffice to make a social choice function effectively immune to strategy: in particular domain restrictions and subtler solution concepts in game theory than the dominal,ed strategy equilibrium, in which each player’s strategy is best regardless of the actions of the others. E. Maskin in particular d-e&@ the latter idea of a social choice correspondence which could be realized for *me game using a specifiedgame-theory solution concept. This is called implelion and it is, generally speaking, the main topic of this book. The book also up d:Jmain restriction (not many positive results have been achieved in and in mrticular proves the result that if a domain permits a neutral,