Hilborn, R.

International Institute for Applied Systems Analysis (IIASA)

A Formal Method for the Construction of Cross-Impact MatricesHilborn, R.IIASA Working PaperWP-74-0641974 Hilborn, R. (1974) A Formal Method for the Construction of Cross-Impact Matrices. IIASA Working Paper. WP-74-064 Copyright © 1974 by the author(s). http://pure.iiasa.ac.at/107/ Working Papers on work of the International Institute for Applied Systems Analysis receive only limited review. Views or opinions expressed herein do not necessarily represent those of the Institute, its National Member Organizations, or other organizations supporting the work. All rights reserved. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage. All copies must bear this notice and the full citation on the first page. For other purposes, to republish, to post on servers or to redistribute to lists, permission must be sought by contacting repository@iiasa.ac.at A FORMAL METHOD FOR THE CONSTRUCTION OFCROSS IMPACT MATRICESRay HilbornNovember 1974 WP-74-64Working Papers are not intended fordistribution outside of IIASA, andare solely for discussion and infor-mation purposes. The views expressedare those of the author, and do notnecessarily reflect those of IIASA.A ｆｏｩｬｬｾｾｌ ｾｬｬｩｔｈｏｄ FOR THE CONSTRUCTIONOF CROSS IMPACT MATRICESRay· HilbornAs resource scientists explore a wider and wider rangeof techniques to test the consequences of human activity, oneｴ ･ ｣ ｨ ｮ ｩ ｾ ｵ ･ frequently used is cross impact analysis. Thisinvolves the construction of a matrix listing all of the im-ｰ ｯ ｾ ｴ ｡ ｮ ｴ variables in the system under consideration as bothrows and columns. You then designate that one axis, for instancethe rows, represents the "impacts upon" list, and the otheraxis represents the "impacts of" list. Proceeding down therows and collL"1lns you repeatedly ask the question "how doesthis variable affect the other?" This question can beanswered in a variety of ways; either by yes or no, or posi-tively, negatively, or not at all, or some indication of themagnitude of the effect may be expressed depending upon theuse to which the cross impact matrix will be put. We fre-quently use these matrices as the first step to buildingsiflulation models of ecological system. The matrix is usedto isolate the relationships between variables which willhave to be defined in the model. Sometimes the matrix is anend in itself; it is used as the data base for a qualitativesimulation or assessment procedure which can be used to pre-dict trends or impacts of manipulation of the variables.-2-(Kane, 1972; Gallopin, 1974). In these cases any errors informulating the cross impact matrix will have effects upon thefinal conclusion. The rest of this paper will be devoted tothese types of situations ..We have found from actual experimentation that formulatingcross ｩ Ｚ ｾ ｰ ｡ ｣ ｴ matrices is extremely qualitative and the sameperson will rarely produce the same matrix on two separateoccasions. Although this is not surprising when consideringsystems which are poorly understood, it nevertheless seems tohappen just as frequently when dealing with systems theformulator knows well. In a recent experiment one researcherwho had built several simulation models of a lO-variablesystem, produced two very different cross impact matrices onthe same day. I believe that this is a product of the mental?rocedure used to formulate the cross impact matrix, and Iwill propose a mental procedure which should eliminate thisproblem. The usual mental method used when_asking how doesA affect B, is to ask what effect will there be on B if Ais increased slightly (or decreased slightly) -- basically amental partial differentiation. The problem stems primarilyfroD situations in which A affects B which affects C. If youask what effect will a slight increase in A have on C, youmust decide if you will say yes because B will go up whichcauses C to go up, or if you will say no because the actionis mediated by B. There is general agreement that you shouldtake the second choice, "no," but experience has shown thatpeople do not act in this fashion while formulating their matrices.-3-I propose that we should redesign our mental method for askingthe questions by formulating our knowledge as a simulation ofthe system and then looking at the equations formulated andsimply copying onto the cross impact matrix the relationshipsused in the model. Cross impact analysis is generally usedfor situations in which there is insufficient knowledge toformulate a working simulation model. The functional rela-tionshi?s are not known, only the general directions andintensities of these relationships. This is not a barrier tothe formulation of the structure of the simulation. You aresimply admitting that you can't guess the parameter values verywell. I believe that you can almost always actually formulatethe equati?ns to the extent required to pullout the neededinformation for a cross impact matrix. Some people may usethis method when they formulate their matrix -- but fromactual experience I think such cases are rare indeed. From aplus-minus matrix you can state the simulation functions. Thefact that these differ between sessions for the same individualsuggests that either the system is poorly understood or theproper method was not used.Xay (1973) has discussed the use of plus-minus matrices infood web modelling and assumes that the formulation followsthe Lotka-Volterra community matrix. This eliminates anyambiguities because the question to be asked when filling ina location in the matrix is very explicit. In this case itis, "if X eats Y, then the effect of X on Y is minus; ifYeats X, then it is plus; and if they compete for limited-4-resources, the effect of X on Y is minus." ne is rarelyfortunately enough in environmental systems to have such asimple structure. We hope that the basic structural elementsof the system will emerge by formulating the system as asimulation model.There are two additional advantages.of the procedure. Theprocess of trying to formulate the system as a simulationmodel is quite useful in defining new state variables whichreduce the complexity of the formulation, and it provides thebasic starting format for actually building a simulation ifthe information required about the system is available. Thestudy of cross impact analysis is in its infancy and one couldargue that it is most useful in situations where the informa-tion about the system is so limited that it is impossibleeven to formulate the basic structure of the simulation.If this is true, I believe that the value of the analysiswould. lie solely in the discussion generated while constructingthe matrix and not in the final matrix produced.I have carefully avoided discussing the techniques and meritsof cross impact analysis in general. It is certainlyquestionable whether even the most sophisticated algorithmsfor prediction from cross impact matrices are useful at all.If these matrices are ever to be of use, I believe we mustproduce some formal methods for generating them.-5-ReferencesGallopin, G., 1974. Reference unknown at present.Kane, J., 1971. Reference unknown at present.May, R.M., 1973.ecosystems.Stability and complexity in modelPrinceton Univ. Press. 235pp.

A Formal Method for the Construction of Cross-Impact Matrices

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