¹ Simon, H. A. and Ando, A., “Aggregation of Variables in Dynamic Systems,” Econometrica, Vol. 29 (April, 1961), pp. 111–138CrossRefGoogle Scholar and A. Ando and F. M. Fisher, “Near-Decomposability, Partition and Aggregation, and the Relevance of Stability Discussions,” in detail (forthcoming) present and prove the theorems. The relation of the sort of system involved to ordinary notions of causation is discussed in Simon, H. A., “Causal Ordering and Identifiability,” ch. 3 in Studies in Econometric Method (Hood, W. C. and Koopmans, T. C., eds., New York, 1953Google Scholar; Cowles Commission Monograph No. 14), reprinted as ch. 1 in Simon, H. A., Models of Man (New York, 1957)Google Scholar; while the bearing of this sort of problem on the estimation of parameters in one of a set of interrelated systems is covered in Fisher, F. M., “On the Cost of Approximate Specification in Simultaneous Equation Estimation.” Econometrica, Vol. 29 (April 1961), pp. 139–170CrossRefGoogle Scholar.

² If values in the further past are relevant, a formal redefinition can always be made to eliminate time before t − 1, without changing anything. Simultaneous dependencies can also be treated, though, for reasons of clarity and intuitive appeal, we restrict ourselves to cases where there is some time lag somewhere in the system. Further, we could consider time as continuous without essential changes. Strictly speaking, however, the theorems under discussion are only known to hold for cases in which the relations involved are linear. Since this was written, the principal results of the theorems have been shown to hold for at least one important class of nonlinear systems in F. M. Fisher, “Decomposability, and Balanced Growth Under Constant Returns to Scale” (forthcoming).

³ Hereafter, to simplify terminology, by the term “set” instead of “subset,” we mean a subset within the set of all variables in the system under consideration.

⁴ Simon and Ando, op. cit.

⁵ Here and later what is meant by “sufficiently weak” depends on what standard of approximation one wishes to impose on the results. The more closely one insists that the behavior of the system must approximate that of the corresponding completely decomposable one, the weaker must “sufficiently weak” be. What the theorem guarantees is that whatever standard of approximation is required (so long as it is an approximate and not an exact standard) a non-zero degree of weakness always exists which is sufficient to produce results satisfying that standard.

⁶ Even before the full system settles down to its ultimate behavior, variables within any one set will move proportionally, so that inter-set influences can be analyzed as influences among indices, each index representing all variables in a particular subset of the system, rather than as among the individual variables themselves. This point, while a little aside from the main drift of our discussion in the text, is useful and important and will show up in our examples below. It is true of both theorems.

⁷ This phrase is not used accidentally. The influence of the outside roles will eventually be on group behavior as a whole rather than on individual behavior within the group (remember that such influences are small relative to withingroup forces). See the preceding footnote.

⁸ Indeed, it would be odd (although not impossible) for both statements to hold exactly.

⁹ Ando and Fisher, op. cit.

¹⁰ This is not to deny, of course, the usefulness of inter-disciplinary work. The point is that the results of intra-disciplinary studies need not be vitiated because the real world is not so neatly divided as the academic one.

¹¹ Again, the use of past time is not a restriction; the past date can be last year or yesterday or an hour ago. Similarly, influences in the further past can be formally subsumed under this model by redefinition. Finally, the formulation includes the case in which it is the rate of change rather than the level of the armament stocks which are dependent on past armaments. This sort of treatment of arms races originates with L. F. Richardson. See his Arms and Insecurity (Chicago, Quadrangle Books, 1960)Google Scholar. Of course the stocks of armaments depend on other things such as economic resources; we are trying to keep the example as simple as possible. The functions in question are supposed to represent the strategic choices made by governments.

¹² They need not be functions of armament stocks in all lower-numbered races; one will do. Similarly, not every country in the given arms race need have such links; all that is required is that at least one does, so that the system, while nearly decomposable, is not nearly completely decomposable. It is also possible to analyze a case in which a part of a nearly decomposable system happens to be nearly completely decomposable.

¹³ A discussion of this general sort of model which aggregates from the individual level is given in Anderson, T. W., “Probability Models for Analyzing Time Changes in Attitudes,” Chapter 1 in Mathematical Thinking in the Social Sciences, Lazarsfeld, P. F., ed. (Glencoe, 1954)Google Scholar. It would be possible to abandon the probability interpretation and to let higher strength in one party lead to lower strength in another, but this would lead to an example without some of the nice features of the present one. It would also be possible to build in other influences on party voting strength. Again, we are striving for an illuminating example rather than for a full-blown “realistic” theory. The assumption that every individual votes for some party is innocuous, since we can count all those not voting for any formal party as voting for a party of their own, the “Non-such” Party.

¹⁴ Technical footnote. Precisely, cohesiveness is given by the largest characteristic root of the submatrix of A's corresponding to the parties in the set. See Fisher, F. M., “An Alternate Proof and Extension of Solow's Theorem on Nonnegative Square Matrices,” Econometrica, Vol. 30 (July, 1962)CrossRefGoogle Scholar forthcoming. The fact that the root lies between the largest and smallest column sums is well known—see Solow, R. M., “On the Structure of Linear Models,” Econometrica, Vol. 21 (01, 1952), pp. 29–46CrossRefGoogle Scholar.

¹⁵ We are perhaps making thia result sound a bit more paradoxical than it is. Different tendencies to lose voters to major parties do in fact matter and get taken into account in an implicit fashion. Since every party's voters end up some where (the sum of the A's in any column is one), a minor party with a high propensity to lose voters to major ones will have—other things being equal— a lower propensity to retain voters than a minor party with a lower propensity to lose voters to major ones. This will show up in considering their relative strength in isolation; however, it can be nullified by other tendencies if other things are not equal.