DRAFT: Summer, 2001

TYPE OF ARTICLE: Research Paper, mobile-1.mws

WORD COUNT: ~4000

Michael A. Faia,
College of William & Mary

Address for correspondence:
Michael A. Faia,
327 Richmond Rd.,
Box 8795,
College of William & Mary,
Williamsburg, VA 23187-8795;
mafaia@wm.edu,
(757) 221-2593

Abstract
 

``Frictions of space," although hard to measure, impinge on many forms of mobility in society. George K. Zipf's principle of least effort, emphasizing cost minimization, is a fruitful way to conceptualize the idea that high mobility costs tend to be avoided. Zipf's work parallels many elements of rational choice theory (RCT) and linear programming (LP). Further exploration of least-effort, least-cost, and LP principles leads to the claim that LP inquiries into cost minimization have implications for chaos theory.

We explore a migration process with two unknowns, facilitating geometric illustrations. Chaos events involve potential migratory streams that remain empty, along with abrupt shifts of corner solutions.Post-optimality experiments give further evidence of such events. In a model involving school enrollments, we show that apparently minor policy constraints, construed as emergent social norms, may create large shifts of optimal LP solutions.

Finally, we develop an LP model of social mobility containing utilities to be maximized. This analysis---although more experimentation is needed---leads to an illustration in which a mobility pattern with realistic utilities tends to eliminate nearly all ``stayers."

Introduction: Tautologies and falsifiable assertions
 

To attack such a formidable defense ``all along the line," without detailed reconnaissance, without specific objectives, and without a follow-up plan, was an act of colossal misjudgment. By the time Grant called off the attack, that was clear to every general and every private. They understood what had happened, and after belatedly inspecting the field, the generals understood how. Now, some in outrage, some in puzzlement, some in shame, they tried to understand why. But to soldiers deepening their trenches and clutching the earth between the lines, the first question was whether they would survive.

The killing was far from over.

---Furgurson (2000:168)

Zey (1998) provides a perceptive analysis of rational choice theory (RCT) and its prospects---not strong, in her view---of subsuming organizational theory. She defines ``rational actions of rational individuals" as having several indispensable dimensions (Zey, 1998, pp. 2-3): First, such behavior distributes scarce resources (such as money, power, time, energy, and information) that have a measurable value. Second, such behavior forces decisionmakers to suffer opportunity costs, i.e., valuable options foregone in favor of those chosen. Third, such behavior responds to social norms that attempt to manipulate rewards and punishments associated with particular options. Fourth, rational behavior embodies a serious effort to obtain and process accurate information about options available, about utilities associated with each option, and about the probability that a given utility will materialize [note 1]. Fifth (Zey, 1998, p. 34),
 

Consistency [ordinality] demands that it must be possible for all of the decision maker's options to be rank ordered. ... RCT does not require that numerical values be attached to preferences ... It only assumes the possibility of rank-ordered preferences.

Sixth, ``... preference orderings are transitive. If A is preferred to B, and B is preferred to C, then consistency requires that A is preferred to C ..." Finally, individuals ``maximize the expected value of their payoff measured on some utility scale ..."

In summary, Zey identifies at least seven ideal-type dimensions of rational behavior (cf. Abell, 1992, pp. 188-204; Green and Shapiro, 1994, pp. 14-17; Ghosh, 1995; Rambo, 1995):
 

(a) Rational behavior deals with scarce resources;

(b) it assesses opportunity costs;

(c) it responds to social norms;

(d) it is information intensive;

(e) it orders all available options;

(f) it assumes transitivity among options;

(g) it maximizes measurable payoffs.

Zey argues consistently (1998, pp. 42-50,70) that a serious incompatibility arises between RCT and organizational theory because of RCT's emphasis on developing syntactical/deductive models with limited empirical support; organizational theory, by contrast, seeks empirical evidence of conformity to and departures from rational behavior, but often lacks an adequate operational definition of rationality. Presumably, any attempt to integrate the two bodies of theory must address the issue of falsifiability through empirical testing (Smelser, 1992).

As an example, Zey (1998, pp. 76-77) cites empirical work suggesting that modern bureaucratic organization may fall far short of Weberian rationality. In a second example, Green and Shapiro (1994, Chapter 4) argue cogently that RCT does not provide adequate explanations of basic political behavior, such as the willingness of many, perhaps most, citizens to go to the polls and cast votes. As a third example, we cite a classic form of the prisoner's dilemma game, in which prisoners with limited information (and limited mutual trust), deciding ``rationally" whether to confess a crime, consistently create ``Nash equilibria" rather than the more desirable, trust-based ``Pareto equilibria" (Kreps, 1990, p. 29; Green and Shapiro, 1994, p. 25). In this latter instance one sees what is arguably a tautological relationship between items (d) and (g) above, in that the existence of limited information explains entirely the acceptance of a less than maximal result. Each of these illustrations implies that RCT has a strong tendency to focus on the syntactical/deductive interrelationships among the several dimensions of rationality, while neglecting departures from these ideal-type behaviors other than those implicit in the syntactical/deductive model itself.

Presumably, organizational theory deftly handles what appear to be many empirical instances of ``irrationality," while discerning little need to invoke the elaborate syntactical/deductive models of RCT. To paraphrase Dawkins (1986, p. 9), ``there are many more ways of being irrational than of being rational" [note 2]. Both sides of this quandary may suffer serious opportunity costs: RCT perhaps remains underdeveloped in its ability to theorize about departures from rationality, and organizational theory presumably remains deficient in conceptualizing rationality models from which individuals and organizations may deviate. In any case, several critics have made huge exertions in support of Zey's first argument, i.e., that RCT does not make a serious effort to develop a strong empirical grounding. Zafirovski (1999, p. 77; cf. Goode, 1997), for instance, has compiled an elaborate analysis of the RCT literature, largely in order to show that
 

In addition to generally minimizing non-utilitarian or non-economic goals, rational choice theory fails to envision that social actors can pursue such goals, ranging from power and status to religious salvation and ethical punctuality, that are inconsistent with utility or profit optimization ...

A similar argument appears in Green and Shapiro (1994), with reference to the extensive RCT literature dealing with several problems in political science, and in our discussion below of production quotas and work schedules at Anheuser-Busch.

A major purpose of this paper is to enlist established techniques of linear programming (LP) to clarify and help resolve the above conundrum; we refer to Zey's items (a) through (g) in several places, interpreting them with reference to several specific applications of LP modeling. In a typical LP problem (Strum, 1972, pp. 8-11; Dantzig and Thapa, 1997, pp. 12-15, 37-41) one may have to decide upon an appropriate ``mixture," e.g., a certain combination of various types of commodities to be produced, given that several constraints impinge on the production process and given also that each type of commodity has a unique market value. Regarding constraints, a particular commodity may require x units of work of type A, y units of work of type B, and z units of work of type C, and the relative amounts of A, B, and C differ substantially among various commodities currently in production. Due to budgetary factors, social norms, or other limitations, each type of work may be available only for a specific amount of time per week. The typical problem for LP is to find solutions of simultaneous equations representing the constraints and thereby to determine how many units of each type of product should be produced, while remaining within the constraints and while maximizing marketability and ultimate profit. Notice, finally, that this decisionmaking process meets all the criteria of rational action as developed above: Constraints often define scarce resources; production schedules not selected become opportunity costs; institutional norms---say, a forty-hour work week---may create additional constraints; LP problems, defined mathematically, almost certainly would demand complete information about options available, associated utilities, and so forth; numerical scores permit rank ordering and have clear transitivity. In LP models, finally, maximal or minimal solutions are almost invariably sought, in what will soon be defined as the established Zipfian mode.

In perusing several additional introductory examples in Dantzig and Thapa (1997, pp. 2-7), one sees that LP models are highly abstract and generalizable, with applications that are sometimes surprising. For instance, the program to suppress drug trafficking in Colombia involves all sorts of social control strategies intended to bring about a range of outcomes. The many potential outcomes defy definition and measurement, and their costs and utilities are hard to ascertain. The social control activities themselves are highly complex, variable, and multi-dimensional. Various social factors act to limit or constrain access to essential resources.

In this paper we suggest that LP models be seen as metaphors with wide applicability, in the spirit captured nicely by an LP classic (Dorfman et al., 1958:121) that addresses several problems simultaneously:
 

It should be noted that the international-trade example, while an illustration of the transportation problem, has nothing to do with transportation costs. Neither, in any fundamental sense, does the transportation problem, in spite of its misleading name. The essential characteristic of the transportation problem is that restraints are imposed on both inputs and outputs and that, subject to those restraints, some number which depends on the allocation of the inputs is to be either minimized ... or maximized. Any economic or business problem of this general structure is a transportation problem.

This statement seems to inspire the sorts of ``mad-libs" that lead to the perception of new problems [note 3].

LP models appear to be ideal-type representations of many features of RCT. It would be most edifying to design experiments in which clearly defined LP models were described as vignettes to panels of decisionmakers, who would then attempt to make rational choices replicating the optimal selections usually arrived at deterministically by standard LP procedures. It would be possible, for instance, to design an LP problem for which there were no solution at all, and then to determine whether presumably rational decisionmakers would be able to discern this unique Arrowesque impossibility (Sen, 1993; Goode, 1997). In more typical instances, in which there exists a wide range of ``feasible" solutions, it would be possible to ascertain whether judges using some sort of vignette approach (SELFREF, 1980) would be able to arrive at solutions close to the optimal solutions identified by LP modeling.

Throughout this paper, then, we address the range of issues identified by Zey. We use LP models to clarify aspects of RCT essential to Zey's definition of rational behavior. Beyond that, we attempt to clarify concepts such as consistency, transitivity, and optimization (maximization or minimization) within an LP context, and to reveal possible contributions of LP modelling both to RCT and to chaos theory. In brief, we attempt an operational definition of rational choice within an LP framework, while invoking what appear to be non-rationalistic features of chaos theory.

Maxima and minima

The least-effort principle postulates that in social life we search for and adopt ways of minimizing work and other costs, while simultaneously trying to maximize highly valued outcomes. The terminological overlap of least-effort theory with linear programming [note 4] is striking, and it is perhaps not entirely coincidental that Zipf's famous book on the least-effort principle was published at the inception of linear programming in the late 'forties [note 5]. In Dantzig and Thapa (1997, p. 1) [note 6] we read that ``mathematical programming (or optimization theory) is that branch of mathematics dealing with techniques for maximizing or minimizing an objective function subject to linear, nonlinear, and integer constraints on the variables." Social scientists define large numbers of distributive values that become the ``benefits" of benefit /cost analysis, and the contemporary opportunity to use powerful LP programs to pursue Zipfian hypotheses on such matters is most promising and most alluring. LP models operate within boundaries (Dubin, 1978, Chapter 6), within limitations of resources, time, money, etc., that appear to be the bane of social existence, and this makes them highly realistic. Oftentimes there are infinitely many potential solutions, from which no small sampling is likely to contain very good prospects. There may be a small number of optimal solutions---usually only one [note 7]---and these optimal solutions are highly elusive. For those adept at the search, the basic strategies are highly demanding, be they geometric, algebraic, or both (Strum, 1972, p. vii). Most important of all, a successful LP model demands a willingness to run the risks of trial-and-error, of post-optimality analyses that may (or may not) make it possible to render even better what appears for the moment to be the best of all possible solutions.

In developing the minimization principle, Zipf (1949, pp. 6-7) invokes literally the notion of intervening obstacles, asking us to consider
 

... two towns, A and B, that are separated by an intervening mountain range. Here we can see the enormous amount of work that could be saved in travel and trade if the two towns were connected by a tunnel ...; we can also see the enormous amount of work that it would take to construct such a tunnel. We are simply arguing that when the probable cost in work of digging the tunnel is estimated to be less than the probable work of not having the tunnel, then ... the tunnel will be dug.

The above type of argument will also apply to a path of least time over the mountain. Thus the enormous cost of flying munitions over the mountain to save time in supplying an army ... may be more than justified ...

Zipf, we suspect, did not ``fly the hump"; if he had done so, he would have realized that in that famous instance of intervening obstacles there were at least two variables to be minimized and one to be maximized: The first, as he says, was time; the second was getting ``over" the hump at the lowest altitude possible without colliding with it; the third was to carry the largest reasonable payload. These goals, perhaps combined with others, necessitate some sort of balance; they make it clear that maximization and minimization problems often turn out to be multidimensional and highly complex.

Similarly, as we work through Zipf's extraordinary Chapter 9 (1949, pp. 347-415), entitled ``The Economy of Geography," we realize that the many graphs depicting ecological relationships, such as number of intercity bus passengers (Zipf, 1949, p. 395) as a function of the size of two cities and a distance measure, neglect many undefined and unspecified independent variables: In Zipf's tables, we invariably read y, an effect, as a function of

,

the product of two interacting populations divided by their distance from each other. But migration researchers know that populations of origin and destination contain myriad push and pull factors not at all captured by size. Further, the frictions of space, considered metaphorically, involve far more than mere distance (Lee, 1966). In Zipf's graphs one cannot tell, for instance, whether the relatively limited ability to predict airline traffic between pairs of cities (circa 1933), compared to the higher predictability of the number of bus or rail passengers, was due primarily to characteristics of populations or to some multidimensional set of frictions of space, or to other unknown factors including possible interaction among independent variables. (Nor does one know how these relationships have changed over the last half-century or more.)

The contours and corners of chaos

According to Medio (1992, p. 4), chaos involves apparently ``... stochastic behavior occurring in a deterministic system." The stochastic appearance, however, is deceptive. In a study by Perrow (1989), for instance, we encounter a configurational theory of accidents that has a highly deterministic flavor; thus, accidents are said to be normal. Every disaster, actual or potential, described by Perrow's book involves a chaos process, an extremely dangerous series of surprises that develop rapidly due to the interaction of several events that may never before have occurred together. The most rapid and dangerous escalation occurs when the several events have a property called ``tight coupling" (Perrow, 1984, pp. 89-90), i.e., they interact so closely that causal processes reverberate immediately throughout the system and even a small change in one place quickly creates larger, unpredicted problems in a contiguous place, in the style stipulated by chaos theory.

In interpreting LP output, ``range analysis" tells us ``... how much the objective function values or constraint limits would have to change before the optimal selection of activities would change" (Pannell, 1997:188). One cannot fail to notice that, in instances where changes over a small range have a huge impact on optimal solutions, we are essentially in the presence of tight coupling. The difference is that in Perrow's disasters the components that produce tight coupling are not known a priori to the degree that they must be known in writing a linear program.

Consider a simple illustration: If many human bodies, pieces of luggage, food trays, and other paraphernalia already have become airborne inside a large aircraft that is wildly out of control (NTSB, 1993), it is almost impossible even to begin a process of securing badly injured passengers, passengers not yet injured, or the many objects flying around the cabin; flight attendants, trying to apply their admittedly minimal training, cannot secure anything while they themselves float and bounce and undergo constant bombardment by all sorts of heavy or hard-edged missiles. A more subtle example: A few years ago an airliner was taken way off course because the pilot, very long of the torso, could not see the top of the number 7 on a digital display indicating the frequency of a nearby navigational radio transmitter (VOR) that he wished to use to establish a course. Instead of tuning to the frequency 108.1 he had unknowingly selected 108.7, and the airplane crashed into a mountain. Perrow's disasters, all involving modern high technology of high complexity such as nuclear power plants or Boeing 747's, do indeed give the appearance of stochastic, not deterministic, behavior. And yet, the National Transportation Safety Board rightly studies such disasters and attempts to implement safety programs, on the assumption that the same configuration of causes will almost invariably produce a comparable accident.

In this paper, the corner solutions of LP are regarded as discrete configurations often subject to change due to the impact of small causes. The corner point principle defined such that ``if a linear programming problem has an optimal solution, then that solution is at a corner point of the feasible region" (Radlow, 1979, p. 239); see Figure 1, for an instance of a feasible region. We show the sorts of circumstances under which very small changes introduced into linear programs may bring about shifts of corner solutions. Although most of our data are hypothetical, we spell out implications for many real-world processes.

Consider, for instance, the way Anheuser-Busch makes beer. Our own qualitative investigations of this enterprise suggest that the most intriguing questions have to do with centralized vs. decentralized control of production schedules, and ways in which these schedules reflect price shifts for beer components. For the most part, production is controlled and coordinated from the St. Louis plant, which is corporate headquarters. Elaborate computer programs, almost certainly using LP or comparable procedures in order to minimize costs, assign quotas to each of the eleven plants of this multinational firm, with the consequence that, for a time, beer consumed in the vicinity of the Newark plant may well be brewed in Fairfield, California, due to a transitory benefit/cost configuration. This arrangement presumably makes for high efficiency in the continental operation of Anheuser-Busch, but it has the effect that individual plants are expected to make rapid shifts of production quotas, with resultant stress on employees. Such stress, which may be substantial in comparison to small cost reductions, is not necessarily factored into routine, short-term benefit/cost calculations. To organization theorists such neglect may be ``irrational," especially if worker resistance eventually were to raise costs.

It would be easy to set up an LP problem comparable to the ``labor scheduling" example developed by Pannell (1997:89-93), and to manipulate small features of it in ways that would have large consequences in changing the optimal solution. The mathematical logic of the situation would remain linear, but such experiments nonetheless produce results similar to those obtained by Israel (1996:62-66) using more typical non-linear, iterative functions:
 

The sensitive dependence on initial conditions has practical significance for the predictability of systems in real life. In principle, if you know the function f and the initial value , this determines all the [results]. However, in a real system you never know exactly, but can only measure it to a certain degree of accuracy. In systems that exhibit sensitive dependence on initial conditions, even a very small initial inaccuracy grows over time, until eventually it completely ruins any predictions made about the system. This is believed to be the reason that accurate long-range weather forecasts are impossible.

A problem defined
 

Between the desire to move and the actual decision to do so there ... may be intervening obstacles... The distance of the expected destination, the cost of getting there, poor health, and other such factors may inhibit migration. These obstacles are hard to predict on any wide scale, however, and so we tend to ignore them and concentrate our attention on explaining the desire to move.

---John R. Weeks (1999, p. 239)

Let's create a problem in specific detail [note 8].

There are two sources or origins for migrants (or_1 and or_2), and three destinations (d_1 through d_3). Each of the two sources provides a number of migrants, in thousands, as indicated below. Employment opportunities at the three destinations are just adequate to absorb the migrants [note 9]. ``Frictions of space" are such that, in a table for which the sources provide row headings and the destinations provide column headings (see matrix_1, below), the dollar costs and other costs of migration for each of the six table cells are indexed according to the values given below by through . Costs are subtle: Along with the factors specifically mentioned above by Weeks, we would surely have to consider costs related to ``culture shock" and other aspects of psychic adaptation. For the present problem there are fascinating ecological complexities such as the fact that, on the essentially two-dimensional plane defining the earth's surface, a number of migratory paths are likely to cross each other [note 10]. This means that migrants are likely to pass through various transportation ``hubs" developing at crossing points, hubs that provide ``intervening opportunities" (Hawley, 1950, pp. 242-43). Given, however, that in the present instance we assume that all migrants arrive at their ultimate destinations, the intervening opportunities are more appropriately considered to be ``opportunity costs," hardly distinguishable from intervening obstacles. They are so regarded in this paper---another instance of the subtlety of costs.

The simplex algorithm, an essential part of the LP armamentarium, finds the frequency of migrants for all cells such that migration costs due to frictions of space will be minimized. Our prediction is that, given an adequate migration-registration system, these cell frequencies would be observed as a result of least-effort selectivity (Zipf, 1949).

Citing Zipf liberally, Bogue (1959, pp. 502-504) presents an early elaboration of the ``migration stream" metaphor. He develops a formula for ``relative stream velocity," an interval measure, that does not incorporate distance measures or other aspects of the frictions of space. He then argues that ``the important thing accomplished by this expression is control of the common elements [e.g., size of origin and destination populations] in the above formulations, thereby permitting the expression to become a dependent variable rather than an explanation." In our analysis of migration streams and comparable phenomena, we assume that all relevant characteristics of origin and destination populations already have been held constant so that the stream-intensity data, as it were, consist of residuals. We then assess the effects of multidimensional frictions of space in a situation where we wish to test Zipfian notions about cost minimization. In brief, we analyze the denominator of the typical benefit/cost formula. This strategy is altered slightly in the final section of the paper.

Mathematical software

This article uses the mathematical software package known as Maple. As is typical of such packages, Maple incorporates an extraordinary diversity of symbolic, numeric, and graphic capabilities that create an effective problem-solving environment for modeling social phenomena and, indeed, for learning mathematics from simple to advanced. Each of these capabilities is used liberally throughout this article. Beyond the powerful mathematical features of the package, Maple provides word processing as well as automatic formatting for technical Internet (HTML) or LaTeX documents.

The ``restart" command, immediately below, clears computer registers. All input commands after the initial restart line are preceded by the ``input" prefix. Any input statements preceded by the # symbol are interpreted as comments. Input commands followed by a colon rather than the usual semicolon are executed but do not display output. Virtually all output involves standard mathematical formatting and notation.

> restart; interface(prompt="input: "); interface(warnlevel=0);

input: with(simplex): # An LP software package

Here we list the origin and destination frequencies; they are the marginals of matrix_1, below. We make sure that the two sets of frequencies have the same sum. (Expressions such as 110/100 are useful in setting up experiments with input values.)

input: or_1 := 47; or_2 := round((110/100)*60);

input: d_1 := 51; d_2 := round((105/100)*27);

input: d_3 := or_1 + or_2 - d_1 - d_2;

A boolean check, making sure that :

input: evalb(or_1 + or_2 = d_1 + d_2 + d_3);

Matrix representation

In matrix_1 we have what is essentially a cross-tabulation, or mobility table, containing the basic data for migration frequency. The third row and fourth column show marginal frequencies for the six cells of the table. In cells a through f are cell frequencies for migrants, expressed in terms of two unknowns and .

input: matrix_1 := array([[x, y, or_1 - (x + y), or_1],
[d_1 - x, d_2 - y, d_3 - (or_1 - (x + y)), or_2],
[d_1, d_2, d_3, or_1 + or_2]]);

In a moment we write constraints that force each of the unknowns, and , to be equal to or greater than zero, and force all cell frequencies to remain within limits set by the table marginals. If x and y have a sum that is less than 13, then cell f has a negative value---and this result will be disallowed by means of an additional constraint.

Contexts and constraints

A few extra algebraic steps are necessary to prevent a negative value for cell f. If this term were positive, the constraint would no longer be necessary. The negative term in cell f needs to be isolated for use in a constraint. We call the cell frequency cell_f1, and the isolated negative value, f_constant.

input: cell_f1 := d_3 - (or_1 - (x + y));

input: f_constant := cell_f1-(x+y);

The list of constraints below, then, arises from the fact that all cell frequencies, for this sort of problem, must be positive and remain within the marginals. It is almost invariably true that constraints sharply define a range of ``finite resources," and constraints therefore comply with and illustrate dimension (a) of Zey's definition of rational behavior. In many instances constraints entail budgetary considerations or the quantity of some component to be added to a mixture as defined earlier; in the present example, constraints have to do with the carrying capacity of each of several migratory streams, as influenced by marginal frequencies.

input: constraints := {x >= 0, y >= 0, x+y <= or_1, x <= d_1, y <= d_2, x + y >= abs(f_constant)};

The region containing feasible solutions for this problem is shown in Figure 1. In most instances there will be an indefinitely large number of feasible solutions. In this sense, linear programming provides a way of resolving one of the more intractable problems of RCT, namely, that of providing a mutually exclusive and exhaustive list of options available to decisionmakers (Zey, 1998, pp. 33-35). Experimentation involving the ability of subjects to perceive correctly the ``feasible solution" geometry of Figure 1, or at least to answer the question immediately below regarding the existence of solutions, would edify the ``information intensive" aspects of rationality, another of Zey's ideal-type dimensions.

The following command asks essentially whether the constraints allow any solution at all:

input: feasible(constraints);

There is at least one solution. We may proceed.

Costs and their minimization

Now we list the estimated costs of the frictions of space for each of the six potential migratory streams. These costs may be measured as thousands of dollars per person, hundreds of dollars per person per day, etc. As we have seen, they may also have non-monetary dimensions, e.g., the dangers and weather risks of the Cimarron cutoff of the Santa Fe trail.

input: k1 := 7; k2 := 2; k3 := 1; k4 := 6; k5 := 4; k6 := 2;

Again these costs are estimates, and it is perhaps a characteristic of RCT that such estimates tend to be employed in lieu of empirically-based utilities; this appears to be the modal complaint of critics. In LP, by contrast, costs are often measured empirically with considerable accuracy, and the measures are usually monetary. There is, however, no reason why utilities, involving cardinal-number scores, should not be developed in instances where costs are multi-dimensional, highly complex, and not entirely monetary. Conjoint analysis, for instance (www.sawtooth ...; www.spss.com/software), is a method whereby respondents evaluate a complex series of designs, perhaps multiple-attribute designs for a new commodity, for which the producer/manufacturer must decide on packaging, brand name, price, a Good Housekeeping seal of approval, and some sort of guarantee, along with several properties that may actually make the commodity perform well. Clear utilities are an essential feature of ``information-intensive" rationality---item (d) of Zey's ideal-type dimensions.

The following objective equation, multiplying the number of migrants anticipated for each of the six cells by the estimated cost of the frictions of space for each cell, will capture total migration cost, a macro-structural feature. Once we solve for the two unknowns, we have a good understanding of the temporal dynamics and ``rationality" of the entire migratory network . We first write the equation in unevaluated form,

and then we evaluate it; it simplifies considerably.

input: obj := k1*x + k2*y + k3*(or_1 - x - y) + k4*(d_1 -x) + k5*(d_2 - y) + k6*(or_2 - ((d_1 - x) + (d_2 - y)));

Now we are ready to solve for the unknowns, and , finding values that will minimize migration costs. We arrive at these values by means of the efficient simplex algorithm, but it would have been possible to find the solution by solving simultaneously the equations defining each corner of Figure 1, testing each solution by means of the objective equation, and then selecting the solution that minimizes costs (Strum, 1972, Chapters 1-3). In either case, costs will be minimized by the following solution set, substituted into the objective equation.

input: sol := minimize(obj,constraints);

Theories implicit within graphics: Corner solutions as chaos phenomena

Let us represent the process graphically. Begin with a plot of the feasible region (Figure 1), a region within which there is likely to be a specific solution for and that will minimize total migration costs. Any feasible option selected at any distance from the solution corner, given that the latter is almost invariably an optimal singularity, creates costs that presumably could have been avoided. The indefinitely large number of feasible solutions involves an exhaustive set of costs measured at the interval level (not merely the ordinal level, as implied by Zey), and any subset of these costs would be mutually exclusive and transitive---thereby meeting additional criteria of Zey's definition of rationality. In instances where solutions are implemented in a way that violates constraints, it is arguable that decisionmaking has entered a new dimension, perhaps the ethereal realm of ``irrational exuberance." For instance, in the typical mixture problem discussed earlier, Strum (1972, pp. 8-11) lists many alluring options that violate at least one among several constraints.

input: with(plots):

input: inequal({x >= 0, y >= 0, x+y <= or_1, x <= d_1, y <= d_2, x + y >= abs(f_constant)}, x=-5..55, y=-15..50, optionsfeasible = (color=turquoise), optionsexcluded = (color=white), optionsclosed = (thickness=3), labels=[`x-axis`, `y-axis`], title = `Figure 1: Feasible Region`, font=[TIMES,ROMAN,8], titlefont=[TIMES,ROMAN,10]);

At this point we show geometrically how the objective function relates to the feasible region (Radlow, 1979, pp. 230-35), as plotted above. In brief, the objective function is very likely to intersect one and only one of the ``corners" of the above diagram. But this, as we explain in note 6, is not always the case (Strum, 1972, p. 40; Radlow, 1979, p. 246).

First, without knowing the specific value for obj, we solve the objective function for , and call it :

input: eq1 := solve(obj, y);

The slope for is correct, but the -intercept, now enhanced by the numerical value of obj, has to be lowered to a level where it will touch the optimal solution corner. We know from the minimization solution for and that the -intercept of the objective equation is 28 at the correct corner. Therefore below, after we write into it the correct slope and intercept, should touch this corner. We can anticipate that the numerical value for obj will be .

input: eq2 := 2*x + 28;

Figure 2 contains a complete graphic representation of this problem. It presents the same two-dimensional graphic as that found in Figure 1, but it now includes the objective function, which does indeed touch the optimal solution corner where and .

input: inequal({y = eq2, x >= 0, y >= 0, x+y <= or_1, x <= d_1, y <= d_2, x + y >= abs(f_constant)}, x=-10..55, y=-15..50, thickness=3, optionsfeasible = (color=turquoise), optionsopen = (color=brown), optionsclosed = (color=red),optionsexcluded = (color=yellow), labels=[`x-axis`, `y-axis`], title = `Figure 2: Feasible Region Intersected by Objective Function`, font=[TIMES,ROMAN,8], titlefont=[TIMES,ROMAN,10]);

If the objective function happened to have the same slope as the constraint for cell f, which creates a line segment from and to and , it might simultaneously touch each of these two corners along with the line between them. All solutions along this line, including the corners, then would be optimal.

One would also be able to show that tiny changes in the slope of the objective function could be used to switch back and forth between two relevant corner solutions, which would alternate as the isolated optimum; this appears to be essentially what happens at Anheuser-Busch, as noted earlier. In creating experiments for this purpose one should remember that it is possible, by manipulating the cost schedule only (k1 through k6 for the present problem), to change the objective equation without disturbing the geometry of the feasible region. An extremely small change of the objective function could then be used to create very large consequences: Such relationships are an essential dimension of chaos theory (Brown, 1995, pp. 14-21; Israel, 1996, p. 66; Korsch and Jodl, 1994; Nicolaides and Walkington, 1996, pp. 144-46; Shone, 1997, pp. 209-18; Sparrow, 1982; Thesen and Travis, 1992).

Predicted stream intensity and associated costs

Using the ``assign" command we isolate the solutions for and , and make appropriate substitutions into the six expressions for migration frequencies. The minimal cost of this complex migration process, as shown below and as anticipated earlier, turns out to be units.

input: assign(sol);

Through substitution we obtain the optimal size of the six migratory streams, along with their associated marginals:

input: matrix_2a := array([[x, y, or_1 - (x + y), or_1],
[d_1 - x, d_2 - y, d_3 - (or_1 - (x + y)), or_2],
[d_1, d_2, d_3, or_1 + or_2]]);

The frequencies above must sum to the original number of migrants.

Here is the cost estimate, after substitutions into obj:

input: obj;

As a check, notice that the objective equation, , when solved for , is equivalent to the equation derived above for insertion into Figure 2, i.e., .

Even within the region of feasible and therefore tempting solutions, costs could have been much higher than the minimum. Suppose, for instance, that we inadvertently had ended up at the point within the feasible region where and . Substitute the new values into the objective function (assignments such as delete a pre-existing value):

input: x := 'x': y := 'y': x := 10: y := 10:

Take a look at the result. It raises costs considerably. It may at one time have been an optimal solution, but it has presumably become ``extinct" and is now surviving solely due to inertia.

input: obj;

Experiments

Strum (1972, p. 160) suggests that in experimental ``post-optimality analysis" it is wise to introduce only the smallest experimental variations, lest one be overwhelmed by the substantial labor (and increased risk of error) involved in calculating new results. With modern computers and mathematical programs, carrying out elaborate calculations is no longer a burden. However, even when we have the advantage of fast and accurate calculation, we find---as chaos theory leads us to anticipate---that small experimental changes may have surprisingly large effects for which a clear interpretation is no easy task.

First series

In the following experiment we began by introducing minor changes into the marginal totals for in-migrants and out-migrants. Eventually these changes created a sudden shift in the location of the solution corner, with both an and a value greater than zero. There remained, however, two potential migratory streams without migrants, and we tried to change this result by manipulating the schedule of costs of the frictions of space. Changing costs is tantamount to changing the objective function, and eventually this function generated a second, distinctive optimal solution corner with a very high value for . The zero-frequency streams stubbornly remained---just as they seem to have persisted in airline transportation routes (although in rapidly changing locations) since the phasing in of deregulation.

input: restart; interface(prompt="input: ");

input: with(simplex):

Warning, the protected names maximize and minimize have been redefined and unprotected

input: or_1 := 47; or_2 := round((75/100)*60);

input: d_1 := 51; d_2 := round((105/100)*27);

input: d_3 := or_1 + or_2 - d_1 - d_2;

input: evalb(or_1 + or_2 = d_1 + d_2 + d_3);

Again, matrix_1:

input: matrix_1 := array([[x, y, or_1 - (x + y), or_1],
[d_1 - x, d_2 - y, d_3 - (or_1 - (x + y)), or_2],
[d_1, d_2, d_3, or_1 + or_2]]);

Adjustments for cell f:

input: cell_f1 := d_3 - (or_1 - (x + y));

input: f_constant := cell_f1-(x+y);

A new set of constraints:

input: constraints := {x >= 0, y >= 0, x+y <= or_1, x <= d_1, y <= d_2, x + y >= abs(f_constant)};

A new list of costs:

input: k1 := 4; k2 := 4; k3 := 2; k4 := 2; k5 := 1; k6 := 2;

And a new objective function, again simplified. Inspection of this function tells us that the linear relationship between y and x is negative, and therefore this function is likely to intersect the feasible region at a location that will differ substantially from that of the preceding example.

input: obj := k1*x + k2*y + k3*(or_1 - x - y) + k4*(d_1 -x) + k5*(d_2 - y) + k6*(or_2 - ((d_1 - x) + (d_2 - y)));

And, finally, a new solution set:

input: sol := minimize(obj,constraints);

input: with(plots):

Warning, the names changecoords and display have been redefined

input: inequal({x >= 0, y >= 0, x+y <= or_1, x <= d_1, y <= d_2, x + y >= abs(f_constant)}, x=-5..55, y=-15..50, optionsfeasible = (color=turquoise), optionsexcluded = (color=white), optionsclosed = (thickness=3), labels=[`x-axis`, `y-axis`], title = `Figure 3: Feasible Region`, font=[TIMES,ROMAN,8], titlefont=[TIMES,ROMAN,10]);

As in the preceding case, we associate the objective function with the feasible region.

input: eq1 := solve(obj, y);

Again the slope of eq1 is correct, but the y-intercept is distorted because we have not yet obtained a numerical value for obj. The second term of eq2, as before, is the manually corrected intercept for the objective equation. We anticipate that the correct value for obj will be .

input: eq2 := -(2/3)*x + (224-156)/3;

input: inequal({y = eq2,x >= 0, y >= 0, x+y <= or_1, x <= d_1, y <= d_2, x + y >= abs(f_constant)}, x=-10..55, y=-15..50, thickness=3, optionsfeasible = (color=turquoise), optionsopen = (color=brown), optionsclosed = (color=red),optionsexcluded = (color=yellow), labels=[`x-axis`, `y-axis`], title = `Figure 4: Feasible Region Intersected by Objective Function`, font=[TIMES,ROMAN,8], titlefont=[TIMES,ROMAN,10]);

As before, we insert solutions for x and y into the matrix:

input: assign(sol):

Matrix_2b shows the optimal set of migratory streams,

input: matrix_2b := array([[x, y, or_1 - (x + y), or_1],
[d_1 - x, d_2 - y, d_3 - (or_1 - (x + y)), or_2],
[d_1, d_2, d_3, or_1 + or_2]]);

for which minimal costs are

input: obj;

We make sure that the sub-experiment following, which raises costs by implementing a non-optimal set of migratory streams, remains within the new boundaries of the feasible region as shown in Figure 4; we see, then, another instance of a feasible solution that presumably is irrational.

input: x := 'x': y := 'y': x := 30; y := 10;

By substitution, we find a relatively costly outcome:

input: obj;

Second series: Constraints involving social norms

input: restart; interface(prompt="input: ");

input: with(simplex):

Warning, the protected names maximize and minimize have been redefined and unprotected

Suppose that the places of origin were two counties from which students were drawn for a unified school district. Suppose further that there is a requirement that the number of students from county or_1 who attend school d_2 cannot exceed the students attending school d_1 by more than 7 units---representing perhaps 700 students. This requirement embodies social norms pertaining to school funding arrangements, to standards regarding student population densities, to policies regarding busing, and so forth; it involves item (c) in Zey's multidimensional definition of rationality. The new requirement will generate an entirely new constraint, along with the more typical constraints.

input: or_1 := 47; or_2 := round((110/100)*60);

input: d_1 := 51; d_2 := round((100/100)*32);

input: d_3 := or_1 + or_2 - d_1 - d_2;

input: evalb(or_1 + or_2 = d_1 + d_2 + d_3);

For this problem, again, most of the constraints pertain to individual cell frequencies for the following matrix.

input: matrix_1 := array([[x, y, or_1 - (x + y), or_1],
[d_1 - x, d_2 - y, d_3 - (or_1 - (x + y)), or_2],
[d_1, d_2, d_3, or_1 + or_2]]);

As before, we have a problem with cell f.

input: cell_f1 := d_3 - (or_1 - (x + y));

input: f_constant := cell_f1-(x+y);

The list of constraints below, then, arises from the fact that all cell frequencies, as usual, must be positive while respecting the marginals, and from the imposition of a new requirement regarding the discrepancy allowed between y and x.

input: constraints := {x >= 0, y >= 0, x+y <= or_1, x <= d_1, y <= d_2, x + y >= abs(f_constant), y-x <= 7};

The region containing solutions for this problem is shown in Figure 5. Now we list the estimated costs:

input: k1 := 7; k2 := 2; k3 := 1; k4 := 6; k5 := 4; k6 := 2;

The objective equation:

input: obj := k1*x + k2*y + k3*(or_1 - x - y) + k4*(d_1 -x) + k5*(d_2 - y) + k6*(or_2 - ((d_1 - x) + (d_2 - y)));

We are ready to solve for the two unknowns, and , finding values that will minimize costs while meeting new conditions. Given that LP, which occasionally behaves as a chaos model, is a game of extremes and abrupt changes (Strum, 1972, p. 26), it is not surprising that the solution for this system occurs at the margin of a rule violation regarding the allowable discrepancy between y and x (Aday, 1990): There may be substantial pressure for y to become even larger. In Zipfian terms, this situation is likely to maximize the costs of social control, as perhaps occurs in the case of Anheuser-Busch when the workforce becomes restive in the face of frequent cost-reducing shifts of work schedules.

Our solution set:

input: sol := minimize(obj,constraints);

Next, as usual, we represent the process graphically.

input: with(plots):

Warning, the names changecoords and display have been redefined

Again we show geometrically how the objective function relates to the feasible region as plotted above. First, assign :

input: eq1 := solve(obj, y);

As occurred earlier, the slope for is correct, but the -intercept has to be raised or lowered to a level where it will touch the optimal solution corner. Knowing that the desired -intercept must be the term of , and knowing the value of both and , we see that . Therefore below, containing the correct slope and intercept (Figure 6), should touch the solution corner.

input: eq2 := eq1 - 447 + 2;

Finally, the new constraint is the final member of the set of constraints listed in the following command, .

input: inequal({y = eq2,x >= 0, y >= 0, x+y <= or_1, x <= d_1, y <= d_2, x + y >= abs(f_constant), y-x <= 7}, x=-10..55, y=-15..50, thickness=3, optionsfeasible = (color=turquoise), optionsopen = (color=brown), optionsclosed = (color=red),optionsexcluded = (color=yellow), labels=[`x-axis`, `y-axis`], title = `Figure 6: Feasible Region Intersected by Objective Function`, font=[TIMES,ROMAN,8], titlefont=[TIMES,ROMAN,10]);

We now isolate the solutions for and , and make appropriate substitutions into the six expressions for school-attendance frequencies. Remember:

input: sol;

input: assign(sol):

input: x,y;

And we now know the optimal size of the six student streams:

input: matrix_2c := array([[x, y, or_1 - (x + y), or_1],
[d_1 - x, d_2 - y, d_3 - (or_1 - (x + y)), or_2],
[d_1, d_2, d_3, or_1 + or_2]]);

The cost estimate, finally, is shown by substitution to be .

input: obj;

Notice that the new constraint, combined with a small adjustment of the marginals, raises costs substantially. It also has a major impact on the ratio of and , and it has interesting effects elsewhere. For instance, only one stream now remains empty, and stream intensities have changed considerably compared with those of matrix_2b.

Another small escalation with large results

input: restart; interface(prompt="input: ");

input: with(simplex):

Warning, the protected names maximize and minimize have been redefined and unprotected

There is, however, a special problem in analyzing the Slomczynski-Krauze data. For our preceding applications, we assumed that migration and mobility streams had been residualized, so that the dependent variable to be explained (Bogue, 1959, pp. 502-504) was likely to reflect costs only due to factors such as frictions of space. By contrast, the Slomczynski-Krauze data presumably reflect simultaneously both costs and benefits of social mobility. Accordingly, in a table for which row headings provide sources and column headings provide destinations (see matrix_1, below), the benefit/cost ratios for each of the nine table cells are estimated as the values given below for through . The simplex algorithm will find the frequency of movers and stayers for all cells, such that mobility benefits will be maximized.

Again, a new set of data, derived from the Slomczynski-Krauze study:

input: or_1 := 199; or_2 := 259; or_3 := round((100/100)*542);

input: d_1 := 309; d_2 := round((100/100)*381);

input: d_3 := or_1 + or_2 + or_3 - d_1 - d_2;

The usual check:

input: evalb(or_1 + or_2 + or_3 = d_1 + d_2 + d_3);

Matrix representation with new dimensions

For this problem the constraints pertain to cell frequencies for movers and stayers, stated in terms of four unknowns. Cells a through i, in matrix_1, contain various expressions involving the unknowns. As before, the bottom row and the most rightward column contain marginal sums for the nine cells of the table, each of which contains at least one symbolic term:

input: matrix_1 := array([[r, s, or_1 - (r + s), or_1],
[t, u, or_2 - (t + u), or_2],
[d_1 - (r + t), d_2 - (s + u), d_3 - (or_1 - (r + s) + or_2 - (t + u)), or_3],
[d_1, d_2, d_3, (d_1 + d_2 + d_3)]]);

In a moment we write constraints that force each of the unknowns to be equal to or greater than zero while remaining consistent with marginal frequencies. As occurred before, a given cell, in this case cell i, may have a negative numerical term, and because we're dealing with ratio variables we must force such frequencies to be positive by means of additional constraints.

Adding constraints

A few extra algebraic steps:

input: cell_i1 := d_3 - (or_1 - (r + s) + or_2 - (t + u));

The numerical term is isolated for use in a constraint.

input: i_constant := cell_i1-(r+s+t+u);

And the set of constraints:

input: constraints := {r >= 0, s >= 0, or_1 >= (r + s),
t >= 0, u >= 0, or_2 >= (t + u),
d_1 >= (r + t), d_2 >= (s + u), (r+s+t+u) >= abs(i_constant)};

Utilities and their maximization

Now we list estimated benefit/cost ratios for each of the nine stayer or mover cells.

input: k1 := 3;k2 := 1;k3 := 1;

input: k4 := 5;k5 := 2;k6 := 3;

input: k7 := 6;k8 := 2;k9 := 3;

The benefit/cost schedule, i.e., the utility schedule, has a surface plausibility. Utilities sum to 14 units for white-collar sons, 5 units for blue-collar sons, and 7 units for farm sons.

input: k1+k4+k7;

input: k2+k5+k8;

input: k3+k6+k9;

Utilities for stayers sum to units. For all forms of mobility, utilities are the sum of off-diagonal values.

The following objective equation, multiplying the number of stayers and movers anticipated across the nine cells by the benefit/cost ratio for each cell, will capture a maximal utility value once we solve for the four unknowns, r, s, t, and u:

As usual, the equation simplifies considerably.

input: obj := k1*r + k2*s + k3*(or_1 - (r + s))
+ k4*t + k5*u + k6*(or_2 - (t + u))
+ k7*(d_1 - (r + t)) + k8*(d_2 - (s + u)) + k9*(d_3 - (or_1 - (r + s) + or_2 - (t + u)));

Now we solve for the four unknowns, finding cell values that will maximize utilities:

input: sol := maximize(obj,constraints);

It appears that many mobility channels remain empty.

Predicted stream intensity and the costs of non-optimal feasibility

We isolate the solutions for , , , and , and make appropriate substitutions into the nine expressions for cell frequencies. The maximal utility for this complex mobility process is given below, where we substitute values into .

input: assign(sol):

Examine the optimal size of the nine cells. Maximizing the utilities of social mobility, in this instance, requires that circulation mobility be raised to a high level: Matrix_2d below has few stayers (51), while the Slomczynski and Krauze (1987, p. 603) table shows 619 stayers. On the other hand, this matrix in many experiments has unrealistic features, such as the number of farm fathers whose sons end up in white-collar occupations. In further experimentation, constraints would rule out unrealistic patterns.

input: matrix_2d := array([[r, s, or_1 - (r + s), or_1],
[t, u, or_2 - (t + u), or_2],
[d_1 - (r + t), d_2 - (s + u), d_3 - (or_1 - (r + s) + or_2 - (t + u)), or_3],
[d_1, d_2, d_3, (d_1 + d_2 + d_3)]]);

Substituting r, s, t, and u into gives us our maximal utility, which we name max_util.

input: max_util := obj;

Once again we show that utilities, within the region of feasible solutions, could have been much lower than the maximum. Suppose, for instance, that we inadvertently had arrived at a point within the feasible region where the four unknowns, although (by definition) meeting the criteria of the various constraints, were non-optimal. We begin by deleting the current values assigned to the four unknowns:

input: r := 'r': s := 's': t := 't': u := 'u':

We select a set of non-optimal values for the four unknowns of cells a, b, d, and e, making sure that they are feasible, i.e., that they do not violate any of the constraints stated earlier. To ensure this result, the values entered below are taken directly from Slomczynski and Krauze (1987, p. 603) and pertain, again, to the case of Spain [note 12]. We merely substitute the new values into the objective function.

input: r := 150; s := 42; t := 69; u := 178;

Within this model, the Spanish mobility pattern reduces utilities to a point well below the maximum attainable---the percentage reduction is evaluated below as .

input: obj; reduction := evalf(100*(obj - max_util)/max_util, 4);

Perhaps it would be possible to reduce this disparity by inserting into the model more realistic utilities and/or a better set of data. In the meantime, we have presumptive evidence of considerable macro-level irrationality.

Conclusions

Dialectical materialism (Engels, [1972]; Haldane, [1969]) is a philosophical forerunner of contemporary chaos theory. In explaining it, a nearly perfect metaphor would be a kaleidoscope: As one rotates the device very slowly, for long moments nothing at all happens that one is able to see. But forces or ``contradictions" accumulate steadily and stealthily, and instability rises invisibly and inexorably. We know the outcome with certitude, but we have little hope of predicting the precise instant of its occurrence. Then, suddenly, the frail mosaic collapses and creates a nearly new configuration. Not, of course, totally new, for the transformation is not done by means never before observed: What we see is a tiny landslide, tumbling precipitously and predictably. Yet, again, the timing and the precise pattern of change are largely unpredictable, like a change in the work schedule at Anheuser-Busch [note 13].

The same story occurs in Gilbert and Troitzsch (1999, pp. 9-10), but with a slightly different metaphor:
 

... consider pouring a steady stream of sand out of a pipe so that it mounts up into a pyramid. As you pour on more sand, there will be little landslides down the side of the pile. While the pyramidal shape of the pile and, in particular, the angle of the side are predictable, depending on the properties of the average sand grain, the timing, location, and scale of the landslides are unpredictable because the slippage is nonlinear. Once a grain of sand starts sliding, it pulls others along with it and there is positive feedback leading to a mass of sand slipping. Similar nonlinearities are thought to cause stock market crashes.

These metaphors, although fascinating, are mere analogies; as such, they must inevitably arrive at a point where they mislead. In both instances we realize that if nearly unattainable controls were introduced---the size, shape, and density of the sand grains or the stones, the rotation speed of the kaleidoscope, the speed of the pipefeed, etc., all appearing to be crucial---we would have better prospects of establishing predictability. In human (and many physico-chemical) settings, of course, such control is impossible. If it were possible to exercise it, then we would observe something like the deterministic predictability of Lorenz' famous system of differential equations governing weather dynamics (Boyce and DiPrima, 1997, pp. 533-41): Each chaotic run, involving tiny changes of initial conditions, looks impulsive and unpredictable, and yet each is entirely replicable, understandable, predictable, i.e., deterministic. The problem? First, an anomalous beat of a butterfly's wing in Madagascar---taken to represent a tiny change of initial conditions---eventually may create an entirely new concatenation of weather events that, due to its complexity, cannot be entirely understood or effectively predicted unless one could trace accurately every step of the process, starting with the errant butterfly. Second, little is known about the interaction, the feedback, among the many meteorological grid points---grains of sand, as it were---whose dynamics are revealed by the Lorenz system. The Lorenz equations, arguably, are micro-level: They tell us about the behavior of a single stone or sand grain, on the strong assumption that each grid sector has an independent dynamic. For social phenomena the stones and the sand grains (bourgeoisie and proletariat?) are not sufficiently uniform, nor do they think alike, act alike, or tumble alike.

The strategy of this paper, accordingly, has been to take a relatively simple instance of linear determinism---linear programming---and show that the diversity of sand and stone will create measurable departures from deterministic rationality, seemingly to the delight of organization theory.

Beyond that, we have addressed the macro-micro issue, providing at least a candidate for an operational definition of it: Returning to the Gilbert-Troitzsch metaphor, the sand grains are micro-social, the sand heaps---held together by an infinitude of network connections based either on power engineering or communication engineering---are macro-social. LP makes this distinction entirely transparent: Everything is macro-social except for the activities and perceived utilities reported by individuals. In conducting a benefit/cost analysis within the framework of RCT/LP, it is almost mandatory that one carry out, at some point, micro-social inquiries with an empirical emphasis. Again, if we wish to eliminate the apparently disastrous division of labor (Zey, 1998) between RCT and organizational theory, thereby causing more broadly-gauged scholars to work harder, it may turn out that the appropriate way to reconcile these new exorbitant demands with the realities of time constraint would be to form a new division of labor between macro-social organization theory and whatever theories generate the micro-social probes that tell us what people want and what they are willing to pay or suffer.

Incidentally, another operational definition of macro/micro interaction appears in Gilbert and Troitzsch's discussion of ``emergence" (1999, pp. 10-12).

Last but not least, we demand a renaissance: If RC/organization theory arrives at sharp distinctions between rational decisionmaking and departures therefrom and sharp distinctions between macro- and micro-social analysis, and also develops a clear conception of chaos both in its linear and nonlinear manifestations, then it is reasonable for us to expect from RC/organization theory a far better understanding of the determinants and consequences of rationality and irrationality. What would it matter, for instance, if Spanish social mobility processes departed substantially from benefit/cost rationality?

NOTES:

(1) In this paper, utilities will be regarded as deterministic. For a discussion of the relative advantages of deterministic and probabilistic models, see SELFREF (1989).

(2) Dawkins (1986, p. 9) remarks that in the realm of natural history there are many more ways of being dead than of being alive.

(3) For instance, when office workers (and systems analysts) nowadays talk about ``migrating" from old software to new software, their selections (``streams") probably can be predicted from a least-effort model similar to that exploited herein---Zipf (1949, Part 1), in fact, often discusses maxima and minima for psychic processes, usually with regard to language usage. Using essentially the same shell or template as that developed in the first few paragraphs of this paper, we might describe the new problem as follows:
 

Office workers currently use two obsolescent software packages (or_1 and or_2); three new, improved, highly efficient packages are now available as replacements (d_1 through d_3). Licensing restrictions and company policies require that each worker ``migrate" from one of the older to one of the newer packages. ``Frictions of space" involve several different costs encountered in making the transition: Monetary costs imposed by license formulas, time and work devoted to the learning curve, and employee anxiety presumably vary from one migratory stream to another. In a table for which the sources provide row headings and the destinations provide column headings (see matrix_1), the dollar costs and other costs associated with each of the six table cells are indexed by through . Find the frequency of software migrants for all cells, such that migration costs due to aforementioned frictions of space will be minimized.

Our prediction is that, given an adequate record-keeping system, these cell frequencies would be observed as a result of least-effort selectivity.

With a little practice, these paragraphs flow like mad-libs. One could easily write a mad-lib for airline transportation, thinking of aircraft as entities that carry out more or less diurnal migrations from origins to destinations, and one could ask how it is that routing patterns seek to reduce costs or why it is that so many channels remain entirely empty. And, of course, the obvious analogy between geographic mobility and social mobility leads us to ask about ways in which social mobility patterns might exemplify patterns already implied for migration.

(4) Introductions to LP are found in Dantzig and Thapa (1997), Dorfman et al. (1958), Kolman et al. (1992), Pannell (1997), Radlow (1979), and Strum (1972).

(5) Dantzig (Dantzig and Thapa, 1997, pp. xxi-xxxii), recounting the history of LP, shows the high indebtedness of the field to earlier work by social scientists, especially economists, with whom Zipf was doubtless familiar.

(6) A mathematician, Dantzig was one of the founders of the LP field. Over the years he has adapted the basic LP strategies so that they apply to all sorts of problems in the natural sciences, the social sciences, the humane disciplines, the business realm, etc. He is the ideal-type Comtean scholar, and he is therefore the compleat sociologist.

(7) As we shall see, Zipf (1949, p. 2) potentially would go astray with the belief that ``... no problem in dynamics can be properly formulated in terms of more than one superlative, whether the superlative in question is stated as a minimum or as a maximum... If the problem has more than one superlative, the problem itself becomes completely meaningless and indeterminate." At times, albeit rarely, a given problem has an infinitude of maximal or minimal solutions (Strum, 1972, pp. 27,40; Radlow, 1979, p. 246). In the words of one of our critics, an infinitude of optima can exist in rare situations where ``... one of the edges of the search space happens to be parallel to the graph of the objective function ..." It is easy to envision this prospect with reference, say, to Figure 1. Such situations may be indeterminate, but they are not meaningless: Their indeterminacy, especially within the framework of chaos theory, is highly significant.

(8) Several LP problems developed in this paper are comparable to the ``transportation problem" discussed in the classic textbook by Dorfman et al. (1958). We ran this problem using simplex procedures and modern software, replicating results presented by Dorfman et al. (1958:116 and Table 5-9).

(9) The resultant ``geographical mobility" matrix, then, will be comparable to the national ``circulation" mobility matrices studied by Slomczynski and Krauze (1987, p. 599): Origin and destination sums (not distributions) will be the same.

(10) We have no idea what the likelihood of crossovers would be for a given configuration of origins and destinations, but an analysis of the geometry of intersecting network paths would appear to be a worthwhile undertaking; it also appears to be a neglected problem. In Wasserman and Faust (1994, pp. 94-121), lines connecting the nodes of complicated graphs are often drawn in a way that intentionally minimizes line crossings, but one often wonders what would happen to the geometry of these graphs if the ``ties" among nodes were more substantive than merely mentalistic notions of ``liking," ``knowing," etc., and if lines were allowed to cross in ecologically consequential ways.

(11) Slomczynski and Krauze (1987, p. 601) use linear programming to find the lower limits of structural mobility, i.e., changes of occupational status that must occur across generations in order to satisfy the marginal sums of the table, which typically are not the same for succeeding generations. For the Spanish data, the filial generation has far more white- and blue-collar opportunities than did the parental generation. Circulation mobility may vary within the limits set by structural mobility. Structural mobility in the table for Spain is such that the number of sons from a farm background who end up either in blue- or white-collar occupations must be at least 232. In Slomczynski and Krauze's data for Spain, the actual number is 251 due to the movement of 19 additional sons from farm backgrounds to blue-collar work.

(12) One must be careful: Notice that the data for Norway (Slomczynski and Krauze, 1987, p. 603) do not sum to 1,000.

(13) Claiming that ``... there is a great deal of stability in corner solutions ...," a critic points to the ``... limited applicability ..." of our models. However, the question regarding ``radical" shifts of corner solutions remains essentially empirical. We would be delighted to do surveys of people who use applied LP, e.g., for logistical problems, and see how often they experience large, sudden shifts of optimal solutions. This sort of event seems to happen routinely at Anheuser-Busch.

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