[From Bill Powers (2002.12.18.1131 MST)]

I hope the large attachment proves worth your patience, all you

recipients.

Bill Williams UMKC 12 December

2002 2:15 PM CST

It occurs to me that the Giffen Effect is a special case of a situation

involving multiple goods. Any time a person has a limited budget and is

spending *all of it*, any increase of the price of any good will

necessitate reducing expenditures somehow.

The orthodox prediction – correct me if this is wrong – would be that

when the price of a good is raised, the person will buy less of it. That

may well be true when other sources of the same good or one that has

equal “utility” are available. at the old price. But when the

person is forced to reduce expenditures because the price of one good

went up, the most advantageous move may not be to buy less of that good,

but to buy less of a different good that is valued less, leaving more

money to spend on the more valued good. This could even entail buying

more of the good whose price went up because of getting less of something

supplied by the other good that was curtailed.

When a person has a limited income, the total spent on all goods cannot

be higher than the budgetary limit Therefore the sum of Q[i] times P[i]

(quantity of each good times its price per unit quantity, summed over all

i) must be equal to or less than B, the budget, all in dollars per unit

of time such as a day, month, or year. From the standpoint of budget

alone, there is no reason to suppose that raising the price of one good

will cause less of *that* good to be purchased. Each good, however,

has multiple utilities (I’m avoiding PCT jargon here, but you can do the

appropriate translations), meaning that budget is not the only

consideration. This is why there are indifference curves, for example

between different kinds of candy with different prices and degrees of

tastiness. There can be other kinds of indifference curves; six apples

might be equivalent to 2 oranges in terms of providing the same

tastiness. In fact there can be n-dimensional indifference curves,

surfaces in multiple dimensions.

Bringing in PCT, we can see that multidimensional indifference curves are

produced when a person is controlling for obtaining N goods, each

associated with its own reference level and its own loop gain.The whole

system of controlled variables will be brought as nearly as possible to a

low state of total error, the state that is as close as possible to zero

error in the N-dimensional space. If we say that the overall goal state

is represented by a reference-point in N dimensions, then there will be a

cloud of points around its projections in each dimension representing the

actual state of the system. This represents the closest possible approach

to the ideal solution of the system of control equations, which would be

that every controlled quantity would be exactly at its reference

level.

Now an apparent diversion:

I have recently been looking into multiple control systems (every time I

do this I seem to get just a little farther). This time I tried a set of

systems with N environmental variables and N perceptual variables, each

perceptual variable being obtained from the set of all N environmental

variables as a weighted sum, the weights being selected at random from

numbers between 1 and -1. For convenience, and to further my education a

little, I set the equations up as matrices. So far this isn’t much

different from what I’ve tried before.

Then I tried using an old method of choosing output weights of 1, 0 and

-1 according to the signs of the weights assigned by each control system

in its perceptual input function. With N = 50, about a third of the

trials resulted in all 50 control systems coming close to correcting

their errors. The failures were the cases where the choice of -1, 0, and

1 for the output weightings was not precise enough to prevent direct

conflicts. So instead of that solution, I tried plugging the input

weights into the output functions in such a way as to preserve negative

feedback around all possible loops, and made a startling discovery. The

required matrix of output weightings turned out to the the

“transpose” of the matrix of input weightings! If M[i,j] is a

two-dimensional matrix, its transpose is simply another matrix with the

rows and columns interchanged: M[j,i].

The result of using the transpose of the input matrix as the output

matrix was that for ALL random distributions of input weightings, the 50

control systems converged to a solution – that is, brought their

perceptual signals to a match with the (randomly selected) reference

signals. Sometimes convergence was very slow, when the randomly selected

weights defined directions in hyperspace that were nearly opposite. On

other trials it was very fast. I am now confident that this method will

work with any number of control systems.

HOWEVER, the method would work far faster if the sets of input weightings

for each control system described a direction in hyperspace that was

orthogonal to the directions of all the other sets of input weightings.

In fact, in that case choices of -1. 0, and 1 would probably work as well

as the transpose, or nearly so. This condition is like saying that each

perceptual signal could vary without any of the others varying, so they

all vary independently – that is, it is possible for the environment to

change so that any one of the N perceptual signals can change without any

of the others being disturbed.

The reason I hope we will be able to find a neurally feasible way to

arrive at orthogonal sets of input weightings is that I can’t think of

any believable way in which an input perceptual weighting factor

(synaptic weighting) could be copied into a weighting factor in an output

function – in a nervous system, of course; in a computer it’s easy. I

*can* imagine how a trial-and-error process could result in choices

of output weightings as coarse as -1, 0, and 1, given that the input

weights were orthogonal so this crude adjustment would be sufficient.

Right now I’m using the transpose, but that will not be the final

solution.

So that’s where the multiple-control-system idea is today.

All this is leading up to the attachment, which is a book in progress (I

think). The title may turn out to be just a chapter heading – nothing is

set in stone yet. The book will contain a disk, and on the disk will be a

Turbo Pascal 7.0 program illustrating the control of 50 variables at the

same time, as described in the final pages of the text. The source of the

program is included, as well as the executable .EXE file.

The program starts with controlled perceptions (red circles) and their

reference levels (green circles) in their initial values. Hit the space

bar to watch the control systems operate. Remember that *each*

perception is a different weighted sum of the *same* 50

environmental variables, Control requires finding the values of the 50

environmental variables that will satisfy all 50 control systems at the

same time. If you wait long enough, there will be nothing but red

circles left, showing that all the perceptions match their respective

reference signals within 1 pixel. New random weights are set every time

the program runs. Most of the error, of course, is corrected in the first

few seconds.

So what does this all have to do with Giffen? Well, obviously, it’s about

people who are controlling for many variables at the same time, trying to

get them all to their respective reference levels. The outputs consist of

spending money, and one of the controlled variables will be concerned

with budget. Beyond that I don’t know where we’re going. But I think this

is the direction in which we will find the Generalized Giffen

Effect.

Best,

Bill P.

PCTandEng.ZIP (95 Bytes)