Attractiveness Multiplier

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"Swanson, John"
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Posts: 5
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Attractiveness Multiplier

Post by "Swanson, John" »

Bill Braun and Jim Hines are talking about different things, I think. Jim
refers to attractiveness indicators being used to model market shares. He
offers one formulation, but a better one (Id claim) is based on formal
utility theory, and predicts shares using either a logit or probit model.
The logit model gives the probability that product A is chosen out of n
alternatives as exp(UA)/ sigma( exp(Ui)), where UA is the utility
("attractiveness") of alternative A etc. The utility expression is some
function of the product attributes, commonly linear additive. In practice
it is similar to Jims formulation but uses the exponential of the
attractiveness terms. Its advantage is that it is grounded in utility
theory, and the utility parameters can - usually - be calibrated, using
specialised survey techniques.

The original question was about something rather different. The Urban
Dynamics model calculated attractiveness multipliers for each of several
attributes of the city, and then multiplied them together. The product
(or a lagged version of it) was then applied multiplicatively to one or more
rates, such as migration rate, or construction rate. I have been trying to
use this recently, and have problems with it, mostly to do with calibration.
First, using a product of multipliers has its attractions, but it obscures
the implied tradeoffs between different attributes. For instance, an
increase in land availability of X% will necessarily have the same modelled
effect on construction rates as a change in one of the other multipliers,
vacancies, say, by some other amount, y%. These equivalences are a useful
cross check when testing the credibility of model, but are not explicit in
the formulation and are awkward to tease out.

The second thing is relating the scale of the multipliers to rates of
activity. It seems to be hard for people to think in terms of the effect of
attractiveness on a rate of activity, such as construction. To be credible
to many people - certainly the audience I work to - some evidence or
calibration would be needed, however rough, to demonstrate that credible
orders of magnitude are used for these multipliers, but it is hard to see
how this could be achieved. The original book simply asserted some values -
very bold but not an approach that is easy to sell.

Two ideas occur to me that might improve things. One is to formalise the
notion of attractiveness a little more, probably by tying it in to the
economic notion of utility. The second is to tease out the mechanism linking
attractiveness to activity rates rather more fully, in the hope that
intermediate steps can be identified that can plausibly be
measured/calibrated.

John Swanson
From: "Swanson, John" <
J.Swanson@sdgworld.net>
"Kim Warren"
Junior Member
Posts: 14
Joined: Fri Mar 29, 2002 3:39 am

Attractiveness Multiplier

Post by "Kim Warren" »

"geoff coyle"
Senior Member
Posts: 94
Joined: Fri Mar 29, 2002 3:39 am

Attractiveness Multiplier

Post by "geoff coyle" »

Bill,

Ill be glad to know what you think.

One thing which worries me is the uncertainties in these multipliers. Lets
say we have three:

X=A*B*C

and, as you point out, if all are 0.5, the net effect is 0.125. However, the
data for these things is usually a bit suspect and each might actually be
somewhere between 0.4 and 0.6, say, which is only a 20% uncertainty.
However, if all take their lower values the result is 0.064 and if they are
all at the upper limit it is 0.216. The nominal value of 0.125 might,
therefore be only half the real value or it might be twice the real
value. The real value might vary by a factor of as much as about 3 (its
actually 3.375). That degree of uncertainty, which stems from a mere 20%
uncertainty about the parameters, might have a significant effect on the
model outputs and hence on the confidence one might have in the
recommendations made from study of the outputs.

Obviously there are 6 other combinations of low and high for the three
parameters, and it gets much worse if there are as many as 10. It gets worse
yet when we know that A, B and C come from nonlinearities in which a few
suspect data points have been extrapolated to a continuous curve. Since the
curve is a real function the number of uncertainties is, strictly speaking,
nondenumerably infinite.

Ive just been looking at a model which has no fewer than 16 such curves,
luckily not all used in the same multiplicative fuinction. I think that the
author has not the least notion of what he is doing but HE IS NOT ALONE IN
THAT!!!!!

Geoff (aka Worried of Shrivenham)
From: "geoff coyle" <
geoff.coyle@btinternet.com>
Bill Braun
Senior Member
Posts: 73
Joined: Fri Mar 29, 2002 3:39 am

Attractiveness Multiplier

Post by Bill Braun »

There have been a number of comments about Urban Dynamics (e.g., Sterman,
SD3408) and the debated merits and flaws of the model. Is the use of
multipliers (UD, Productivity Press, pp. 23-25) still considered a valid
approach to represent the dynamics of attractiveness (actual and, with
delays, perceived)?

Bill Braun
From: Bill Braun <medprac@hlthsys.com>
"Jim Hines"
Senior Member
Posts: 88
Joined: Fri Mar 29, 2002 3:39 am

Attractiveness Multiplier

Post by "Jim Hines" »

Bill Braun asks if the Urban Dynamics attractiveness formulation is still
considered useful.

The answer is yes. This formulation (modified just a bit) represents an SD
"molecule" or standard formulation that everyone in the field should have at
their finger tips. (A molecule is a tried and true solution to some
modeling problem. )

The modifications to this molecule involve (1) the use of constants for
normalizing inputs to the multipliers and (2) basing "share" on his
attractiveness divided by "total" attractiveness.

Heres the modern reformulation given in terms of competitive
attractiveness for market share of automobiles.

marketShare = attractivenessUs/ (attractivenessUs + attractivenessThem1 +
attractivenessThem2 + ... + attractivenessThemN)
fraction
attractivenessUs = milageEffectUs*qualityEffectUs*...*effectNUs
dmnl
mileageEffectUs = mileageEffect_f(relativeMileageUs)
dmnl
mileageEffect_f = {a table function}
dmnl
relativeMileageUs = mileageUs/normalMileage
fraction
mileageUs = {some variable -- perhaps smoothed to account perception lags }
miles/gallon
normalMileage = {some constant}
miles/gallon
etc.

The attractiveness for them would be formulated analogously. Note

(1) marketShare is a ratio of attractiveness, so the units in which
attractiveness is measured doesnt matter. The only thing that matters is
the shape and relative magnitudes of the different attractiveness functions
(e.g. mileageEffect_f).

(2) the same functions (e.g. mileageEffect_f) are used for each competitor.

(3) normalMileage is a constant representing some "normal" value in the
range where consumers can perceive differences. The formulation permits the
function to saturate at high levels of the attribute. So, for example, if
the attribute were mileage and our mileage were 2 million miles per gallon,
you could represent the idea that customers doent care that our competitor
has twice the mileage as us: Either auto would more than satisfy customers
desire for thrifty transport.

(3) The same normalization constants are used for each competitor.

Jim Hines
jhines@Mit.edu
Bill Braun
Senior Member
Posts: 73
Joined: Fri Mar 29, 2002 3:39 am

Attractiveness Multiplier

Post by Bill Braun »

For the attractiveness multiplier, are the following valid equations?

If there are three variables that determine overall attractiveness, V[1],
V[2], and V[3], the attractiveness multiplier equation is V[1] * V[2] * V[3].

Similarly, ten variables would be V[1] * V[2] *...* V[10].

With three variables, if each produces a value of 0.9 (through their
respective table functions), overall attractiveness is 0.729. With ten
variables, if each produces a value of 0.9 (through the table functions),
overall attractiveness is 0.349.

If all variables in both cases are exerting the same "pull" (0.9), why is
the overall attractiveness different? (This is not a math question.) The
sensitivity difference is enormous. This suggests that the greater the
number of variables determining attractiveness, the more "convincing" each
has to be to result in the same level of attractiveness.

...
More on the dynamics of attractiveness, an extension of my other post
referring to the number of variables determining attractiveness.

In Urban Dynamics, for AMM (Attractiveness for Migration Multiplier) there
are five multiplier variables. Their maximum contributory effects are:

UAMM: 1.5 (response to UM)
UHM: 2.5 (response to UHR)
PEM: 4.0 (response to TPCR)
UJM: 2.0 response to (UR)
UHPM: 3.0 (response to UHPR)

for a total possible attractiveness of 90.0. How can attractiveness be
greater than 1.0?

Conversationally I can say I find something twice as attractive (2x) or
three times as attractive (3x) as something else.
Quantitatively/dynamically, am I not really saying that I find something
half as attractive (0.5x) or one third as attractive (0.33x), thus choosing
the option closest to 1.0 ("fully attractive")?

.....
Addendum:

Sorry to flood the list...

I think I have partially answered my own question. I neglected to take into
account the variable UAN (Underemployed Arrivals, Normal). Hence, AMM = 2
indicates that UA have doubled and AMM = 0.5 indicates that UA have
decreased by half.

Im still puzzled by the sensitivity based on the number of multiplier
variables.


Bill Braun
From: Bill Braun <medprac@hlthsys.com>
"Jay W. Forrester"
Senior Member
Posts: 63
Joined: Fri Mar 29, 2002 3:39 am

Attractiveness Multiplier

Post by "Jay W. Forrester" »

In Attractiveness Multiplier (SD3448), Bill Braun wrote:

>With three variables, if each produces a value of 0.9 (through their
>respective table functions), overall attractiveness is 0.729. ......
>This suggests that the greater the
>number of variables determining attractiveness, the more "convincing" each
>has to be

The argument here is that the combination of several inputs can
indeed be more influential than the sum of the parts. The cumulative
effect makes the whole matter more visible, as attention is drawn to
a better or to a worse situation, public awareness and publicity
rise, and the influence can be even more than is justified.
Multiplicitive variables do not differ much from additive ones for
small changes, but the the cumulative effect becomes progressive
greater as the individual deviations become larger. The issue is the
psychological and decision-making impact of multiple inputs: are
they merely additive, or does the influence accumulate
disproportionately as the influences pile on top of one another? The
question could lead to an interesting debate, but I tend to vote in
favor of an effect greater than the sum of the parts.

>In Urban Dynamics, for AMM (Attractiveness for Migration Multiplier) there
>are five multiplier variables. Their maximum contributory effects
>total possible attractiveness of 90.0. How can attractiveness be
>greater than 1.0?
>

An attractiveness of "1.0" is merely a defined normal. There is no
reason why attractiveness can not go above that normal value. Also,
combining all the extremes gives an interesting number, but, should
that occur in a simulation, questions should be raised about the
structure of the model. Even individual extreme values usually
should not be reached because that would raise doubts about the
adequate range of a table function. The maximum values of such
functions should lie outside the dynamic range of action that is
being studied.
--
---------------------------------------------------------
Jay W. Forrester
From: "Jay W. Forrester" <
jforestr@MIT.EDU>
Professor of Management
Sloan School
Massachusetts Institute of Technology
Room E60-389
Cambridge, MA 02139
"Michael Bean"
Member
Posts: 22
Joined: Fri Mar 29, 2002 3:39 am

Attractiveness Multiplier

Post by "Michael Bean" »

One nice feature of the multiplicative attractiveness function is that if any
attribute goes to zero, the whole function returns a zero. So, if you are
selling a product a product, the attractiveness of the product could be:

Product Attractiveness = Availability Effect x Wait Time Effect x Price Effect

If the product is unavailable, then the Availability Effect is zero and Product
Attractiveness goes to zero.

As Bill Braun pointed out, one problem with this formulation is the more
attributes you add to the attractiveness function, the more volatile it becomes.

An alternative way to calculate a multi-attribute attractiveness function is to
use a geometric mean.

With the geometric mean, the attractiveness function becomes:

Product Attractiveness = (Availability Effect ^ Availability Effect Strength x
Wait Time Effect ^ Wait Time Effect Strength x
Price Effect ^ Price Effect Strength) ^
(1 / (Availability Effect Strength + Wait Time Effect Strength + Price Effect
Strength)

The geometric mean has several nice features:

Increasing the number of effects does not increase the volatility of Product
Attractiveness.

Its easier to tune your model. For example, if each effect varies between 0 -
3, then the attractiveness output will also only vary between 0 - 3.

If any effect goes to zero, the attractiveness goes to zero (as it did in the
simple multiplicative function).

If you set the strength of any effect to 0, then that effect has no impact on
attractiveness (because any positive number ^ 0 = 1). This makes it easy to turn
off effects.

It also has a few problems:

Compared to a simple multiplication, its harder to explain to others what the
function is doing. I find using examples helps explain the function.

In the special case where an effect is zero, and the strength of the effect is
zero, the function returns an error (because 0 ^ 0 is undefined). I usually
force 0 ^ 0 to return a 1 in this special case, because when I set the strength
to zero, I want that effect turned off.

Regards, Michael
_______________________________________
Michael Bean
Forio Business Simulations
mbean@forio.com
www.forio.com
"George Backus"
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Posts: 23
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Attractiveness Multiplier

Post by "George Backus" »

It is often difficult to separate philosophy from science. It is usually
very easy to use intuition or "feelings" to form an opinion about what is
"good" or "bad." Different value-laden world views produce different
perspectives on how the world works. The reality is that world works as it
does. The world views are only legitimate to the extent that they are
compatible with the reality we can objectively measure (the basis for
rationality). Preferences can be objectively measured as well as cash
flows. Neither measurement is perfect, but it does contain adequate sign and
magnitude information to allow valid decision making. (The information is
sufficient for adequately controlling system behavior.)

In one world view, SD could be perceived as simply a feedback collection of
(only) stock-&-flow and decision components. All decisions affect flows.
The flow affected can only go between 0 and 100% of its potential value. A
negative sign would indicate a different mechanism that would imply the
causal need to add a new (outflow) to the system (level) description
rather than try to make a variable go between (-100% and +100%). This logic
would indicate that the decision process affecting a flow should vary
between
0.0 and 1.0 unless some other mechanism limits the extreme -- in which case,
the flow is 0 to 100% of the limits. (Internal re-scaling of inputs or
intermediate values to make 1.0 "normal" are merely cosmetic and unrelated
to this argument.)

Attractiveness is a preference. A restatment of economics woudl argue that
all decisons are the weighing of preferences (whether preference is in
terms
of the value of $, the taste for the color red, or having the "latest and
greatest.") Daniel McFadden (2000 Nobel Prize in economics) developed the
Qualitative Choice Theory (QCT) in the early 70s. In essence, it simulates
the market share (MS) of a choice (i) based on utility (U) equations we
have seen in other notes.

MS(i)=exp(Ui)/sum(j)(exp(Uj))

The result is a logit curve but the basis is theoretically and empirically
meaningful, i.e., different. (Not all logits are created equal...)

The "U" function an be linear in independent variables (to give additive)
or long-linear to give multiplicative (proportional) impacts. In theory,
the
Utility function can have any defendable form. The primary determinations
of that form can be rigorously determined. For example, see Keeney, R. L.
and
Raiffa, H., Decisions with Multiple Objectives: Preferences and value
Trade-offs, John Wiley & Sons, New York, NY, 1976. Note the use of the
word "preferences" in the title. This is the same as perceptions. Measured
input information comes in and is processed via limited human cognitive
abilities to produce a value-based human decision. The parameters can be
estimated via survey methods or by simply taking the history of past
decisons and the data available to have made those decisions. This latter
estimation process is particularly easy within most firms (where there are
institutionalized information channels). For customers responses, surveys
are need -- along with secondary questions about choices for which the
historical result is known. These secondary questions can then be used to
scale and remove biases in the survey results. (Humans tend to not do
exactly as they say they will do.)

QCT has a secondary validity point in its favor. It is based on the
stochastic and distributive assumption of behaviors. A group does not
unanimously produce the same decision when polled. Not everyone sees the
same
"attractiveness" in a set of conditions. The recognition of this
uncertainty (distribution) in information content, the interpretation of
the data, and the decisions derived form that data, is explicitly contained
in the parameters produced by the regression. These parameters then have a
policy related meaning that can be used to leverage (or mitigate) the impact
of information that greatly affects the outcome decisons.

If one looks at the table functions in Urban Dynamics or other SD efforts
using "attractiveness", the curves are sigmoidal (logit) or asymptotic (one
side of a logit). The QCT easily encompasses (plus defendably quantifies)
both of these mathematical features.

So what does this have to do with multiplying attractiveness together
(V1*V2*.. VN). One way SDers like to dump on (simplistic) econometric
models is to point a finger at them when they either calculate market share
(or any allocation)

as

MS=1-V1*V2*V3..*VN

when in this case MS is a residual and the Vs are often in the elasticity
form of (X/X0)^E

or as

MS=K*V1*V2..*VN

In both instances, the MS is not bounded properly (logically or
numerically). Like the SD attractiveness approach, the asymptotic values
can be arbitrarily limited to keep values in bounds -- but that approach
makes for more arbitrary guesses, distortion of theory, a greater certainty
that logical errors have been introduced, and a greater certainty that
reviewers will require some "convincing" before accepting the results. The
"old" SD attractiveness approach appears to have more utility than
validity -- just like its
econometric forbearers.

In summary, I think that the additive versus multiplicative debate may be
archaic. A rethink of how information causally affects decisons, in the
context of more recent research, may be a more productive path.

George
From: "George Backus" <
George_Backus@ENERGY2020.com>
"Ray on EV1"
Member
Posts: 29
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attractiveness multiplier

Post by "Ray on EV1" »

The latest issue of The Industrial Physicist had an article on how to apply
the predator-prey model to sales. One of the three parameters of the model
can be viewed as attractiveness.

Ray
From: "Ray on EV1" <rtjoseph@ev1.net>
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