There has been a discrepancy between what mathematical modeling is assumed
to be, and what it explicitly is. There is an interesting diagram by Dr. Professor Zalman
Usiskin (2011) that defines the mathematical modeling process in a pictorial fashion.
Here, we will list the steps, and his interpretation of the steps: The real problem,
Simplified problem, Mathematical model, Solution to mathematical model of problem,
Solution to simplified problem, Checking of feasibility, and repeat as necessary. Usiskin
then defines the step in a one-to-one correspondence: 1) Simplify, 2) Find model, 3)
Solve, 4) Translate, 5) Check, 6) Decide whether to repeat the process, it the solution is
valid for the model. During the tenure of this modeling course, the focus was primarily
on behavioral sciences and how modeling is used to solve posed situations. The
previous modeling course focused on the natural sciences. Thus, an apparent need for
modeling in all areas that require situations modeled and solved. In particular, we will
focus on papers that developed a model in the behavioral or social sciences, use
Usiskin’s model as a guide when necessary to identify any key components, and
summarize what the authors conclude implementing a model for their findings.
The first paper is on modeling in decision analysis. The founder of decision
analysis is Ronald A. Howard and he defines it in his first paper entitled “Decision
Analysis: Applied Decision Theory” in the following paragraph:
Decision analysis is a logical procedure for the balancing of the factors that
influence a decision. The procedure incorporates uncertainties, values, and
preferences in a basic structure that models the decision. Typically, it includes
technical, marketing, competitive, and environmental factors. The essence of the
procedure is the construction of a structural model of the decision in a form
suitable for computation and manipulation; the realization of this model is often a
set of computer programs.
The definition is simply to provide background as we confirm the modeling methods. In
this case, the paper uses modeling to assess profit lottery, “the quantified expression of
our beliefs about the likelihood of occurrence of real-world events beyond our
immediate perception” (Tani, p. 1500). In this case, Tani (1978) goes into background
information about probabilities and truths. That is, the difference between authentic and
operative probability. The former being probabilities we ideally want, yet cannot achieve
and the latter being probabilities we can work with physically. Moving away from the
background work, Tani explicitly writes out the intended modeling process as follows:
1.
Identify a set of real-world factors on which profit is believe to depend
and represent them as sate variables, denoted by vector s.
2.
Encode our uncertainty about the future behavior of the state
variables as a probability distribution {s | ε}.
3.
Encode our uncertain understanding of the dependence relationship
between profit v and the decision and state variables d and s as a
conditional probability distribution, {v | d, s, ε}.
4.
Using external means of calculation, determine the profit lottery via
the expansion equation: {v | d, ε} = ∫s {v | d, s, ε} {s | ε}.
Tani states the modeling strategy does not relieve the necessity of assessing
probabilities; moreover, it is a system used to assess the probabilities in an easier
fashion. As labeled clearly by Usiskin’s model, the need to check the solution is crucial.
As it turns out, an abridged model was used as instead of the initial proposed model.
The reason was there were too many constraints with the first model. Here is a case in
Usiskin’s strategy where the initial model did not provide what we needed, and an
altered model needed to be formulated. The conclusion of the article is the
“dissatisfaction” of the model. As Tani stated initially, “in theory, we do not need to us
mathematical modeling to perform decision analysis” (p. 1500). As it turns out, the
model gives not an accurate readings of probability; moreover, an approximation with
uncertainty. “The better the approximation, the closer the modeled profit lottery is to the
authentic profit lottery” (Tani, p. 1506), which is the ideal situation desired.
The next paper is about of modeling choice behavior. This paper is more
extensive and goes through several definitions before reaching the heart of the
modeling aspect. We will need several definitions to understand the summary of this
modeling process, and even then, this is simply grazing the surface to what insight this
model is providing. Included are two figures that went along to demonstrate the models
hypothesized and used. To understand these pictorial depictions, we will define
definitions. Ben-Akiva et al. (1999) defines Behavioral Decision Theory having to do
with uncertainty and game theory which had two major affects: (a) formal, axiomatic
analysis fashionable in economics and psychology, and (b) invited laboratory
experimentation to test the descriptive validity of the axioms. Economics and
psychology have different views of the cognitive process for decision-making (DM)
process. The former focuses on the mapping from information inputs to choice while
the latter focuses to understand the nature of the decision elements, how they are
established and modified by experience, and how they determine behavior. These are
the associated definitions to provide information about figure one. For both figures,
“ellipses represent unobservable (i.e., latent) constructs, while rectangles represent
observable variables” (Ben-Akiva et al., p. 194). Figure one is the theoretical framework
for modeling choice behavior or the DM process. Figure two is the integrated choice
and latent variable model or the modeling framework. The objective is to model the DM
process in figure one. The authors had several criticisms about the models respectively
and related tasks. For one, they needed to understand “whether there is a fixed order
of the processes of preference construction and choice determination...search for
background variables, such as familiarity with the decision problem, that permit
imposing structure on Figure 1” (Ben-Akiva et al., p. 199). Another task the authors
wanted to partake on in a future event is to take their current model and branch into
simpler models. This is significant as even an appropriate model was discovered by the
authors to map out their situation, there are generally areas for improvement for
modeling aspects. In this case, creating a model within a model. Simplifying the
already solved model of the real problem as Usiskin describes. Ben-Akiva et al. (1999)
conclude their paper “outlines a view of how to augment extant choice modeling
frameworks, survey data and discrete choice methods to explicitly capture in practical
empirical models the insights into the decision-making process available from behavior
decision theories and research” (p. 200).
The last
paper is about prospects and problems in modeling group decisions., specifically, group
choice modeling literature in marketing and consumer choice. Group decision is
precisely what one may predict - The ability to make decisions when apart of a group.
“In the early 1970s, researchers in marketing and consumer decision making realized
through studying families and organizational buying centers that the group was an
important unit of analysis” (Steckel, Coffman, Curry, Gupta, & Shantey, p. 231). Steckel
et al. (1991) describe explicitly the questions that arise from group decision modeling,
specifically: “‘What have we done,’ ‘What have we learned?’ ‘What do we need to do to
learn more?’ and ‘How do we go about learning?’” (p. 231). The significance of this is
the model that may answer one of these questions, it may not be a fit model to answer
the other questions. In contrast, creating one single model may not be sufficient to
answer all the questions. As it turns out, the questions are checkpoints in determining if
a model is fit, analogous to checks and balances of the desired model. From these
checks and balances, two types of predictive models: linear and normative. The former
focusing as if the groups followed the linear model and whether the model predicted
well, whereas the latter describe behavior that reasonable, rational people should follow
(Steckel et al., 1991). Between the linear and normative models, some familiar
situations to what the modeling course covered revealed itself. For example, in the
linear model, the weight of each individual mattered and combinations of those
preferences did not matter. Like most suggestions, equal weights were considered for
each preference in a group, but we know this implies all group members have an equal
impact on the outcomes of the group’s decision. This is clearly not true. The normative
models reflect fairness, efficiency, and people within the group should follow these
principles. The answers a normative model produce determine “correct” or
“appropriate” decisions that should be made by the group, providing a reasonable
benchmark for descriptive modeling (Steckel et al., 1991). An interesting point is how
John Nash is quoted for his bargaining theory on the conflict point or the no settlement
point. This point is a default outcome which the bargainers must accept if they are
unable to agree. The follow axioms Nash proposed almost coincide with the content we
learned during the course!:
1. Individual Rationality - Both players should be better off at the outcome
than they would at the conflict point;
2. Feasibility - The outcome should be chosen from the set of possible
outcomes of the negotiation;
3. Independence of Utility Function Scale - The outcome should not be
depend on how utility is measured;
4. Pareto Optimality - No other settlement which both bargainers prefer shoul
exist;
5. Independence of Irrelevant Alternatives - if X is the outcome of a bargaining
situation, it should also be the outcome of any situation where the possible
outcomes are a subset (including X) of the original ones; and
6. Symmetry - If the set of feasible outcomes is symmetric (i.e., if one
alternative has a utility distribution a, b for the two bargainers, than another
has a distribution b, a) then the outcomes will provide both bargainers with
equal utility.
Steckel et al. (1991) continues describing which model works when and why,
hypothesizing linear models work, and two assertions: “(1) If group decisions have a
formal structure (e.g., bargaining, majority rule, etc.), models that account for this
structure will out predict those that do not, and (2) Models which incorporate behavioral
reality should out predict those that do not” (p. 236). The assertions are proven using
the following conditions: “(a) each input variable has a conditionally monotone
relationships with the output; (b) there is error of measurement; and (c) deviations from
optimal weighting do not make much practical difference” (p. 235). Steckel et al. (1991)
concludes within the modeling realm, the one special difficulty is collecting “good data,”
which was an issue for the second paper we briefly analyzed. The case-based
reasoning approach is more about data collection and the use of data as oppose to
modeling a situation altogether. The paper concludes with obstacles that arise when
using case-based reasoning and strategies on how to compensate and manipulate the
data meaningfully to model.
Mathematical modeling is still unfamiliar to many mathematicians and especially
those educating in the field of mathematics. Many think of it as a derivation to problem
solving, but Usiskin’s depiction is sufficient to see there is more occurring within a
mathematical model than simply problem solving. We were able to discover some
examples of mathematical models through papers specializing in the behavioral and
social sciences. The behavioral and social sciences are only two of many fields
mathematical modeling could be applied. If mathematical modeling is anything like
contemporary physics, it should be considered to be mainstreamed more so within our
educational standards.References
Ben-Akiva, M., McFadden, D., Gärling, T., Gopinath, D., Walker, J., Bolduc, D.,...Rao,
V.
(1999). Extended Framework for Modeling Choice Behavior. Marketing Letters,
10(3). Retrieved from http://www.jstor.org/stable/40216534
Howard, R.A. (1966). Decision Analysis: Applied Decision Theory. Proceedings of the
Fourth International Conference on Operational Research, pp. 55-71,
Wiley-Interscience,. Reprinted in Howard, R.A. & Mathewson, J.E. (Eds.)
READINGS on the Principles and Applications of Decision Analysis. Strategic
Decisions Groups, Menlo Park, California.
Steckel, J.H., Coffman, K.P., Curry, D.J., Gupta, S., & Shantey, J. (1991). Prospects
and Problems in Modeling Group Decisions. Marketing Letters, 2(3). Retrieved
from http://www.jstor.org/stable/40216218
Tani, S.N. (1978). A Perspective on Modeling in Decision Analysis. Management
Science, 24(14). Retrieved from http://www.jstor.org/stable/2630604
Usiskin, Z. (2011). Mathematical Modeling in the School Curriculum.
[PowerPoint Slides]. Retrieved from [email protected]