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.
2.
3.
Identify a set of real-world factors on which profit is believe to depend and represent them as sate variables, denoted by vector s .
Encode our uncertainty about the future behavior of the state variables as a probability distribution { s | ε}.
Encode our uncertain understanding of the dependence relationship
4.
between profit v and the decision and state variables d and s as a conditional probability distribution, { v | d , s , ε}.
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 o bservable 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 z-usiskin@uchicago.edu