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Article Excerpt
Observing reality is especially valuable. However, without models, every situation at every time on every variable would be unpredictable. Assumptions allow models and theories to assert constancy. Assumptions distill and simplify reality by dismissing the conspicuous but irrelevant. Criticizing assumptions as unrealistic is absurd. Abstraction is the precise virtue of an assumption. For example, seldom are we prisoners facing interrogation, yet the prisoner's dilemma remains relevant. The adage "A bird in the hand is worth two in the bush" is relevant for more than birds. Unrealistic assumptions that deny current beliefs breed great new theories.
Assumptions are analogous to the basic ingredients in a gourmet recipe. Only the final product of the recipe dictates whether the ingredients suffice. Similarly, assumptions are realistic when they produce good theories, satisfactory predictions, valuable implications, and correct recommendations. Output matters far more than input. Realism is only an issue when creatively diagnosing poorly performing models, not when judging model performance.
Assumptions are the source of value in empirical analyses. If data sets were truly the source of value, empirical research studies would only greatly devalue the raw data by dramatically reducing rich observations to a few meager summary statistics or estimated parameters. Most empirical research makes a contribution by ignoring (assuming away) most information in the data.
We must dramatically shift our attention far away from the hopeless pursuit and sophistry of realistic assumptions to the contribution those assumptions produce. There are scientific methods for evaluating model output (i.e., predictions, findings, implications, recommendation) on criteria such as accuracy, reliability, validity, robustness, and so on. No corresponding objective scientific methods exist for evaluating realism. Realism depends only on personal taste.
Key words: models; mathematical models; realistic assumptions; instrumentalism; empirical research
********* Why Study Assumptions?
The role of explicit assumptions in building models is widely misunderstood. In fact, observations, from the peer review process at scholarly journals, chronicle widespread misunderstandings, causing mistakes and inconsistencies in the peer review process. Misguided evaluators sadly focus on evaluating assumptions (i.e., the input) rather than the findings (i.e., the output) or the process (i.e., the logic creating the output). Dismissing models based on only the realism of their early assumptions is facile and invalid compared with the correct approach of carefully evaluating the entire model development, the novelty of the findings, and the contribution to the literature. Given that all assumptions are unrealistic per se, the claim of unrealistic assumptions is empty. Assessing the realism of the assumptions might be important for creatively diagnosing a poorly performing model, but it is an inappropriate measure of performance.
Problems with the peer review process alone warrant careful examination of the role of explicit assumptions. Beyond that, however, understanding the role of assumptions is critical for the advancement of knowledge, understanding, and application. Assumptions allow us to focus our modeling effort, to build theory, to extract information from numerical data, and to interpret qualitative observations. Without unrealistic assumptions, most research would add no value to merely reporting raw numeric data or detailing qualitative observations.
We argue that properly evaluating research (and most professional activities as well) requires a focus on output rather than on input. Focusing on input (the assumptions, the observations, the numerical data, etc.) is wrong. All high-quality research makes explicit and implicit assumptions based on either explicit or implicit past observations. Evaluation of input, without regard to output, is an exercise in personal taste. In contrast, scientific methods exist for evaluating output (e.g., predictions, findings, implications, recommendations) on criteria such as predictive accuracy relative to extant models, reliability, validity, robustness, interobserver agreement, and generality. Moreover, we have criteria for evaluating the scientific research process itself, such as logical validity, objectivity, clarity, meticulousness, replicability, verifiability, and compatibility with scientific procedures.
We also argue that explicit assumptions provide the value from all analyses in all research, whether empirical or theoretical. Models are only useful because they remove the irrelevant aspects of reality. Without models, every situation at every time on every variable would be unpredictable. Assumptions allow models and theories to assert constancy and to predict in new situations.
Finally, for purely empirical research, we require assumptions to reduce a vast quantity of numerical data points to a very small meager set of summary statistics or estimated parameters. It is impossible to add value to a data set by ignoring parts of it, unless ignoring parts of it reveals the predictive or explanatory power of the assumptions. Otherwise, empirical research would simply diminish the value of the raw data by ignoring some of it.
What Are Assumptions?
Virtually all scholarly research, benefiting from mathematical models or not, begins with both implicit and explicit assumptions, but the term "assumption" is often ill defined. Although dictionaries define assumptions as something taken as true without formal proof, assumptions need not be true or false. Assumptions can be approximations, limitations, conditions, or merely premises. Most often, assumptions are sufficient conditions that guarantee the validity of the subsequent findings but whose violation by no means necessarily invalidates those findings. Published research often assumes that random error causes unpredicted outcomes, rather than wrong theory (Shugan 2006). Published research often assumes that results are invariant to the time, place, and sample. Most published research assumes continuity when virtually no variables are infinitely divisible. If assumptions could be false, we would dismiss most research.
An Example
Consider, for example, mathematician Albert W. Tucker's famous prisoner's dilemma game (Tucker 1950, Goeree and Holt 2001, Lev 2006). We assume two prisoners, only four possible outcomes, insufficient evidence to convict, guilty prisoners, prisoners acting in their own best interest, prisoners unable to communicate, prisoners knowing four possible outcomes, police who can lessen sentences for confessing, no possible retribution, no possible reputation effects, severe sentences for conviction without confessing, among still more assumptions. These assumptions are sufficient to find that both prisoners will confess, resulting in a worse situation for both than had the prisoners coordinated their decisions.
The assumptions are seldom true (i.e., if truth is even possible). Scholars are seldom prisoners, advising prisoners or advising the police. Nevertheless, there are thousands of references to the prisoner's dilemma. These apparently limiting assumptions reveal practical, remarkably general, and extraordinary powerful implications. It is the generality of the implications that matters, not the realism of the assumptions. Good theory makes good predictions regardless. As Trefil (2003) states "good theory ... will do more than incorporate facts already known--it will make predictions about phenomena that haven't been seen before (p. xv)."
Why Make Assumptions?
If the essence of a well-executed competitive strategy is deciding what not to do (e.g., Porter 1996), the essence of mathematical or analytical modeling is deciding what not to model. As it is deceptively easy to add one more strategic activity (e.g., see Porter 1996), it is deceptively easy to add still another variable, relationship, or parameter to a mathematical model. However, the "no free lunch" principle applies. As diversifying competitive activities diverts resources in strategic development, diversifying mathematical models also diverts resources and often hinders the original objective of the entire modeling effort. In this way, assumptions define the research question by dictating what is endogenous (in the research) and exogenous (outside of the research). Assumptions exclude variables, exclude possible relationships, and exclude possible causality. The assumptions are the foundation of proposed models, hypotheses, theories, forecasts, and so on. They dictate which variables to observe, not to observe, and the relationship between them.
Finally, when models become theories, assumptions dictate the...
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