Monday, June 20, 2011

Three types of assumption in analytical (theoretical) models

Because there's a lot of confusion between different types of assumption in analytical models --  the kind of models that are not calibrated on data (sometimes called theoretical models; analytical here in the sense of mathematical analysis, not data analytics) -- here are the three types and their implications:

Conditions are the assumptions that drive the result of the model: the game form of a IO model, for example. The important point here is that a change in the conditions will change the result in a significant way and they must be identified as such. A common condition in many management models is that utility is concave, a major assumption that tends to drive results yet is treated as a trivial thing.

Simplifying assumptions are used to make the point as simply as possible. Instead of using a complicated payoff function, for example, one can use linear as long as the linearity itself is not crucial to the result (a fact that most linear models overlook). These assumptions are usually preceded with "WLOG" (without loss of generality), but it is really important to determine whether there's loss of generality or not. The important point here is that a change in a simplifying assumption should not change the results significantly.

Technical assumptions allow the modeler to use certain mathematical tools in the model development. For example, if one assumes continuity then derivatives can be used instead of differences. This allows for much smaller proofs, typically without loss of generality. Like simplifying assumptions, these technical assumptions are a convenience, and changing technical assumptions should not lead to significant changes in the results.

A lot or research papers in management get these types of conditions confused; many use linearity as the main driver of the result, yet describe it as WLOG. Others put inconveniently unrealistic conditions in footnotes or endnotes under the guise of "technical assumptions," which they clearly are not.

One should always be careful of what assumptions are conditions, as these are the ones that matter in the end.