probabilistic dependencies are not accounted for properly. There are many reasons, including simplicity and familiarity of naive independent-event calculations, which many people can perform adequately with a spreadsheet. A deeper reason is that untutored intuition is often poor on issues involving probabilistic calculations. Indeed, probabilistic effects are sufficiently nonintuitive that workers often revert to simplified and naive calculations (including deterministic calculations when they are clearly inappropriate) even though they were once sensitive to the subtleties.

A third reason for failure to treat dependencies is that past computing capabilities did not permit workers to handle them effectively. This, coupled with the intuitive plausibility of many correlated effects averaging out or at least greatly simplifying, led to dependence on aggregate expressions such as the Lanchester equations described elsewhere in this report. That simplification has sometimes been valid and sometimes not. The conclusion here should be that without detailed analysis, and in many cases detailed simulation akin to “experimentation” with a real system, we should be skeptical about the validity of formulations that do not treat statistical dependencies.

Where might we find instances of such problems? The answer is “in virtually every part of combat modeling.” Indeed, whenever calculations of effects such as attrition are effectively multiplying together a set of planning factors, the effect is to assume an independence of events that may not be correct. Current-day assessment of precision-strike effectiveness against an invading army is a prime candidate for errors.


The panel was asked a number of questions about configural effects in mine warfare. It was surprised that there continues to be controversy about the importance of treating correlation effects such as those that are manifest in mine warfare. When two ships move through a mine field, the mines do not re-randomize their locations between ships. Nor do the ships move independently (e.g., one may follow the trail of another). Nor, in fact, is the pattern of mines in a given waterway random in many circumstances. And so on. The probabilistic effects reported for mine warfare are real, and Navy doctrine and decision aids for dealing with mine and countermine warfare should reflect them. This is without prejudice to how that is accomplished, since there are a variety of possible modeling approaches. In particular, one approach is analytical and has advantages for moving from requirements (e.g., a maximum loss rate to mines) to estimates on how many mines to buy or how to lay them. A second approach involves simulation, which has many advantages (and is in some respects easier), but which is not easily able to answer the questions analytical models are best for. Also, analytical models and the related theory can clarify the structure of the problem so as to illuminate the dependence on controllable variables.

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