Summary

Validity of Aggregation. In this report I illustrate some generic subtleties of aggregation and disaggregation in combat models by starting with an analytically tractable model at one level of detail and then attempting to derive an aggregate model. In particular, suppose that a Lanchester square law is valid for ground combat in each of a number of individual sectors. What equations then describe events at a higher, more aggregate, level? What factors determine whether a closed aggregate-level model exists (i.e., a reasonably accurate model dependent only on aggregate-level variables and with any coefficients being independent of time)? The answer is that what matters is "outside" the "detailed model" altogether, notably (1) higher-level strategy, (2) command and control, and (3) the relative durations of several time scales for battle and maneuver. These factors have major effects on whether a valid aggregate-level model exists and, if it does, what values its coefficients should have.

The 3:1 Rule. A bonus of this analysis is a clarification of when the famous 3:1 rule applies. If it applies at the sector level, then it may or may not apply at a more aggregate level. Indeed, in a theater with multiple corps sectors (e.g., the old Central Region of Europe), the theater-level break-even ratio will typically be more like 1.5:1 than 3:1. By contrast, it is possible for the same 3:1 rule to apply at several lower levels (e.g., corps, division, brigade, and even battalion). In mobile combat in which there is no particular defense advantage, the theater-level break-even force ratio may be about 0.8 or 0.9.

Maneuver, Tempo Control, and Reequilibration. One of the major factors determining outcomes at lower levels is the relative ability of the sides to control where and when to have decisive engagements. If a side can readily break off battle, collect forces, and reengage, then temporary concentrations by the opponent will be less significant. Conversely, the side can itself choose to have decisive engagements under favorable circumstances. These considerations are very important quantitatively and help explain why operational commanders have long tended to focus more on maneuver than, say, on the advantages of static defenses so often praised by analysts. They also illustrate again the disadvantages of military strategies that constrain the defender to fight in particular places and times (e.g., the old NATO forward-defense strategy). Such defenses are quite feasible, but they require more forces.

Aggregation and Disaggregation in Distributed Simulation. Although the specific analysis presented here is narrow, it suggests a broader conclusion of particular interest for distributed simulation, including distributed interactive simulation (DIS). A recurring question is whether it is legitimate and desirable to disaggregate and reaggregate processes during the course of a given simulation run (e.g., decomposing a battle into a higher resolution view, noting the results, reaggregating to a higher level, and later disaggregating again). Extrapolating from the example worked out in detail here, it seems that such temporary disaggregation would be most defensible if the real-world forces are able to reequilibrate at the aggregate level in between the events the simulation describes at a disaggregated level. A simple example might be a division fighting a battle, maneuvering to a different position, taking up new positions, and fighting again. Typically, this sequence would include reequilibration such as combining partly degraded units, balancing across subunits, and assigning new functions. Thus, it might be legitimate to use a disaggregated description for the battles and an aggregate description for the maneuver. By contrast, if the same unit underwent two periods of intensive battle separated by only a few hours (division level) or a day (corps level), the unit's initial state at the time of the second battle might be much the same as at the end of the first battle, in which case aggregating and disaggregating would sacrifice important information.

Families of Models. The simplified analysis of this report also motivates a number of other generic principles. In particular, and contrary to current trends in developing families of models, it is desirable to work top-down rather than bottom-up (or, more accurately, to work both top-down and bottom-up rather than only the latter). The reason is fundamental: The allegedly detailed models are only selectively detailed. In particular, they typically do not contain the information most critical in developing valid aggregate expressions. By contrast, top- or aggregate-level issues such as strategy often set context and critical boundary conditions for events at the detailed level.

Validation. This has implications for testing as well. Efforts to validate aggregate-level models should focus on the treatment of strategy, command-control, constraints, time scales, and uncertainties rather than on efforts to calibrate aggregate results against those predicted by a detailed model in which these factors are not even well represented. Interestingly, this is why comparing with historically based and insightful commercial board games of combat can sometimes be more useful to validation than comparisons with high resolution models. Detailed models, however, can be very useful in selectively calibrating specific parameters of higher-level models. Further, they are essential in understanding cause-effect relationships and determining which aggregate-level variables are important. Although I do not discuss such matters in this report, detailed models can also help generate statistical distributions that should be used to inform the calibration of deterministic or stochastic aggregate models. Ultimately, then, developing sound families of models requires giving proper respect to both higher- and lower-level perspectives of the same problem.