## Aggregation, disaggregation, and the 3:1 rule in ground combatPaul K. Davis, United States. Dept. of the Air Force, United States. Dept. of the Army, United States. Dept. of Defense. Office of the Secretary of Defense, Arroyo Center, National Defense Research Institute (U.S.), Rand Corporation This report illustrates a number of basic principles about aggregation and disaggregation in combat modeling by working through the mathematics and phenomenology of a concrete example. In the example, simplified ground combat takes place in a number of sectors and subsectors within a theater. The author assumes that combat at some level of detail is dictated by the Lanchester square law then discusses whether an aggregate law, Lanchester or otherwise, applies at the next level up (that is, one with more aggregation and less detail). The answer depends on the ratios of several time scales related to information, decisions, maneuver, and the duration of a breakthrough battle. The author also discusses how the 3:1 rule does and does not apply at different levels of combat. |

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### Contents

la A Theater with a Network of Independent LOCs | 2 |

DERIVING AGGREGATE MODELS FOR CASES WITHOUT REINFORCEMENT | 6 |

MainSector Force Ratio at the BreakEven Force Ratio | 13 |

Copyright | |

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### Common terms and phrases

3:1 rule applies aggregate equations aggregate level aggregate model aggregate-level models aggregation and disaggregation air forces Ardennes assuming the 3:1 assumptions attacker and defender attacker concentrates attacker-conservative attrition coefficients attrition warfare break off battle break-even force ratio break-even point Bundeswehr calibrating coefficients K^and command-control commit their reserves concentration factor pd counterconcentrate Davis dD I dt Defender Advantage defender strengths DERIVING AGGREGATE detailed model disaggregated description DISAGGREGATION AND REAGGREGATION discussed distributed interactive simulation distributed simulation estimate example Figure force strengths fraction of sectors fractional loss rates functions ground combat higher-level Huber and Helling illustrates intra-battle reinforcement Kdon Ktti Lanchester square law level of combat loss rates RLR lower-level battle main sectors maneuver warfare mathematical analysis mobile warfare Operation Desert Storm parameters ratio of loss reequilibrate reinforcement and maneuver reinforcement and redeployment reserve fractions scores sector level sides commit static defenses stochastic strategy variables temporary disaggregation terrain theater-level