Lanchester Square Research Articles

An experimental test of Lanchester's models of combat in the neotropical termite Nasutitermes corniger (Blattodea: Termitidae).

Lanchester's models of combat have been invoked to explain the mechanics of group fighting in social animals. Specifically, Lanchester's square law posits that the fighting ability of the group is proportional to the square of the number of combatants. Although used to explain a variety of ecological phenomena, the models have not been thoroughly tested. We tested the Lanchester models using group battles between colonies of the termite Nasutitermes corniger. Our main goals were to determine if mortality rates fit the Lanchester models, and if so, whether the behavioural mechanisms underlying a group's success match those used in deriving the model. We initiated battles between pairs of colonies with different ratios of fighters and recorded deaths over time. We found that the numerically larger army has an advantage, but that the advantage is not as pronounced as predicted by Lanchester's square law. We also video-recorded battles to analyse individual behaviour, which did not support the mechanisms invoked by Lanchester. Instead, the killing power of an individual is increased by the presence of nest-mates, giving the larger group a disproportionate advantage. Although the behavioural mechanisms leading to the advantage may differ, our results still support some of the proposed ecological phenomena.

란체스터 전투모형에 기반한 방어작전에서 예비대 편성의 효과 분석

This paper aims at analyzing the effect of organizing reserve force in defensive operations. Though existing studies on troop reassignment assumed that the mutual assistance of the fighting forces was possible. Relocating troops in engagement is unrealistic. Therefore, in this paper, we present a mathematical model based on Lanchester Square Attrition Model with a separate reserve force. The proposed model allows to analyze the change in the number of surviving troops according to the proportion of the reserve force, so that the effectiveness of reserve force in defensive operations can be analyzed.

Методичні положення прогнозування втрат сил протидіючих сторін у загальновійськовій операції (бою)

Прогнозування втрат сил протидіючих сторін звичайно здійснюється органами військового управління у процесі планування операції (бою). За основу під час розробки методичних положень прогнозування втрат загальновійськових формувань, угруповань військ прийнятий квадратичний закон Ланчестера. Втрати загальновійськових формувань, угруповань військ оцінюються залежно від початкового співвідношення сил і засобів протидіючих сторін, яке може визначатися за показниками їхньої бойової могутності та бойового потенціалу. З використанням наведеної в статті методики одержані залежності втрат загальновійськових формувань, угруповань військ в операції (бою) від початкового співвідношення сил і засобів сторін.Методику доцільно застосовувати органам військового управління під час проведення оперативно-тактичних розрахунків.

란체스터 (2, 2) 전투모형의 최적 병력지원규모 해법 연구

란체스터 (2, 2) 전투모형의 최적 병력지원규모 해법 연구

Tensor-Centric Warfare I: Tensor Lanchester Equations

We propose the basis for a rigorous approach to modeling combat, specifically under conditions of complexity and uncertainty. The proposed basis is a tensorial generalization of earlier Lanchester-type equations, inspired by the contemporary debate in defence and military circles around how to best utilize information and communications systems in military operations, including the distributed C4ISR system (Command, Control, Communications, Computing, Intelligence, Surveillance and Reconnaissance). Despite attracting considerable interest and spawning several efforts to develop sound theoretical frameworks for informing force design decision-making, the development of good frameworks for analytically modeling combat remains anything but decided. Using a simple combat scenario, we first develop a tensor generalization of the Lanchester square law, and then extend it to also include the Lanchester linear law, which represents the effect of suppressive fire. We also add on-off control inputs, and discuss the results of a simple simulation of the final model using our small scenario.

Fight the power: Lanchester's laws of combat in human evolution

Lanchester's “Laws of Combat” are mathematical principles that have long been used to model military conflict. More recently, they have been applied to conflict among animals, including ants, birds, lions, and chimpanzees. Lanchester's Linear Law states that, where combat between two groups is a series of one-on-one duels, fighting strength is proportional to group size, as one would expect. However, Lanchester's Square Law states that, where combat is all-against-all, fighting strength is proportional to the square of group size. If conflict has been important in our evolutionary history, we might expect humans to have evolved assessment mechanisms that take Lanchester's Laws of Combat into account. Those that did would have reaped great dividends; those that did not might have made a quick exit from the gene pool. We hypothesize that: (1) the dominant and most lethal form of combat in human evolutionary history (as well as among chimpanzees and some social carnivores) has been asymmetric raids in which multiple individuals gang up on a few opponents, approximating Square Law combat; and (2) this would have favored the natural selection of an evolved “Square Law heuristic” that correlated fighting strength not with raw group size but with group size squared. We discuss the implications for primate evolution, human evolution, coalitionary psychology, and contemporary war.

다중 란체스터 모형에 대한 실용적 해법

We propose a heuristic algorithm for war-game model that is appropriate for warfare in which the maneuver of the attacker is relatively certain. Our model is based on a multi-weapon extention of the Lanchester's square law. However, instead of dealing with the differential equations, we use a multi-period linear approximation which not only facilitates a solution method but also reflects discrete natures of warfare. Then our game model turns out to be a continuous game known to have an -Nash equilibrium for all  ≥. Therefore, our model approximates an optimal warfare strategies for both players as well as an efficient reinforcement of area defense system that guarantees a peaceful equilibrium. Finally, we report the performance of a practical best-response type heuristic for finding an  -Nash equilibrium for a real-scale problem. Keyword:War-Game, Game Theory, -Nash Equilibrium, Heuristic Method

Adopting Lanchester Model to the Ardennes Campaign with Deadlock Situation in the Shift Time between Defense and Attack

General Karl Von Clausewitz believes that attack and defense in warfare are a state of interaction and response. The shift between attack and defense is involved with a short span of time difficult to define in concrete format. This paper extends Chen and Chu's model of the Ardennes campaign of World War II using the Lanchester equations. We use a new variable, deadlocked situation, to represent the shifting time between attack and defense. This article divides the Ardennes Campaign into three periods: (a) the initial period, German attacked while the Allies defended; (b) the middle period, both sides seized the initiative; (c) the final period, Allies attacked while Germans defended. In this paper, we apply Lanchester's Square Law to estimate the casualties by determining the shifting time between attack and defense. We obtain improved goodness of fit for the historical data.

Dynamic Noncooperative Group Formation with Size Advantage and Group Reputation

We analyze a noncooperative dynamic game of group formation with group reputation. It is inspired by observations of adoption patterns in some retail markets. Firms are partitioned into name groups, or affiliation networks, in each period. In a group, all firms share the same group reputation among consumers. After a failed firm exits, an entrant firm chooses which name to adopt or to create a new name. There is no cost of choosing any name, but group size and history matter. Existing firms cannot change names nor prevent an entrant from joining their group, and thus the name choice at the time of entry is a long-term choice.Consumers observe only failure events and not firm actions. Every period there is a small turnover of consumers and newcomers do not have information about group histories. If firm groups have the same record to a consumer, larger groups are much more attractive than smaller groups, which we model as following Lanchester's Square Law (Lanchester, 1916).Empirical patterns show synergy effects: entrants tend to choose one of the existing names, and particularly the biggest untainted name. Second biggest group is not chosen when the biggest group is untainted.We found that, in customer-efficient equilibria, where firm failures are rare because all firms make effort constantly, new name creation does not occur when there is an untainted name group to join. However, a universal name group (that all firms have the same name) is not an absorbing state. Eventually there is a failure and after that a creation of a new name occurs. These are consistent with the empirical patterns. The dynamic process of group structure is recurrent among two name groups and a universal name group, in our stationary size market.

Modeling Choices in Nuclear Warfighting

Two classroom simulations—SUPERPOWER CONFRONTATION and MULTIPOLAR ASIAN SIMULATION—are used to teach and test various aspects of the Borden versus Brodie debate on the Schelling versus Lanchester approach to nuclear conflict modeling and resolution. The author applies a Schelling test to segregate high from low empathic students, and assigns them to hard case positions in three simulations to test whether high empathy students can engage in tactic bargaining and whether low empathetic students are necessarily as escalation prone. He has a bipolar nuclear simulation that is an easy case for the Brodie set of assumptions about nuclear war, avoidance, and Schelling-esque tacit bargaining. He expects the system structure and high empathy leader selection to contain escalation, despite the temptation of relying on accelerated Single Integrated Operational Plan solutions and the counterincentive of diminished tacit bargaining through decapitation attacks. The second simulation is a multipolar nuclear simulation set in the near future of Asia, and emulates the Borden-esque logic of nuclear war as artillery exchanges, with a Lanchester square law logic encouraging rapid escalation, coupled with a selection for the most autistic leadership. The author expects rapid nuclear escalation under these structural and decision-making conditions. His conclusions are anecdotal, but seem to indicate, from student feedback during class discussions, that the failure to model fear may be a factor in undermining successful tacit bargaining by players, suggesting that Borden rather than Brodie better conceptualized nuclear conflict. Therefore, peace is about restraining war initiation, as there are great pressures for escalation once war is initiated.

Differential Game Model and Its Solutions for Force Resource Complementary via Lanchester Square Law Equation

AbstractThis paper investigates the warfare command game problem on both sides with force resource complementary in the fixed time. Under some moderate assumptions, force resource complementary differential game is proposed, which ensures that the optimal strategies are obtained based on Lanchester square law equation, and the optimum conditions are given by the quantitative analysis. In analyzing and solving the optimal strategies of an numerical example, feasibility and effectiveness of the model and the approach is demonstrated. The research results may provide a theoretical reference for combat command decision making and games.

Optimal resource allocation and redistribution strategy in military conflicts with Lanchester square law attrition

AbstractWe address the problem of optimal decision‐making in conflicts based on Lanchester square law attrition model where a defending force needs to be partitioned optimally, and allocated to two different attacking forces of differing strengths and capabilities. We consider a resource allocation scheme called the Time Zero Allocation with Redistribution (TZAR) strategy, where allocation is followed by redistribution of defending forces, on the occurrence of certain decisive events. Unlike previous work on Lanchester attrition model based tactical decision‐making, which propose time sequential tactics through an optimal control approach, the present article focuses on obtaining simpler resource allocation tactics based on a static optimization framework, and demonstrates that the results obtained are similar to those obtained by the more complex dynamic optimal control solution. Complete solution for this strategy is obtained for optimal partitioning of resources of the defending forces. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008

A non-linear inverse Lanchester square law problem in estimating the force-dependent attrition coefficients

A non-linear inverse Lanchester square law problem using the Conjugate Gradient Method (CGM) , i.e. the iterative regularization method, is examined in this study to estimate the unknown force-dependent attrition coefficients in a general reinforcement schedules by using the observed temporal force strengths. The numerical simulations are performed to test the validity of the present non-linear inverse algorithm by using different types of attrition coefficients. Results show that the advantages of applying the CGM in the inverse calculations lie in that (i) the initial guesses of the attrition coefficients can be chosen arbitrarily and (ii) the force-dependent attrition coefficients can be estimated in a very short computer time.

The Lanchester (n, 1) problem

We discuss the background of the Lanchester (n, 1) problem, in which a heterogeneous force of n different troop types faces a homogeneous force. We also present a more general set of equations for modelling this problem, along with its general solution. As an example of the consequences of this model, we take the (2, 1) case and solve for the optimal force allocation and fire distribution in a (2, 1) battle. Next, we present examples that demonstrate our model's advantages over a previous formulation. In particular, we point out how different forces may win a battle, depending on the handling and interpretation of the model's solution. Lastly, we present a variant of the Lanchester square law which applies to the (2, 1) case.

Fitting Lanchester's square law to the Ardennes Campaign

In this research, we try to improve Bracken's and Chen's work to significantly better fit our extended Lanchester model into the Ardennes Campaign live data. Essentially, we adopt the concepts of the tactical factor variable and the shift time variable to improve the original Lanchester's model. Moreover, we use the Lanchester square law model instead of Lanchester linear law model to reflect the fact that the Ardennes Campaign was not an indirect-fire but a direct-fire combat. According to our numerical experimental result, we improved Bracken's work by 39.26%, and Chen's work by 19.51%. The contribution of this research is that we propose a much better qualitative analysis model for the explanation of modern combat.

An empirical test of Lanchester's square law: mortality during battles of the fire antSolenopsis invicta

Lanchester's models of attrition describe casualty rates during battles between groups as functions of the numbers of individuals and their fighting abilities. Originally developed to describe human warfare, Lanchester's square law has been hypothesized to apply broadly to social animals as well, with important consequences for their aggressive behaviour and social structure. According to the square law, the fighting ability of a group is proportional to the square of the number of individuals, but rises only linearly with fighting ability of individuals within the group. By analyzing mortality rates of fire ants (Solenopsis invicta) fighting in different numerical ratios, we provide the first quantitative test of Lanchester's model for a non-human animal. Casualty rates of fire ants were not consistent with the square law; instead, group fighting ability was an approximately linear function of group size. This implies that the relative numbers of casualties incurred by two fighting groups are not strongly affected by relative group sizes and that battles do not disproportionately favour group size over individual prowess.

Chimpanzees and the mathematics of battle.

Recent experiments have demonstrated the importance of numerical assessment in animal contests. Nevertheless, few attempts have been made to model explicitly the relationship between the relative number of combatants on each side and the costs and benefits of entering a contest. One framework that may be especially suitable for making such explicit predictions is Lanchester's theory of combat, which has proved useful for understanding combat strategies in humans and several species of ants. We show, with data from a recent series of playback experiments, that a model derived from Lanchester's 'square law' predicts willingness to enter intergroup contests in wild chimpanzees (Pan troglodytes). Furthermore, the model predicts that, in contests with multiple individuals on each side, chimpanzees in this population should be willing to enter a contest only if they outnumber the opposing side by a factor of 1.5. We evaluate these results for intergroup encounters in chimpanzees and also discuss potential applications of Lanchester's square and linear laws for understanding combat strategies in other species.

An optimal control problem in determining the optimal reinforcement schedules for the Lanchester equations

An optimal control problem for the Lanchester equations by utilizing the iterative regularization method, i.e. a nonparametric conjugate gradient method (NP-CGM), with adjoint equations is illustrated to determine the optimal reinforcement schedules based on the desired (or required) force strengths at some specified time. The numerical simulations are performed to test the validity of the present algorithm by using three different types of force strength requirements. Results show that the method is powerful enough to yield desired strengths at all points during the battle. Moreover, the advantages of applying the NP-CGM in the optimal control calculations lie in that (i) the initial guesses of the reinforcement schedules can be chosen arbitrarily and (ii) the optimal time-dependent reinforcement schedules can be determined within a very short computer time. Scope and purpose The Lanchester square law has been used as a warfare model for a long time. Many research papers in determining the attrition coefficients can be found in the literature. For instance, Robert L. Helmbold has discussed the inverse problem for the Lanchester equation in estimating the constant attrition coefficients. More recently, Hsi-Mei Chen used a nonparametric conjugate gradient method (NP-CGM) in predicting the time-dependent attrition coefficients and obtained good estimation. However, the discussion regarding the determination of optimal reinforcement schedules for the Lanchester square law has never been found, to the author's best knowledge, in the literature. For this reason, an optimal control algorithm for the Lanchester square law based on the NP-CGM is examined in the present study. The optimal reinforcement schedules are to be defined such that the desired force strengths at any specified time in the battle can be satisfied. Moreover, constraint for the reinforcement schedules is also required. For this reason, the objective function should be defined as the combination of the square of difference between desired and computed forces and the square of reinforcement schedules. Finally, three numerical experiments will be illustrated to show the validity of the present optimal control algorithm in estimating the optimal reinforcement schedules for the Lanchester square law.

An Inverse Problem of the Lanchester Square Law in Estimating Time-Dependent Attrition Coefficients

This paper considers the inverse problem of estimating time-varying attrition coefficients in Lanchester's square law with reinforcements, using observed data on some or all of the battle's strength histories and the reinforcement schedules. The method employed is a nonparametric extension of the parametric conjugate gradient method (P-CGM). We use hypothetical strength histories and reinforcement schedules that are known to be without error at several points in time to illustrate the method. However, the method has application in other circumstances. The problem of estimating the time-dependent attrition coefficients that best fit a set of given strength histories is inherently a nonparametric inverse problem. In this paper we cast it into a nonlinear optimization problem, and show how to solve it numerically by using a nonparametric conjugate gradient method (NP-CGM). Two numerical test cases are provided to illustrate the application of the method.

Attrition models of the Ardennes campaign

This paper revisits the modeling by Bracken [3] of the Ardennes campaign of World War II using the Lanchester equations. It revises and extends that analysis in a number of ways: (1) It more accurately fits the model parameters using linear regression; (2) it considers the data from the entire campaign; and (3) it adds in air sortie data. In contrast to previous results, it concludes by showing that neither the Lanchester linear or Lanchester square laws fit the data. A new form of the Lanchester equations emerges with a physical interpretation. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 1–22, 1998