Operations Research & Strategy
Lanchester’s Square & Linear Laws Formula: Concentration Advantage & Remaining Force
Model competitive market-share and operations-research engagements using Frederick Lanchester’s Square Law (alpha * A_0^2 vs beta * B_0^2), parity threshold, and surviving resource capacity.
Why 2x market concentration delivers 4x competitive power under Lanchester’s Square Law!
Derived by Frederick W. Lanchester (1916) and adapted into Japanese & global market-share strategy by Nobuo Taoka, Lanchester’s two differential equations distinguish fragmented one-on-one competition (Linear Law: alpha * A_0 vs beta * B_0) from concentrated multi-channel competition (Square Law: dA/dt = -beta*B, dB/dt = -alpha*A => alpha * (A_0^2 - A(t)^2) = beta * (B_0^2 - B(t)^2))!
Under the Square Law, numerical scale squares while unit quality/effectiveness enters linearly! Thus Force A (A_0 = 100, alpha = 0.09 -> 900 power) defeats a higher-quality Force B (B_0 = 60, beta = 0.16 -> 576 power) with Exact Surviving Capacity A_final = SQRT(A_0^2 - (beta / alpha) * B_0^2) = SQRT(10,000 - 6,400) = 60 units intact!
| Competitor B Scale (A_0=100, alpha=0.09, beta=0.16) | Square-Law Power Ratio | Outcome & Surviving A Capacity |
|---|---|---|
| B_0 = 60 (Power = 576 vs 900) | 1.5625 (> 1.00) | A Wins with 60.00 surviving units |
| B_0 = 75 (Exact Parity: 0.16*5625 = 900) | 1.0000 | Mutual exhaustion (0.00 remaining) |
| B_0 = 85 (Power = 1,156 vs 900) | 0.7785 (< 1.00) | Insufficient Force A — Below Square-Law Parity |
Surviving Force A Capacity Under Lanchester’s Square Law
Evaluates SQRT(A_0^2 - (beta / alpha) * B_0^2) when alpha * A_0^2 >= beta * B_0^2, or warns when below parity.
=IF(OR(B2 <= 0, B3 <= 0, B4 < 0, B5 <= 0), "Inputs Must Be Positive", IF(B3 * (B2 ^ 2) < B5 * (B4 ^ 2), "Below Square-Law Parity — Adversary Prevails", ROUND(SQRT((B2 ^ 2) - (B5 / B3) * (B4 ^ 2)), 2)))Lanchester Square-Law Competitive Power Ratio (alpha * A_0^2) / (beta * B_0^2)
Measures relative concentrated strength (> 1.00 indicates decisive advantage; >= 3.00 is Taoka’s dominance threshold).
=IF(OR(B4 <= 0, B5 <= 0), 0, ROUND((B3 * (B2 ^ 2)) / (B5 * (B4 ^ 2)), 4))Try the example
Use the original values or change them to test your understanding.