BV = sqrt(Attack × HP). One number to compare any two units across any game system.
Links: Gaming, Economics, Risk — The Attrition Constant (the Square-Law counterexample — BV’s sqrt(Attack × HP) derives from Lanchester’s Square Law, which assumes force concentration compounds; Risk caps the engagement frontage at 3-vs-2 dice regardless of stack size and so obeys the linear law instead, with a measured zero concentration advantage. The general lesson for BV: where frontage is capped, the Square-Law premise fails — check it before applying BV to any system), The Multiplayer Coalition Problem, YouTube Migration, The Dominance-Frontier Lens (BV/Cost is this lens’s cost-axis for combat units), D&D Monster Tournament — Exact Markov Chains (the proposed ground-truth oracle for BV — an exact pairwise win matrix is the test this metric has never faced; where √(Attack × HP) fails should be predictable from the MoM complications list. Carries the Hercules warning: Chris’s earlier HeroClix Markov study crowned Hercules on BV/cost purely because range and non-combat abilities were unmodelled — so a combat model and BV that share a melee blind spot will agree with each other for bad reasons, and any BV validation needs a mobility-enabled control run), HeroClix (the Hercules warning’s home project — dial data + official rules now confirmed fetchable, so the melee blind spot is testable rather than merely acknowledged)
Most strategy games use the same combat model descended from D&D (and earlier wargaming): a unit has an attack value (damage per tick) and hit points (how much damage it absorbs before dying). Variations add hit rates, armor, resistances, critical hits, and other modifiers — but the core mechanic is the same: deal numerical damage, subtract from HP, dead at zero.
Given two unit types with different Attack and HP values, which one wins? And if they have different costs, which is the better investment?
The answer is not obvious. A unit with 100 Attack and 10 HP is not obviously better or worse than a unit with 10 Attack and 100 HP. Raw stats don’t compare directly because combat effectiveness depends on the interaction of attack and durability.
This combat model has roots much older than D&D. The original formulation modeled groups, not individuals: 100 hit points meant 100 men, and damage represented how many men could be killed in a given timeframe. The individual-unit version in games is an abstraction of the group model — one character “is” a fighting force with aggregate stats.
This matters because the math was originally designed for exactly the comparison problem BattleValue solves: given two armies with different compositions, which one wins?
Setup: Two units fight. Each deals damage per tick equal to its Attack. Each dies when HP reaches zero.
Unit A: Attack = a, HP = h Unit B: Attack = b, HP = k
Time to kill:
A wins if it kills B before B kills it:
k/a < h/b
Cross-multiply:
k·b < h·a
Rearrange:
a·h > b·k
The product Attack × HP determines the winner. The unit with the higher product wins a 1v1.
But raw products aren’t convenient for comparison — they scale quadratically. To get a linear metric:
BV = sqrt(Attack × HP)
BV₁ > BV₂ means unit 1 wins. The ratio BV₁/BV₂ gives the relative combat effectiveness.
If units have a resource cost, the strategically relevant question isn’t “which unit wins 1v1” but “which unit gives more combat power per resource spent”:
Combat Efficiency = BV / Cost
This is the army composition metric. Given a fixed budget, buy the unit type with the highest BV/Cost ratio. This is directly analogous to the economic concept of return on investment — you’re maximizing combat output per unit of input.
The square root isn’t arbitrary — it linearizes the comparison. Without it:
| Unit | Attack | HP | Attack×HP | BV |
|---|---|---|---|---|
| Soldier | 10 | 100 | 1,000 | 31.6 |
| Glass Cannon | 100 | 10 | 1,000 | 31.6 |
| Tank | 5 | 400 | 2,000 | 44.7 |
| Balanced | 50 | 50 | 2,500 | 50.0 |
Soldier and Glass Cannon have equal BV (they tie in a 1v1 — Soldier kills in 0.1 ticks, Glass Cannon kills in 10 ticks… wait, that’s wrong — Glass Cannon kills Soldier in 100/100 = 1 tick, Soldier kills Glass Cannon in 10/10 = 1 tick. They kill each other simultaneously). The square root preserves this symmetry while making the numbers human-readable and linearly comparable.
A unit with BV 50 is not “twice as good” as BV 25 in a 1v1 — it’s four times as good in terms of raw combat product. But BV makes relative comparison intuitive: bigger number wins, ratio tells you by how much.
Frederick Lanchester (1916) formalized this math for WWI military operations research. His Square Law states:
In concentrated-fire combat (where all units in a force can engage simultaneously), the combat power of a force is proportional to the square of its numbers times its individual firepower.
For a force of N identical units each with attack power a:
Combat Power = N² × a
This means doubling your numbers quadruples your combat power — not just doubles it. This is why concentration of force is a military principle: two forces of 50 lose to one force of 100, even though total numbers are equal.
BattleValue is the individual-unit version of Lanchester. Where Lanchester asks “which army wins?”, BV asks “which unit type wins?” — same math, different scale. And BV/Cost is the resource-allocation version: “given a budget, which unit type produces the strongest army?”
Lanchester also described a Linear Law for sequential combat (where only one unit from each side engages at a time — think single-file through a gate). In this case:
Combat Power = N × a
Numbers matter linearly, not quadratically. This maps to different game mechanics — turn-based combat where only the front unit fights, or narrow chokepoints that prevent concentration.
BV applies to the Square Law case — which is the default in most game combat systems (all units in range attack simultaneously).
The individual BV derivation compares two single units. But games involve armies. Does BV still work when N units fight N units?
Yes — and the proof is elegant.
Setup: 50 units of Type A (1 dmg, 100 HP, BV=10) vs 50 units of Type B (2 dmg, 50 HP, BV=10). Equal BV — should tie.
Round-by-round attrition (simultaneous, concentrated fire):
| Round | A units | A total damage | B units killed | B units | B total damage | A units killed |
|---|---|---|---|---|---|---|
| 1 | 50 | 50 × 1 = 50 | 1 (50/50) | 50 | 50 × 2 = 100 | 1 (100/100) |
| 2 | 49 | 49 | 1 | 49 | 98 | 1 |
| 3 | 48 | 48 | 1 | 48 | 96 | 1 |
| … | … | … | … | … | … | … |
| 50 | 1 | 1 | — | 1 | 2 | — |
Both sides lose one unit per round. Both are eliminated simultaneously. Tie confirmed — BV correctly predicted the outcome.
Note the asymmetry: A units deal exactly enough damage each round to kill one B unit (50 damage vs 50 HP), while B units deal exactly enough to kill one A unit (100 damage vs 100 HP). Different stats, same attrition rate — because A×HP is equal.
Each side’s total damage output over the entire battle follows the triangular number pattern — 50 units attack, then 49, then 48:
Total rounds of fire = N + (N-1) + (N-2) + ... + 1 = N(N+1)/2
For N=50: 50 × 51 / 2 = 1,275 unit-rounds.
The aggregate combat product multiplies this by the per-unit combat product (A×HP):
Aggregate = A×HP × N(N+1)/2
For both sides: 100 × 1,275 = 127,500. Equal — tie confirmed from the aggregate as well.
Now compare the aggregate to the naive linear sum:
Naive sum: BV × N = 10 × 50 = 500
Aggregate BV: sqrt(127,500) ≈ 357
Ratio: 357 / 500 ≈ 71.4%
At larger N, the triangular number N(N+1)/2 approaches N²/2:
Aggregate = A×HP × N²/2
sqrt(Aggregate) = sqrt(A×HP) × N/sqrt(2) = BV × N / sqrt(2)
The ratio of aggregate BV to naive linear BV converges to:
(BV × N / sqrt(2)) / (BV × N) = 1/sqrt(2) ≈ 0.7071 = 70.71%
| N | Triangular N(N+1)/2 | Aggregate (×100) | sqrt(Aggregate) | Naive BV×N | Ratio |
|---|---|---|---|---|---|
| 50 | 1,275 | 127,500 | 357.1 | 500 | 71.4% |
| 100 | 5,050 | 505,000 | 710.6 | 1,000 | 71.1% |
| 1,000 | 500,500 | 50,050,000 | 7,074.6 | 10,000 | 70.7% |
| 10,000 | 50,005,000 | 5,000,500,000 | 70,714.2 | 100,000 | 70.71% |
| ∞ | ≈ N²/2 | ≈ A×HP × N²/2 | BV × N/sqrt(2) | BV × N | 1/sqrt(2) |
This IS Lanchester’s Square Law derived from first principles. The square-law scaling (combat power ∝ N²) appears naturally from the triangular attrition pattern. The 1/sqrt(2) factor is the bridge between the individual metric (BV) and the army-scale metric.
The 1/sqrt(2) correction is a constant — it applies equally to both sides. If BV₁ > BV₂ for individual units, then:
BV₁ × N / sqrt(2) > BV₂ × N / sqrt(2)
The constant cancels. BV correctly predicts the winner at every scale. The absolute aggregate values differ from naive BV×N by the sqrt(2) factor, but the relative comparison — which is all that matters for deciding who wins — is preserved identically.
This is why BV works as a universal metric: it captures the combat product that determines outcomes, and the multi-unit scaling is a constant correction that doesn’t change the ranking.
The multi-unit convergence proves BV works for equal forces. The real power shows when forces are unequal — BV gives you exact survival rates.
When two forces of unequal total BV engage, the winner’s surviving combat power is:
Surviving BV = sqrt(Total_BV_A² - Total_BV_B²)
Where:
And the surviving unit count:
Surviving units = Surviving BV / BV_individual
This is a Pythagorean relationship — the two forces’ total BVs form the legs, and the difference of squares gives the surviving power. The sqrt() is what makes this work. With raw A×HP products, you’d need to track quadratic terms that don’t compose. The square root linearizes individual BV so that total BV is additive (BV × N), and the survival formula falls out as a clean difference of squares.
Example 1: Numerical advantage, equal unit types
100 A-units (1 dmg, 100 HP, BV=10) vs 50 B-units (2 dmg, 50 HP, BV=10):
Total BV_A = 10 × 100 = 1,000
Total BV_B = 10 × 50 = 500
Surviving BV = sqrt(1,000² - 500²) = sqrt(750,000) ≈ 866
Surviving A units = 866 / 10 ≈ 87
A 2:1 advantage costs only ~13% losses. This is the Lanchester concentration-of-force principle quantified: doubling numbers doesn’t double your advantage — it quadruples your combat power (because BV² scales with N²), so losses are disproportionately small.
Example 2: Quality vs quantity
30 Elite units (10 dmg, 90 HP, BV=30) vs 100 Militia units (2 dmg, 18 HP, BV=6):
Total BV_Elite = 30 × 30 = 900
Total BV_Militia = 6 × 100 = 600
Surviving BV = sqrt(900² - 600²) = sqrt(450,000) ≈ 671
Surviving Elites = 671 / 30 ≈ 22
30 Elites beat 100 Militia and lose only 8 units. But if Militia cost 1/5 the price of Elites, they have equal BV/Cost — and at equal budget, the Militia win because they field more total BV.
Example 3: The break-even
When Total_BV_A = Total_BV_B, surviving BV = sqrt(0) = 0. Mutual annihilation. This is the tie case from the multi-unit convergence section, confirmed from the survival formula.
The survival formula reveals why certain strategies dominate:
Concentration of force is always correct under Lanchester. Splitting your army in half doesn’t halve your power — it quarters it (each half has (N/2)² = N²/4 of original power). Never fight two battles when you can fight one.
First strike / alpha strike is devastating because units killed before they fire contribute zero BV to the fight. Eliminating 10% of the enemy force before combat starts doesn’t reduce their power by 10% — it reduces it by ~19% (1 - 0.9² = 0.19). This is why initiative, range, and ambush are consistently the strongest mechanics.
Attrition wars favor the larger force non-linearly. A force with 20% more total BV doesn’t win by 20% — it wins overwhelmingly with minimal losses. This is why smart strategy games make comeback mechanics (terrain, morale, logistics) to prevent the snowball.
Cost efficiency (BV/Cost) is king. The survival formula shows that total BV determines outcomes. Given a fixed budget, the player who buys more total BV wins. This makes BV/Cost the single most important number in any strategy game’s unit roster.
The base BV formula assumes the simplest possible combat model. Real games add complications:
If attacks have a probability p of hitting:
Effective Attack = Attack × p
BV = sqrt(Attack × p × HP)
If armor reduces incoming damage by a flat amount r:
Effective HP = HP × (Attack_enemy / (Attack_enemy - r))
This gets circular (your effective HP depends on the enemy’s attack), which is why armor makes comparison harder — BV gives the clean case, and armor requires knowing the matchup.
Area damage multiplies effective attack by the number of targets hit. This is why AoE is disproportionately powerful — it’s a force multiplier that scales with enemy concentration (Lanchester’s Square Law in reverse: the more concentrated the enemy, the more effective your splash).
Healing effectively multiplies HP by extending the time to kill. A unit with 100 HP and 10 HP/tick regeneration against an enemy dealing 15 HP/tick damage has effective HP of:
Effective HP = HP / (1 - heal_rate/incoming_damage) = 100 / (1 - 10/15) = 300
First strike (attacking before the enemy can respond) is a massive force multiplier that BV doesn’t capture. A unit that kills before being attacked has infinite effective HP for that first exchange. This is why range advantages and initiative bonuses are consistently among the strongest mechanics in strategy games.
| Game/Genre | Attack Analogue | HP Analogue | BV Application |
|---|---|---|---|
| D&D/RPGs | DPS (damage per round) | Hit Points | Character build optimization |
| RTS (StarCraft, AoE) | DPS | HP + Armor | Unit composition, build order efficiency |
| 4X (MOO, Civilization) | Ship/unit firepower | Hull/HP | Fleet composition, tech investment |
| Auto-battlers (TFT) | DPS | HP + shields | Team composition, item optimization |
| Card games (MTG) | Power | Toughness | Creature evaluation, trade analysis |
| Wargames | Combat Factor | Steps/strength | Force comparison, the original use case |
The table above is genres; these are the in-vault pages that actually use BV, with what each contributes (the reciprocal of the up-links they already carry):
sqrt(Melee × Hits × Figures); the multi-figure/Lanchester extension; BV/Slot vs BV/Cost dual constraint; the hard-counter matchup-matrix where effective BV drops to 0.BV/Cost is structurally identical to return on investment. The strategy game player faces the same problem as the entrepreneur:
The complication — in both games and markets — is that BV/Cost is only the local metric. The global optimization includes:
The BV formula gives you the floor of analysis. Everything above it is game-specific strategy — but without the floor, you can’t even start comparing.