lanchester’s laws in space. overview introduce lanchester’s laws show a game demo go through the...
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Lanchester’s Laws in Space
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Overview• Introduce Lanchester’s Laws• Show a game demo• Go through the math• How to apply to game design
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Lanchester’s Laws• Devised by Frederick Lanchester in WW I• Describe the relative strengths of two fighting
armies in a war of attrition.
What’s better: more troops or bigger guns?
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“…a man with a firepower of 2X is not worth twice the value of a man with firepower X, but rather as much, after taking into account one hit on either kills just as dead.”
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See AlsoThe Designer's Notebook: Kicking Butt by the Numbers: Lanchester's Laws– Ernest Adams
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Game Demo
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Lightspeed Brigade• Lots of different tradeoffs between firepower,
hitpoints and # of units.
• Need to balance units, but also create R/P/S relationship.
• Leave room for future content expansion.
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Lanchester’s LawsModel relationship between numerical advantage (N) and firepower advantage (F)
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Example: Fighters vs. NinjasSuppose:• Everyone has 10 hitpoints• Ninjas do 1 hitpoint per round.• Fighters do 10 hitpoints per round.
How many Ninjas can a Fighter defeat?
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Case 1: Thunderdome
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Case 2: Gun Fight
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Case 1: ThunderdomeNinja HP
1 2 3 4 5 6 7 8 9 10 Fighter HP
Start 10 10 10 10 10 10 10 10 10 10 10
Round 1 10 10 10 10 10 10 10 10 10 9
Round 2 10 10 10 10 10 10 10 10 8
Round 3 10 10 10 10 10 10 10 7
Round 4 10 10 10 10 10 10 6
Round 5 10 10 10 10 10 5
Round 6 10 10 10 10 4
Round 7 10 10 10 3
Round 8 10 10 2
Round 9 10 1
Round 10 0 0
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Case 1: ThunderdomeEquilibrium:
(Lanchester’s Linear Law)
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The Math:Ninjas Fighters
Damage (kills/round)
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The Math:Ninjas Fighters
Damage (kills/round)
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The Math:Ninjas Fighters
Damage (kills/round)
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The Math:Ninjas Fighters
Damage (kills/round)
Army Size
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The Math:Ninjas Fighters
Damage (kills/round)
Army Size
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The Math:Ninjas Fighters
Damage (kills/round)
Army Size
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The Math:Ninjas Fighters
Damage (kills/round)
Army Size
Casualties Per Round
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The Math:Ninjas Fighters
Damage (kills/round)
Army Size
Casualties Per Round
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The Math:Ninjas Fighters
Damage (kills/round)
Army Size
Casualties Per Round
% Army Lost per Round
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The Math:Ninjas Fighters
Damage (kills/round)
Army Size
Casualties Per Round
% Army Lost per Round
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% Army Lost per Round
The Math:
=
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The Math:
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The Math:
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The Math:
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• It could be multiple Thunderdomes• 1-on-1 fighting is the important property
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Case 2: Gun Fight • Combatants gang up for maximum attrition.• No Overkill• No “Underkill”
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4 Ninjas vs. 1 FighterNinja HP
1 2 3 4 Fighter HPStart 10 10 10 10 10
Round 1 10 10 10 6Round 2 10 10 3Round 3 10 1Round 4 0 0
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32 Ninjas vs. 10 FighterNinjas Fighters
Start 32 10Round 1 22 6.8Round 2 15 4.6Round 3 10 3.1Round 4 6 2.1Round 5 3 1.5Round 6 1 1.2Round 7 0 1.1
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One More NinjaNinjas Fighters
Start 32 10Round 1 22 6.8Round 2 15 4.6Round 3 10 3.1Round 4 6 2.1Round 5 3 1.5Round 6 1 1.2Round 7 0 1.1
Ninjas FightersStart 33 10Round 1 23 6.7Round 2 16 4.4Round 3 11 2.8Round 4 8 1.7Round 5 6 0.9Round 6 5 0.3Round 7 4 0
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Case 2: Gun FightEquilibrium:
(Lanchester’s Square Law)
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The Math:Ninjas Fighters
Weapon Damage (kills/sec)
Army Size
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The Math:Ninjas Fighters
Weapon Damage (kills/sec)
Army Size
Total Kills Per Second
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The Math:Ninjas Fighters
Weapon Damage (kills/sec)
Army Size
Total Kills Per Second
% Army Lost per Second
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The Math:
% Army Lost per Second =
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The Math:
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The Math:
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The Math:
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The Math:
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Lanchester’s Laws
• Thunderdome
• Gun Fight
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Point Value of a Unit• How many Ninjas is a unit worth?• Solve for N.
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Point Value of a Unit
• Thunderdome
• Gun Fight
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Variable Hitpoints
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Variable Hitpoints
• Thunderdome
• Gun Fight – 2 guys can beat 2x firepower OR 2x health– But not both.
• Works for heterogeneous armies.
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Two extremes
• Thunderdome
• Gun Fight
• In General
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k represents inefficiencySources of inefficiency:• Limited range and/or movement• Limited space• Overkill• Underkill
Wikipedia: Modern calculations use k =⅔
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Possibly two different k values
• Point Value
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Factors that Don’t Affect Efficiency• Multipliers on F– fire rate– to-hit chance
• Multipliers on H– dodge chance– damage reduction
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Factors that Don’t Fit the Model• Healing/Regeneration• Per-hit damage thresholds• Etc.
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Back to Lightspeed Brigade• Different ship sizes and ship costs• Rock/Paper/Scissors relationship
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Ship Values
Ship Cost Fire Rate DamageHit
Points
Fighter 0.2 1 1 5
Cruiser 2 3 5 35
Dreadnought 3 0.45 35 75
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Point Values (k = .5)
Ship Cost Fire Rate DamageHit
PointsPoint Value PV/Cost
Fighter 0.2 1 1 5 2.2 11.2
Cruiser 2 3 5 35 22.9 11.5
Dreadnought 3 0.45 35 75 34.4 11.5
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Point Values (k = 2/3)
Ship Cost Fire Rate DamageHit
PointsPoint Value PV/Cost
Fighter 0.2 1 1 5 2.9 14.6
Cruiser 2 3 5 35 65.1 32.5
Dreadnought 3 0.45 35 75 111.7 37.2
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Creating Rock/Paper/Scissors• Create match-ups that are inefficient for each
side.
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Overkill
Ship Cost Fire Rate DamageHit
Points
Fighter 0.2 1 1 5
Cruiser 2 3 5 35
Dreadnought 3 0.45 35 75
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Range
Ship Cost Fire Rate DamageHit
Points Range
Fighter 0.2 1 1 5 50
Cruiser 2 3 5 35 65
Dreadnought 3 0.45 35 75 100
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AI changes were needed• Dreadnoughts: Don’t bother shooting at
fighters. • Cruisers: Change targets quickly. Only steer at
one target.
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Explicit R/P/S Mechanics Avoided• One less rule to learn • Leave room for future expansion
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If it’s R/P/S, why do Point Values matter?
• You could end up with a “lopsided” R/P/S: – Rock beats Scissors handily– But Paper beats Rock just barely.
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To Sum Up• “Thunderdome” and “Gun Fight” have perfect
solutions.• Your game lies between them. • Lanchester’s laws can give you a head start on
point values. • Still no substitute for playtest. • Spreadsheets are awesome.
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Questions?