Pentagon Nash Equilibrium

The Scoring Formula

The core mechanic of 5xFaction is the Pentagon Nash Equilibrium scoring system.

Score = Σ(Target TVLs) - Σ(Predator TVLs)

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Mathematical Definition

For any faction F:

Variable	Definition
Targets(F)	Set of 2 factions that F defeats
Predators(F)	Set of 2 factions that defeat F
TVL(X)	Total Value Locked in faction X
Score(F)	Net advantage of faction F

Score(F) = TVL(Targets[0]) + TVL(Targets[1]) - TVL(Predators[0]) - TVL(Predators[1])

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Complete Scoring Table

Faction	Formula	Score
SHADOW	(Spirit + Wind) - (Blade + Pillar)	TVL₃ + TVL₅ - TVL₂ - TVL₄
BLADE	(Shadow + Pillar) - (Spirit + Wind)	TVL₁ + TVL₄ - TVL₃ - TVL₅
SPIRIT	(Blade + Wind) - (Shadow + Pillar)	TVL₂ + TVL₅ - TVL₁ - TVL₄
PILLAR	(Shadow + Spirit) - (Blade + Wind)	TVL₁ + TVL₃ - TVL₂ - TVL₅
WIND	(Blade + Pillar) - (Shadow + Spirit)	TVL₂ + TVL₄ - TVL₁ - TVL₃

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Visual Example

Winner: BLADE with score of +60

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Nash Equilibrium Properties

Zero-Sum Nature

The sum of all faction scores always equals zero:

Score(SHADOW) + Score(BLADE) + Score(SPIRIT) + Score(PILLAR) + Score(WIND) = 0

This ensures the system is balanced — for every advantage, there's an equal disadvantage.

Dynamic Equilibrium

As TVL shifts, optimal faction choice changes:

No Dominant Strategy

Unlike rock-paper-scissors with 3 options, the 5-faction pentagon creates:

• More complex strategy space

• Multiple viable counter-strategies

• Richer metagame evolution

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Strategic Implications

1. Contrarian Value

When a faction is popular (high TVL), its predators gain automatic score advantage.

2. Minority Advantage

Backing an unpopular faction means fewer predators are attracted to counter you.

3. TVL Momentum

Large TVL swings during the 2-day deposit phase can flip optimal strategy entirely.

4. Game Theory Application

Skilled players must model:

• Other players' likely choices

• Second-order effects (what others think others will choose)

• Time-weighted deposit patterns

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The 7-Day Cycle Strategy

Phase	Duration	Strategic Focus
Deposit Phase	2 days	Analyze, stake, adjust
Lock Phase	5 days	Hold position, yield generates
Resolution	End of Day 7	Winner determined, yield distributed

The extended 5-day lock phase means:

• More yield accumulates

• Commitment matters more

• Strategic decisions have larger impact