# Game-Theoretic Equilibrium Analysis for Web4 Trust Systems

**Version**: 1.0
**Date**: 2026-02-05
**Author**: Autonomous Research Session
**Status**: Research Draft

---

## Executive Summary

This document formalizes the game-theoretic foundations of Web4's trust and reputation systems. The core question: **Is honest behavior a Nash equilibrium?**

Current analysis suggests: **YES, under current parameters**, but the equilibrium is sensitive to detection effectiveness and depends on specific conditions that need ongoing validation.

---

## 1. The Trust Game: Formal Model

### 1.1 Players and Strategies

**Players**: N agents participating in the Web4 ecosystem, each identified by an LCT.

**Strategy Space** S for each agent:
- **H (Honest)**: Execute actions truthfully, build reputation through genuine quality
- **M (Malicious)**: Execute various attack strategies (Sybil, reputation gaming, collusion)
- **G (Greedy)**: Optimize for short-term gain without explicit malice
- **S (Strategic)**: Long-con patient adversary, builds trust then exploits

**Information Structure**:
- Imperfect information: Agents cannot directly observe others' types
- Asymmetric information: Attackers know their strategy, defenders must infer
- Signals: Reputation (T3/V3), coherence metrics, behavioral fingerprints

### 1.2 Payoff Functions

Let π(s_i, s_{-i}) be the payoff to agent i given their strategy s_i and others' strategies s_{-i}.

**Honest Agent Payoffs**:
```
π_H = R_base + R_reputation × T3_score - C_effort
    = 6000 + 6000 × 0.8 - 2000
    = 8800 ATP/month (certain)
```

**Malicious Agent Payoffs** (Expected Value):
```
π_M = P(undetected) × G_attack - P(detected) × L_penalty - C_setup
    = 0.3 × 50000 - 0.7 × 75000 - 5000
    = 15000 - 52500 - 5000
    = -42500 ATP/month (expected)
```

Where:
- R_base = Base reward for participation
- R_reputation = Reputation-gated rewards
- T3_score = Trust tensor composite score
- C_effort = Honest effort cost
- P(detected) = Detection probability (currently ~70%)
- G_attack = Gain from successful attack
- L_penalty = Loss if detected (stake + reputation damage)
- C_setup = Attack setup cost

### 1.3 Game Type

This is a **repeated Bayesian game** with:
- Incomplete information (agent types unknown)
- Signaling (reputation signals type)
- Enforcement mechanism (detection + penalties)
- Long-term relationships (repeated interactions affect future payoffs)

---

## 2. Nash Equilibrium Analysis

### 2.1 Pure Strategy Equilibrium

**Proposition 1**: Honest is a Nash equilibrium when:
```
π_H > max(π_M, π_G, π_S)
```

Given current parameters:
```
π_H = 8,800 ATP/month
π_M = -42,500 ATP/month
π_G = variable, typically < π_H due to reputation decay
π_S = long-term negative (patience doesn't change underlying economics)
```

**Result**: Honest is weakly dominant at current parameters.

### 2.2 Mixed Strategy Equilibrium

In a mixed strategy equilibrium, agents randomize between strategies. The indifference condition:
```
π_H = p × G_attack - (1-p) × L_penalty - C_setup
```

Solving for p (probability of non-detection needed for indifference):
```
8800 = p × 50000 - (1-p) × 75000 - 5000
8800 = 50000p - 75000 + 75000p - 5000
88800 = 125000p
p = 0.71 (71% non-detection)
```

**Result**: If detection drops below ~29%, malicious becomes profitable. Current detection at 70% provides significant margin.

### 2.3 Equilibrium Stability

The equilibrium is **stable** if small deviations return to equilibrium:

1. **Self-enforcing**: Detection of attackers → penalties → reputation damage → fewer attackers → less value in attacking
2. **Network effects**: More honest participants → better detection via witnessing → higher P(detected)
3. **Reputation memory**: Past malicious behavior creates permanent record

**Instability conditions**:
- Detection drops significantly (parameter shift)
- Coordinated attack overcomes diversity requirements
- System-wide shock (e.g., oracle failure)

---

## 3. Attack-Specific Analysis

### 3.1 Sybil Attack Game

**Players**: Attacker vs System (represented by detection mechanism)
**Attacker strategy**: Create N fake identities to inflate influence
**System strategy**: Detection via hardware binding, behavioral correlation, diversity requirements

**Payoff Matrix**:
```
                    | System: Weak Detection | System: Strong Detection |
Attacker: Small N   |    (+small, -small)    |     (-setup, +caught)     |
Attacker: Large N   |    (+large, -large)    |     (-large, +caught)     |
```

**Current parameters**:
- N=5 Sybils: Setup cost 75,000 ATP, Expected gain 0 (100% detection)
- Hardware binding raises floor significantly
- **Equilibrium**: Don't create Sybils

### 3.2 Collusion Ring Game

**Players**: K potential colluders forming coalition
**Mutual benefit**: Cross-validate to inflate reputation
**Detection**: Behavioral correlation, graph analysis

**Coalition stability condition** (from coalition game theory):
```
v(coalition) > Σ v(individual_i) + coordination_cost
```

Where v() is the value function. Collusion is stable when joint gain exceeds sum of individual honest gains plus coordination cost.

**Current parameters**:
- Coordination cost: High (need secure channel, identity risk)
- Detection: Behavioral correlation catches ~80% of collusion rings
- **Equilibrium**: Collusion unstable at N < 10 participants

### 3.3 Long-Con (Patient Adversary) Game

This is a multi-stage game:
1. **Building phase**: Invest in legitimate reputation (cost C_build)
2. **Exploitation phase**: Cash out via malicious action (gain G_exploit, risk of losing everything)

**Present value calculation** (discount rate δ):
```
NPV_longcon = -C_build × (1 + δ + δ² + ... + δ^T) + δ^T × [P(undetected) × G_exploit]
```

For this to beat honest:
```
NPV_longcon > NPV_honest = R_honest × (1 + δ + δ² + ...)/(1-δ)
```

**Key insight**: Long-con is defeated by:
1. Trust velocity limits (can't build too fast)
2. Behavioral authenticity analysis (robotic patterns detectable)
3. Trust decay (built reputation decays if not maintained)
4. One-shot exploitation detection (sudden betrayal is suspicious)

**Current parameters** (with T=12 months building, δ=0.95):
```
NPV_longcon ≈ -100,000 + 0.54 × 0.25 × 500,000 = -32,500 ATP
NPV_honest ≈ 8,800 × 20 = 176,000 ATP (over same horizon)
```

**Equilibrium**: Long-con is NPV-negative

---

## 4. Mechanism Design Insights

### 4.1 Optimal Stake Levels

The stake S must satisfy:
```
P(detected) × S > (1 - P(detected)) × G_attack
S > G_attack × (1 - P(detected)) / P(detected)
```

At P(detected) = 0.7, G_attack = 50,000:
```
S > 50,000 × 0.3 / 0.7 = 21,429 ATP minimum
```

Current stakes (75,000 ATP base) provide 3.5x safety margin.

### 4.2 Detection Investment ROI

Let C_detect be cost of detection mechanism. Optimal investment satisfies:
```
∂(Total System Value) / ∂(C_detect) > 0
```

Where:
```
Total Value = Σ(honest rewards) - Σ(attack damages × P(undetected)) - C_detect
```

Marginal value of detection:
```
dV/dC = attack_damages × dP(detected)/dC - 1
```

Detection is worth investing in while marginal damage reduction exceeds cost.

### 4.3 Penalty Structure Design

**Principle**: Penalties must exceed expected gains, adjusted for detection probability.

**Formula**:
```
Penalty > Expected_Gain / P(detected)
       > 50,000 / 0.7
       > 71,429 ATP
```

Current 75,000 ATP satisfies this condition.

**Graduated penalties** based on attack severity:
- Minor gaming: 1× base penalty
- Reputation manipulation: 2× base penalty
- Sybil attack: 3× base penalty
- Coordinated attack: 5× base penalty + collateral to sponsors

---

## 5. Equilibrium Conditions Summary

### 5.1 Necessary Conditions for Honest Equilibrium

| Condition | Current Value | Threshold | Status |
|-----------|---------------|-----------|--------|
| Detection rate | 70% | > 29% | ✅ OK |
| Stake/Gain ratio | 1.5× | > 1× | ✅ OK |
| Collusion ring detection | 80% | > 50% | ✅ OK |
| Long-con NPV | negative | < 0 | ✅ OK |
| Diversity requirement | 3+ platforms | ≥ 3 | ✅ OK |

### 5.2 Sufficient Conditions (Stronger)

For robust equilibrium:
1. Detection rate > 50% (current: 70%)
2. Stake > 1.5 × Expected gain (current: 1.5×)
3. Trust velocity < natural growth rate
4. Reputation decay > 0 for inactive agents
5. Penalty contagion to sponsors

### 5.3 Monitoring Indicators

Track these metrics to detect equilibrium drift:
- Detection rate (monthly rolling)
- Successful attack count
- Collusion ring formations
- Average reputation velocity
- Stake adequacy ratio

---

## 6. Attack Vector → Equilibrium Mapping

| Attack Vector | Equilibrium Impact | Defense Mechanism | Margin |
|---------------|-------------------|-------------------|--------|
| Sybil | Increases attack utility | Hardware binding | 3× |
| Reputation gaming | Increases G_attack | Velocity limits | 2× |
| Collusion | Reduces P(detected) | Behavioral correlation | 1.5× |
| Long-con | Time-shifted gain | Trust decay, authenticity | 5× |
| Eclipse | Network partition | Diversity requirements | 2× |
| Governance interface (DU) | Process exploitation | TCB, BOI matching | 1.2× |

---

## 7. Open Questions and Future Research

### 7.1 Unsolved Problems

1. **Coalition Formation Dynamics**: When do N rational agents form a cartel?
2. **Information Cascade Effects**: How does one successful attack change beliefs?
3. **Cross-System Arbitrage**: How do multi-system attacks affect equilibrium?
4. **Behavioral Economics**: How do cognitive biases affect strategy choice?

### 7.2 Recommended Empirical Studies

1. **Agent-Based Simulations**: Run 10,000+ agent simulations with mixed strategies
2. **Historical Attack Analysis**: Study real attack patterns from blockchain data
3. **Parameter Sensitivity**: Map equilibrium boundaries across parameter space
4. **Adversarial Red Team**: Have skilled attackers attempt to break equilibrium

### 7.3 Formal Verification Goals

1. **TLA+ Specification**: Encode game as TLA+ and verify safety properties
2. **Z3 Proof**: Prove Nash equilibrium formally with SMT solver
3. **Coq Formalization**: Machine-verified proof of equilibrium existence

---

## 8. Conclusion

The Web4 trust system exhibits a **robust Nash equilibrium favoring honest behavior** under current parameters. The equilibrium is:

- **Self-enforcing**: Detection mechanisms create feedback loops
- **Stable**: Small perturbations return to equilibrium
- **Parametrically bounded**: Specific conditions must hold

Key insight: **The equilibrium is designed, not emergent**. It requires careful calibration of:
1. Detection mechanisms (70%+ effectiveness)
2. Stake levels (1.5×+ expected gains)
3. Penalty structures (graduated, contagious)
4. Trust dynamics (decay, velocity limits)

**Recommendation**: Establish monitoring dashboard for equilibrium health indicators and trigger alerts when any condition approaches threshold.

---

## References

- `/home/dp/ai-workspace/web4/game/ATP_ECONOMIC_ANALYSIS.md`
- `/home/dp/ai-workspace/web4/attack_vectors_reputation_gaming.md`
- `/home/dp/ai-workspace/hardbound/adversarials/TAXONOMY.md`
- Fudenberg & Tirole, "Game Theory" (1991)
- Myerson, "Game Theory: Analysis of Conflict" (1991)
- Roth & Sotomayor, "Two-Sided Matching" (1990)

---

*This document is a living analysis. Update as parameters change or new attacks are discovered.*
