Nash Equilibrium Analysis

Theorem 8.1 (Honest Equilibrium): Under the condition α > R/Stake, all validators voting honestly constitutes a Nash equilibrium.

Proof:

Assume other validators vote honestly; consider validator vᵢ's optimal response:

Case 1: vᵢ votes honestly

• Expected utility: E[u_honest] = p·R (p is correct judgment probability)

Case 2: vᵢ deviates from honest strategy

• Short-term gain: May avoid some loss

• Long-term loss: Slashed α·Stake after 3 consecutive deviations

Expected utility comparison:

E[u_deviate] = q·R - (1-q)³·α·Stake

When α > R/Stake and q < 0.9:
E[u_honest] > E[u_deviate]

Therefore, honesty is the optimal strategy, and all validators being honest constitutes Nash equilibrium. ∎

Corollary 8.2 (Slashing Parameter Optimization): The optimal slashing ratio is:

α* = argmin_{α} [False_positive_rate + False_negative_rate]
subject to: α ≥ R/Stake

Through numerical simulation (Section 12.3), we find α* ≈ 1% is optimal in most cases.