Wager Game Theory
The Fundamental Wagering Inequality
A wager is +EV if and only if:
T₁ / T₂ > V₁ / V₂
Where:
T₁= expected turns for opponent to identify your NFT (your survival time)T₂= expected turns for you to identify opponent's NFT (your attack time)V₁= value of your NFTV₂= value of opponent's NFT
In plain english: Your NFT must be proportionally harder to guess than it is valuable.
Win Probability
Under equal skill:
P(win) = T₁ / (T₁ + T₂)
Your win probability is proportional to how long your NFT survives.
Expected Value
EV = P(win) × V₂ - P(lose) × V₁
= [T₁ × V₂ - T₂ × V₁] / (T₁ + T₂)
Example: The Whale Trap
| P1 (Whale) | P2 (Shark) | |
|---|---|---|
| NFT Value | 10 ETH | 1 ETH |
| GI | 1.6 (Critical) | 0.7 (Low) |
| E[Turns to crack] | 6.25 | 14.29 |
| Win probability | 30.4% | 69.6% |
| EV | -6.66 ETH | +6.66 ETH |
The whale holds a 10x more valuable NFT but loses 70% of the time. The rarity that made it expensive at mint makes it dangerous to wager.
Kelly Criterion for Repeat Wagerers
Optimal fraction of bankroll to risk:
f* = P(win) - P(lose) / b
where b = V_opponent / V_own
Only wager when f* > 0 — which reduces to the fundamental inequality above.