1. The Multiplicative Gamble
Ensemble mean
Median
Individual agents
t=0
E[W]: 1.00
median: 1.00
%broke: 0%
2. Two Growth Rates
Ensemble growth ge
Time-avg growth gt
Zero line
ge = +5.0%
gt = -5.3%
gap: 10.3%
3. Additive (Ergodic) vs Multiplicative (Non-Ergodic)
Additive mean
Additive median
Multiplicative mean
Multiplicative median
Add mean: 1.00
Add med: 1.00
Mul mean: 1.00
Mul med: 1.00
4. Multi-Agent: Diverse vs Correlated
7 diverse agents (independent)
7 correlated agents (same coin)
Single best agent
Diverse: 1.00
Correlated: 1.00
Single: 1.00
The ergodicity gap: Expected value (ensemble average) says this gamble is great: +5% per round.
But YOUR trajectory (time average) shrinks ~5% per round. Over 100 rounds, the mean of 500 players
explodes to ~130x. The median crashes to ~0.005x. A few lucky players hold all the wealth.
This is exactly what happens in multi-agent LLMs: running 7 copies of the same model
(correlated) behaves like a non-ergodic system — one bad reasoning chain drags all of them.
Running 7 diverse models (independent) approaches ergodicity — the ensemble average becomes meaningful.