At 60% Shared Context, Five Expert Personas Are Worth Fewer Than Two

"You are a panel of five experts. Debate, then agree." The pitch is five viewpoints for the price of one call.

👉 Live, all numbers recomputed in your browser: https://dev48.in/prompt/day82-multi-persona.html

The personas share a prompt, a context window and a model. Their mistakes correlate. And correlated voices do not average.

The arithmetic

For n personas, each with noise σ and pairwise correlation ρ:

Var(mean) = σ² · ( ρ + (1 − ρ)/n )

The second term shrinks with n. The first does not shrink at all.

The usual way to say this is effective sample size: n_eff = n / (1 + (n−1)ρ).

shared context	measured correlation	effective personas	RMSE measured	RMSE closed form
0.00	0.002	5.00	1.3336	1.3416
0.15	0.152	3.13	1.6878	1.6971
0.35	0.351	2.08	2.0710	2.0785
0.60	0.601	1.47	2.4687	2.4739
0.85	0.850	1.14	2.8117	2.8142
1.00	1.000	1.00	3.0004	3.0000

Five personas, identical competence, no bias — only how much of their reasoning is common changes. At 0.6 they are worth 1.47. The other three are decoration you paid tokens for.

The floor

At 40% shared context:

personas	effective	RMSE	floor at n = ∞
1	1.00	3.0000	1.8974
5	1.92	2.1679	1.8974
15	2.36	1.9596	1.8974
50	2.48	1.9124	1.8974

Going from 5 personas to 50 moves the error by about 0.026. The shared component never averages away, so there is a hard floor of σ√ρ and adding voices buys progressively less, then nothing.

Averaging is not a bias correction

This one is simpler and more often ignored. If every persona is wrong in the same direction — which is exactly what a shared system prompt produces — no number of them helps:

personas	RMSE	the shared bias
1	3.6056	2.0000
5	2.2804	2.0000
50	2.0180	2.0000
1000	2.0001	2.0000

A thousand personas, all two units high, give an answer two units high. Only the mean bias matters — biases that cancel do average away, biases that agree never do.

The aggregation rule you left implicit

"Then agree" is a rule. Here is what it costs when one persona out of five is confidently wrong:

aggregation	panel agrees	one persona 12 off	damage
mean	1.5776	2.8667	1.82×
median	1.7971	2.1429	1.19×
trimmed	1.6498	2.0196	1.22×

The mean hands a confidently wrong voice a full 1/n of the answer. The median barely notices. And when the panel agrees, the mean is the best of the three — there is no rule that wins both ways. That is the choice, and leaving it implicit means it gets made for you.

How it is checked

The simulation uses an LCG with Box-Muller. The verifier uses a different generator (xorshift32 with the Marsaglia polar method), derives Var(mean) algebraically from the covariance matrix rather than from the formula above, and checks n_eff against its definition — the n an independent panel would need for the same variance — instead of against its closed form.

Worst relative gap between the second RNG and the algebra: 0.76%.

One honest caveat: this is a model, not a measurement of any particular LLM. What it establishes is that if personas correlate, the arithmetic is unforgiving — and sharing a context window is the most direct way to make them correlate.

141 verifier asserts, 0 failures. 23 in-page checks.