Day 3: Parallel Map-Reduce Agents & Majority Voting Ensembles

1. First-Principles Math: Condorcet's Jury Theorem for Agent Ensembles

Suppose we run $M$ independent reasoning agents (or diverse prompt/temperature samples) on a high-stakes classification or root-cause identification task, where each individual agent has accuracy $p > 0.5$.

By Condorcet's Jury Theorem, the probability $P_{\text{majority}}(M, p)$ that a majority vote of $M$ (odd) agents is correct is: $P_{\text{majority}}(M, p) = \sum_{k = \lceil M/2 \rceil}^{M} \binom{M}{k} p^k (1 - p)^{M - k}$

Example Calculation ($p = 0.80$)

• Single Agent ($M = 1$): $P = 80.0\%$
• 3-Agent Ensemble ($M = 3$): $P = \binom{3}{2}(0.8)^2(0.2) + \binom{3}{3}(0.8)^3 = 0.384 + 0.512 = 89.6\%$
• 5-Agent Ensemble ($M = 5$): $P = 94.21\%$

2. Map-Reduce over Log Shards

When an incident spans 500,000 log lines across N regional clusters, a single context window overflows.

• Map Phase: Partition logs into $N$ regional shards; run $N$ stateless MapperAgent instances in parallel to extract structured ShardAnomalySummary objects.
• Reduce Phase: A single ReducerAgent aggregates the $N$ compact summaries into a global diagnosis.

In Simple English: Wisdom of Crowds Voting ($P_{\text{majority}}$)

Suppose a complex bug classifier is right $p = 75\%$ of the time on a single try:

• Ask 1 agent once: 75% chance of being right.
• Ask 3 independent agents (M = 3) in parallel (with slight prompt/temperature diversity) and take the majority vote (2 or 3 agree):
  • Chance all 3 are right: $0.75^3 = 42.2%$
  • Chance exactly 2 are right: $3 \times 0.75^2 \times 0.25 = 42.2%$
  • Combined Majority Accuracy: $42.2% + 42.2% = \mathbf{84.4%}$ — nearly +10% higher reliability in the exact same wall-clock latency because all 3 ran at once!