Michael Brenndoerfer | Data & AI, Private Equity, Technology

Michael Brenndoerfer

Multi-Agent Systems › Visualizations

Michael BrenndoerferJanuary 3, 20264 min read

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Part 48 of 67Tool Use and Agents

Chapter 371Multi-Agent Systems › Visualizations

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To concretize the quantitative differences between coordination strategies, let's visualize three aspects: how task completion time scales with parallelism, how error rates compound through pipeline stages, and how agent vote count affects answer accuracy.

Out[16]:

Visualization

Line chart showing wall-clock time vs number of tasks for 1, 3, and 5 parallel agents — Theoretical wall-clock time for multi-agent pipelines as the number of parallel subtasks grows, modeled using Amdahl''s law with 30% serial work (analysis and writing stages). The sequential baseline scales linearly with task count, while adding 3 or 5 parallel agents flattens the growth substantially. The curves demonstrate diminishing returns: going from 1 to 3 parallel agents provides the largest speedup, with additional agents offering progressively smaller gains.

Out[17]:

Visualization

Line chart showing pipeline output accuracy vs number of stages for three per-stage accuracy levels — End-to-end accuracy of multi-stage pipelines as a function of stage count, for three per-stage accuracy levels (90%, 95%, 99%). A 95% per-stage accuracy degrades to roughly 77% by the fifth stage, motivating shallow pipeline design. The 99% curve holds above 90% accuracy through seven stages, showing that investment in per-stage quality pays large dividends in end-to-end reliability.

Out[18]:

Visualization

Line chart of collective voting accuracy vs number of agents for three individual accuracy levels — Collective majority-vote accuracy as a function of agent count for three individual accuracy levels (60%, 70%, 80%), derived from the Condorcet jury theorem. Individual accuracy above 70% rapidly yields collective accuracy above 90% with 5-9 agents, but individual accuracy below 50% makes voting counterproductive. The chart illustrates that collective voting provides the largest gains in the first few agents added, with diminishing returns beyond 9-11 agents.

The three charts above illustrate key quantitative relationships in multi-agent system design. The parallelism speedup chart shows that the gains from adding parallel agents are most dramatic in the early additions (going from 1 to 3 agents), with diminishing returns thereafter due to the serial portion of the work. The error compounding chart makes concrete why shallow pipelines matter: even 95% per-stage accuracy degrades to about 77% accuracy after five stages. The voting accuracy chart applies the Condorcet jury theorem to show that majority voting improves collective accuracy only when individual agents are better than chance, and the improvement saturates after about 9 to 11 agents.