The New Mathematics of Democracy

Key points

• The paper has no original experiments of its own; instead, it interprets theory, real election data, institutional adoption cases, and open research problems across three fields.
• Among about 3,000 IRV and STV election analyses cited in the paper, IRV and Condorcet methods choose the same winner more than 99 percent of the time, although voter behavior could also change if different rules were actually used.
• Budgeted voting has been used in 32 real participatory-budgeting elections, and the equal-share method reduced the share of voters who received no approved project in Wieliczka, Poland, from 28 percent under the prior method to 18 percent.
• Determining whether there exists a citizen panel that satisfies overlapping population quotas is NP-complete, and the related tool Panelot was used on 1,071 citizen assembly lotteries over 294 distinct days in 2024 and 2025.
• The equal-share method can leave budget unused, and proportionality can conflict with total satisfaction, so AI should serve as an aid that translates citizen expression into a verifiable mathematical form rather than replacing decisions.

Paper links

External research summaries. These are not HDATF publications or measured product results.