Virtual Balances Prevent Slim Majorities From Monopolizing Participatory Budgeting

Virtual balances work by dividing a city's total participatory budget equally among all voters, ensuring a 51% majority can only spend 51% of the funds. This mathematical constraint leaves the remaining 49% of the budget strictly reserved for the priorities of minority voters.[1][3]

Most municipalities use a "greedy" or knapsack voting method for their participatory budgets, simply funding the projects that receive the most overall votes until the money runs out. While this sounds democratic, it mathematically allows a slim, coordinated majority to sweep the board.[3]

"During the process of voting one can imagine a situation where 51 percent of the voters choose only all 'blue' projects and 49 percent only the 'green' ones," explains Piotr Skowron from the University of Warsaw.

"Through the standard voting method, the resources in the participatory budget would be allocated to the 'blue' projects," Skowron notes. The 49% minority goes home empty-handed, effectively wasting their votes and discouraging future civic participation.

How Virtual Balances Change the Math

The Method of Equal Shares, developed by researchers including Skowron and Dominik Peters, replaces this winner-take-all system with proportional representation. Instead of treating the budget as a single communal pot, the algorithm assigns every voter an equal virtual share.[1][2]

If a city allocates $1 million and 10,000 residents vote, each voter receives a virtual balance of $100. This money cannot be traded or cashed out; it exists solely as a mathematical constraint within the voting software to cap individual influence.[1]

Voters still cast their ballots exactly as they always have, selecting the projects they want to see funded. The change happens entirely on the backend, where the algorithm uses these virtual balances to determine which projects actually get built.[1][4]

The system begins by identifying the most popular project overall. However, instead of simply paying for it out of the general fund, the algorithm divides the project's cost equally among the specific voters who supported it.[3][4]

The Cost-Sharing Mechanism

When a project is selected, its cost is deducted directly from the virtual balances of its backers. If a $50,000 playground is approved by 1,000 voters, each supporter pays exactly $50 from their virtual account to fund the construction.[3][4]

Crucially, voters who did not support the playground pay nothing. Their virtual balances remain untouched, preserving their full voting power for the projects they actually care about, such as library renovations or bike lanes.[1][3]

The algorithm then moves to the next most popular project and repeats the process. As the majority bloc continues to win projects, their virtual balances steadily deplete until they can no longer afford to fund anything else.[3]

Once the 51% majority exhausts their 51% share of the budget, they are mathematically locked out of the remaining funds. The algorithm bypasses their remaining preferences and begins funding the top choices of the 49% minority.[3]

Guaranteeing Proportional Representation

This mechanism ensures what computer scientists call "extended justified representation." In practical terms, it means that any cohesive group comprising exactly 10% of the electorate is mathematically guaranteed to control exactly 10% of the total budget.[3]

By capping the spending power of any single coalition, virtual balances prevent popular project categories from monopolizing the ballot. A highly organized neighborhood cannot drain the citywide budget at the expense of less coordinated districts.[1][3]

The system also naturally handles overlapping preferences. If a voter supports both a universally popular transit project and a niche community garden, their virtual balance will contribute to both, provided they have enough funds remaining.[1][4]

If a voter cannot afford their equal share of a project's cost because their balance is too low, they contribute whatever they have left. The remaining cost is then divided among the project's other supporters who still hold sufficient funds.[3][4]

Real-World Implementation and Results

The Method of Equal Shares is not just a theoretical exercise; it is actively reshaping civic technology. Cities like Aarau in Switzerland and Wieliczka in Poland have already implemented the algorithm for their participatory budgeting elections.

In these municipalities, the transition required no changes to the voter experience. Residents simply checked boxes on a digital ballot, while the open-source algorithm handled the complex cost-sharing calculations behind the scenes.[1]

Data from these real-world deployments shows a marked increase in overall voter satisfaction. Because the budget is distributed proportionally, a significantly larger fraction of the electorate sees at least one of their chosen projects fully funded.[1]

Researchers analyzing past voting data from Warsaw found that applying virtual balances would have systematically reduced the bias toward dominant project types. Road infrastructure would receive less disproportionate funding, elevating underrepresented categories like education and green spaces.[1]

The Future of Civic Technology

As participatory budgeting grows globally, the demand for fairer aggregation rules is accelerating rapidly. The traditional greedy algorithm is increasingly viewed as a legacy system that fails to capture the nuance of diverse, modern urban populations.[2]

Virtual balances offer a scalable, mathematically rigorous alternative that builds trust in democratic institutions. When citizens know their votes directly control a specific fraction of the budget, they are more likely to engage with the process.[1][2]

The European Research Council recently awarded a prestigious grant to Piotr Skowron's team to further develop these proportional algorithms. This funding signals strong institutional support for upgrading democratic decision-making tools across the continent and beyond.

By simply changing how votes are counted on the backend, cities can transform a majoritarian popularity contest into a genuinely collaborative budgeting process. The algorithm ensures that every demographic weight translates directly into purchasing power, leaving no voter empty-handed.[5]