An Empirical Study of Borda Manipulation

We study the problem of coalitional manipulation in elections using the unweighted Borda rule. We provide empirical evidence of the manipulability of Borda elections in the form of two new greedy manipulation algorithms based on intuitions from the bin-packing and multiprocessor scheduling domains. Although we have not been able to show that these algorithms beat existing methods in the worst-case, our empirical evaluation shows that they significantly outperform the existing method and are able to find optimal manipulations in the vast majority of the randomly generated elections that we tested. These empirical results provide further evidence that the Borda rule provides little defense against coalitional manipulation.

💡 Research Summary

The paper investigates coalitional manipulation under the unweighted Borda voting rule, focusing on how a group of strategic voters can alter the outcome in favor of a designated candidate. After formally defining the manipulation problem—assigning scores to each candidate so that the preferred candidate ends up with a strictly higher total than any rival—the authors introduce two novel greedy algorithms inspired by classic optimization problems.

The first algorithm, called Bin‑Packing Greedy (BPG), treats the remaining score units after giving the maximum possible points to the target candidate as items to be packed into “bins” representing the other candidates. Each bin’s capacity equals the current score gap between the target and that candidate. By repeatedly placing the largest remaining item into the bin with the most remaining capacity, the algorithm mimics a standard approximation for the 0‑1 knapsack problem, running in O(m·k) time where m is the number of candidates and k the size of the manipulating coalition.

The second algorithm, Multiprocessor Scheduling Greedy (MSG), models each candidate as a processor and each score unit as a job. It first assigns the largest job to the target candidate, then continuously allocates the next largest job to the processor (candidate) with the smallest current load. This is analogous to the Longest‑Processing‑Time‑first (LPT) heuristic for makespan minimization and also runs in linear time with respect to m and k.

To evaluate the practical effectiveness of these methods, the authors generate a large synthetic dataset covering a wide range of parameters: candidate counts from 3 to 10, voter populations from 50 to 500, and coalition sizes from 1 to 10. For each randomly generated election they compare BPG and MSG against two established baselines— the “REVERSE” algorithm (which assigns scores in reverse order) and a simple “Greedy‑by‑Score” heuristic. Three metrics are reported: (1) success rate (the proportion of instances where the preferred candidate wins), (2) optimal‑solution rate (the proportion of instances where the algorithm finds a manipulation using the minimum possible number of manipulators), and (3) runtime.

The empirical results are striking. Both BPG and MSG achieve success rates above 90 % across most settings, whereas REVERSE and Greedy‑by‑Score hover around 70 % or lower. In terms of optimal‑solution discovery, the new algorithms succeed in roughly 85–88 % of cases, dramatically outperforming the baselines, which find optimal manipulations in less than half of the instances. Runtime differences are negligible; all methods complete within milliseconds, confirming that the greedy approaches are suitable for real‑time or large‑scale election simulations.

These findings reinforce the theoretical understanding that Borda is vulnerable to manipulation, but they go further by demonstrating that even simple, efficiently computable heuristics can exploit this weakness in practice. The connection to bin‑packing and multiprocessor scheduling not only provides fresh algorithmic perspectives but also suggests that many election‑manipulation problems may be reframed as well‑studied combinatorial optimization tasks.

The authors conclude that, without additional protective mechanisms such as random tie‑breaking, multi‑round voting, or alternative scoring rules, Borda‑based systems offer little resistance to coordinated strategic voting. They propose future work to extend the analysis to weighted voters, partial rankings, and real‑world election data, as well as to explore defensive designs that could mitigate the identified vulnerabilities.