Learning to Manipulate under Limited Information

TL;DR

This work addresses the problem of evaluating manipulation resistance in voting rules when manipulators have limited information. It introduces a neural-network based framework that trains over $10^5$ multilayer perceptrons of varying sizes to learn profitable manipulation against eight voting methods across three probability models for profiles and six information types, in elections with $n\in[5,21]$ voters and $m\in[3,6]$ candidates. The key findings show that Condorcet methods such as Minimax and Split Cycle are comparatively resistant to learnable manipulation, while Plurality and Borda remain highly manipulable under limited information; IRV-PUT also demonstrates notable resistance though not as strong as Condorcet methods. This approach provides a data-driven, operational measure of manipulation resistance that complements worst-case complexity results and points to practical robustness concerns for small committees, with future work including coalition manipulation, broader electorate sizes, and reinforcement-learning based scaling beyond 6 candidates.

Abstract

By classic results in social choice theory, any reasonable preferential voting method sometimes gives individuals an incentive to report an insincere preference. The extent to which different voting methods are more or less resistant to such strategic manipulation has become a key consideration for comparing voting methods. Here we measure resistance to manipulation by whether neural networks of various sizes can learn to profitably manipulate a given voting method in expectation, given different types of limited information about how other voters will vote. We trained over 100,000 neural networks of 26 sizes to manipulate against 8 different voting methods, under 6 types of limited information, in committee-sized elections with 5-21 voters and 3-6 candidates. We find that some voting methods, such as Borda, are highly manipulable by networks with limited information, while others, such as Instant Runoff, are not, despite being quite profitably manipulated by an ideal manipulator with full information. For the three probability models for elections that we use, the overall least manipulable of the 8 methods we study are Condorcet methods, namely Minimax and Split Cycle.

Figures (3)

• Figure 1: Left: the average profitability of submitted rankings by the best performing MLP with any hidden layer configuration for a given voting method and information type, averaging over 3--6 candidates and 5, 6, 10, 11, 20, and 21 voters. Right: the ratio of the average profitability of the MLP's submitted ranking to that of the ideal manipulator's submitted ranking.
• Figure 2: Results using the uniform utility model with 6 candidates and 10/11 voters for MLPs manipulating on the basis of the plurality scores or majority matrix. Error bars indicate twice the estimated standard error of the mean. Hidden layer configurations of trained MLPs are shown on the x-axis. Versions of this figure for the Mallows model, spatial 2D model, and different types of limited information appear in Supplementary Figures A.2--B.3.
• Figure 3: Top: average profitability of submitted rankings by an ideal manipulator. Middle: average profitability by the best performing MLP with any hidden layer configuration using the majority matrix information, averaging over 5, 6, 10, 11, 20, and 21 voters. Bottom: the ratio of the average profitability of the MLP's submitted ranking to the average profitability of the ideal manipulator's submitted ranking. Versions of this figure for the Mallows model and the spatial 2D model appear in Supplementary Figures D.2--3.