Learning Resilient Elections with Adversarial GNNs

Learning Resilient Elections with Adversarial GNNs Hao Xiang Li1* Yash Shah1* Lorenzo Giusti2 1Department of Computer Science, University of Cambridge, Cambridge, United Kingdom 2CERN, Geneva, Switzerland hxl23@cst.cam.ac.uk ys562@cam.ac.uk lorenzo.giusti@cern.ch Abstract In the face of adverse motives, it is indispensable to achieve a consensus. Elections have been the canonical way by which modern democracy has operated since the 17th century. Nowadays, they regulate markets, provide an engine for modern recommender systems or peer-to-peer networks, and remain the main approach to represent democracy. However, a desirable universal voting rule that satisfies all hypothetical scenarios is still a challenging topic, and the design of these systems is at the forefront of mechanism design research. Automated mechanism design is a promising approach, and recent works have demonstrated that set-invariant architectures are uniquely suited to modelling electoral systems. However, various concerns prevent the direct application to real-world settings, such as robustness to strategic voting. In this paper, we generalise the expressive capability of learned voting rules, and combine improvements in neural network architecture with adversarial training to improve the resilience of voting rules while maximizing social welfare. We evaluate the effectiveness of our methods on both synthetic and real-world datasets. Our method resolves critical limitations of prior work regarding learning voting rules by representing elections using bipartite graphs, and learning such voting rules using graph neural networks. We believe this opens new frontiers for applying machine learning to real-world elections. 1 Introduction Although elections are most commonly used to resolve political disagreements, they can be abstracted as a structured tool for resolving disputes and making collective decisions in everyday contexts [1]. They allow groups to aggregate preferences so that every stakeholder has a voice and to ensure fairness such that each participant gets an equal say. Elections also help to avoid conflict by providing a peaceful resolution. Finally, they legitimize outcomes because participants are more willing to accept results when everyone had the chance to participate [2]. When a group of voters with individual preferences face the problem of choosing a single candidate among a set of possible outcomes, elections serve as a mechanism to aggregate those preferences and reach a collective decision that reflects the will of the group with applications beyond political systems, such as in multi-agent robotics, decentralized autonomous agents, and recommender systems [3]. In elections, a voting mechanism refers to an algorithmic process to elicit individual preferences, and choose an outcome with certain criteria [4]. A mechanism ideally possesses socially desirable characteristics: the chosen candidate should be preferred over others in a head-to-head comparison [5]; rational voters express their vote truthfully, or at least, honest opinions should not actively harm the chances of their preferred candidate [6]; the process must be fair, and each voter and candidate should have equal representation. *Equal contribution. Preprint. Unfortunately, selecting a deterministic rule that always selects the socially optimal outcome under a set of seemingly reasonable criteria is often an open problem—or worse, it is sometimes provably impossible for such a rule to exist [7], [8]. Significant research has been directed towards finding improved voting mechanisms, leading to an entire field of study known as voting theory, with contri­ butions exploring fairness criteria [8], strategic manipulation [9], and computational complexity [7]. In this work, we study methods to learn undiscovered voting rules. Our main contribution is a welfare-maximising learning methodology which satisfies the voter anonymity, candidate neutrality, and optionally other miscellaneous criteria, generalises to an arbitrary number of candidates, and is robust to strategic voting. To achieve this, we propose: (a) a combination of permutation-equivariant neural networks composed by: a graph voting network (GEVN) and a graph strategy network (GESN), (b) an algorithmic design of social welfare and monotonicity losses, and (c) adversarial assessment to expect strategic voting. 2 Background and Related Works Graph Neural Networks: Let 𝐺= (𝒱︀, ℰ︀) be a graph, with 𝒩︀𝑣= {𝑢∈𝒱︀: (𝑢, 𝑣) ∈ℰ︀} being the one-hop neighborhood of node 𝑣, having neighborhood features 𝑿𝒩︀𝑣= {{𝒙𝑢: 𝑢∈𝒩︀𝑣}}, where {{⋅}} denotes a multi-set. We define the message passing function, a permutation-invariant function over the 𝑿𝒩︀𝑣, as: 𝑓(𝒙𝑣, 𝑿𝒩︀𝑣) = 𝜙(𝒙𝑣, ⨁ 𝑢∈𝒩︀𝑣 𝜓(𝒙𝑣, 𝒙𝑢)) (1) where 𝜓 and 𝜙 are learnable message, and update functions, respectively, while ⊕ is a permutation- invariant aggregation function (e.g., sum, mean, max). A message passing neural network (MPNN)

This content is AI-processed based on open access ArXiv data.