Real Preferences under Arbitrary Norms

Joshua Zeitlin, Corinna Coupette

Keywords: Spatial Preferences, Social Choice Theory, Voting

TL;DR: We prove generalized upper bounds on the dimensionality required for any preference profile to embed into Euclidean space.

Abstract: Whether the goal is to reach decisions among multiple agents, ensure that AI systems are aligned with human preferences, or design better recommender systems, the problem of translating between (ordinal) rankings and (numerical) utilities arises naturally in many contexts. This task is commonly approached by computing _embeddings_, which represent both the agents doing the ranking (_voters_) and the items to be ranked (_alternatives_) in a shared metric space. Here, ordinal preferences are translated into relationships between pairwise distances. Prior work has established that any collection of rankings with $n$ voters and $m$ alternatives (_preference profile_) can be embedded into $d$-dimensional Euclidean space for $d \\geq \\min\\{ n,m - 1 \\}$ under the Euclidean norm and the Manhattan norm. We show that this holds for _all $p$-norms_ and establish that any _pair_ of rankings can be embedded into $\\mathbb{R}^2$ under _arbitrary norms_, significantly expanding the reach of spatial preference models.

Area: Game Theory and Economic Paradigms (GTEP)

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Submission Number: 84