Proportionality in Thumbs Up and Down Voting
This work addresses a gap in computational social choice for decision-making in settings like constitutional AI, where agents have both approval and disapproval votes, but it is incremental as it builds on existing proportionality concepts.
The paper tackles the problem of defining proportionality in voting systems where agents can express both positive and negative preferences, such as in constitutional AI for selecting ethical principles. It proposes two distinct approaches to interpret proportionality with up and down votes, formalizing axioms and examining their satisfiability by adapting existing voting rules.
Consider the decision-making setting where agents elect a panel by expressing both positive and negative preferences. Prominently, in constitutional AI, citizens democratically select a slate of ethical preferences on which a foundation model is to be trained. There, in practice, agents may both approve and disapprove of different ethical principles. Proportionality has been well-studied in computational social choice for approval ballots, but its meaning remains unclear when negative sentiments are also considered. In this work, we propose two conceptually distinct approaches to interpret proportionality in the presence of up and down votes. The first approach treats the satisfaction from electing candidates and the impact of vetoing them as comparable, leading to combined proportionality guarantees. The second approach considers veto power separately, introducing guarantees distinct from traditional proportionality. We formalize axioms for each perspective and examine their satisfiability by suitable adaptations of Phragmén's rule, Proportional Approval Voting rule and the Method of Equal Shares.