The mathematical study of voting, social choice theory , has traditionally only been applicable to choices among predetermined alternatives, but not to open-ended decisions such as collectively selecting a textual statement. We introduce generative social choice , a design methodology for open-ended democratic processes that combines the rigor of social choice theory with the capability of large language models to generate text and extrapolate preferences. Our framework divides the design of AI-augmented democratic processes into two components: first, proving that the process satisfies representation guarantees when given access to oracle queries; second, empirically validating that these queries can be approximately implemented using a large language model. We apply this framework to the problem of summarizing free-form opinions into a proportionally representative set of opinion statements; specifically, we develop a democratic process with representation guarantees and use this process to portray the opinions of participants in a survey about abortion policy. In a trial with 100 representative US residents, we find that 84 out of 100 participants feel “excellently” or “exceptionally” represented by the set of five statements we extracted.

A key task in certain democratic processes is to produce a concise slate of statements that proportionally represents the full spectrum of user opinions. This task is similar to committee elections, but unlike traditional settings, the candidate set comprises all possible statements of varying lengths, and so it can only be accessed through specific queries. Combining social choice and large language models, prior work has approached this challenge through a framework of generative social choice. We extend the framework in two fundamental ways, providing theoretical guarantees even in the face of approximately optimal queries and a budget limit on the overall length of the slate. Using GPT-4o to implement queries, we showcase our approach on datasets related to city improvement measures and drug reviews, demonstrating its effectiveness in generating representative slates from unstructured user opinions.

We study a matching problem between agents and public goods, in settings without monetary transfers. Since goods are public, they have no capacity constraints. There is no exogenously defined budget of goods to be provided. Rather, each provided good must justify its cost by being utilized by sufficiently many agents, leading to strong complementarities in the "preferences" of goods. Furthermore, goods that are in high demand given other already-provided goods must also be provided. The question of the existence of a stable solution (a menu of public goods to be provided) exhibits a rich combinatorial structure. We uncover sufficient conditions and necessary conditions for guaranteeing the existence of a stable solution, and derive both positive and negative results for strategyproof stable matching.

The mathematical study of voting, social choice theory, has traditionally only been applicable to choices among a few predetermined alternatives, but not to open-ended decisions such as collectively selecting a textual statement. We introduce generative social choice, a design methodology for open-ended democratic processes that combines the rigor of social choice theory with the capability of large language models to generate text and extrapolate preferences. Our framework divides the design of AI-augmented democratic processes into two components: first, proving that the process satisfies representation guarantees when given access to oracle queries; second, empirically validating that these queries can be approximately implemented using a large language model. We apply this framework to the problem of summarizing free-form opinions into a proportionally representative slate of opinion statements; specifically, we develop a democratic process with representation guarantees and use this process to represent the opinions of participants in a survey about chatbot personalization. In a trial with 100 representative US residents, we find that 93 out of 100 participants feel "mostly" or "perfectly" represented by the slate of five statements we extracted. By providing rigorous guarantees through social choice, our work alleviates concerns about AI-driven democratic innovation and helps unlock its potential.

The local properties problem of Erd\Hos and Shelah generalizes many Ramsey problems and some distinct distances problems. In this work, we derive a variety of new bounds for the local properties problem and its variants, by extending the color energy technique---a variant of the additive energy technique from additive combinatorics (color energy was originally introduced by the last two authors [C. Pohoata and A. Sheffer, Combinatorica, 39 (2019), pp. 705--714]). We generalize the concept of color energy to higher color energies and combine these with bounds on the extremal numbers of even cycles. Let $f(n,k,\ell)$ denote the minimum number of colors required to color the edges of $K_n$ such that every $k$ vertices span at least $\ell$ colors. It can be easily shown that $f(n,k,\binom{k}{2}-\lfloor \frac{k}{2}\rfloor +2)=\Theta(n^2)$. Erdös and Gyárfás asked what happens when $\ell=\binom{k}{2}-\lfloor k/2\rfloor +1$, one away from the easy case, and derived the bound $f\left(n,k,\ell\right) =\Omega(n^{4/3})$. Our technique significantly improves this to $f(n,k,\binom{k}{2} - \lfloor k/2\rfloor +1) = \Omega(n^{2- 8/k})$.