UtilityMax Prompting: A Formal Framework for Multi-Objective Large Language Model Optimization

TL;DR

UtilityMax Prompting introduces a formal mathematical framework for multi-objective LLM optimization, improving task precision by explicitly maximizing a defined utility function rather than relying on ambiguous natural language prompts.

Contribution

It formalizes multi-objective prompt design using influence diagrams and utility functions, enabling explicit reasoning and optimization in LLM outputs.

Findings

Consistent improvements in precision and NDCG over natural language baselines.

Validated on MovieLens 1M dataset with frontier models.

Framework directs LLMs toward explicit optimization targets.

Abstract

The success of a Large Language Model (LLM) task depends heavily on its prompt. Most use-cases specify prompts using natural language, which is inherently ambiguous when multiple objectives must be simultaneously satisfied. In this paper we introduce UtilityMax Prompting, a framework that specifies tasks using formal mathematical language. We reconstruct the task as an influence diagram in which the LLM's answer is the sole decision variable. A utility function is defined over the conditional probability distributions within the diagram, and the LLM is instructed to find the answer that maximises expected utility. This constrains the LLM to reason explicitly about each component of the objective, directing its output toward a precise optimization target rather than a subjective natural language interpretation. We validate our approach on the MovieLens 1M dataset across three frontier models (Claude Sonnet 4.6, GPT-5.4, and Gemini 2.5 Pro), demonstrating consistent improvements in precision and Normalized Discounted Cumulative Gain (NDCG) over natural language baselines in a multi-objective movie recommendation task.