# The semi-algebraic geometry of saturated optimal designs for the Bradley-Terry model

## TL;DR

This paper characterizes saturated D-optimal designs for the Bradley-Terry model, showing they correspond to paths in small graphs and providing explicit conditions for optimality regions.

## Contribution

It introduces a semi-algebraic geometric framework to identify and describe saturated optimal designs for the Bradley-Terry model, including explicit inequalities for optimality regions.

## Key findings

- Saturated D-optimal designs are represented by paths in small graphs.
- Explicit polynomial inequalities define optimality regions in parameter space.
- For each parameter point, a D-optimal design can be prescribed.

## Abstract

Optimal design theory for nonlinear regression studies local optimality on a given design space. We identify designs for the Bradley--Terry paired comparison model with small undirected graphs and prove that every saturated D-optimal design is represented by a path. We discuss the case of four alternatives in detail and derive explicit polynomial inequality descriptions for optimality regions in parameter space. Using these regions, for each point in parameter space we can prescribe a D-optimal design.