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Equation 5 · Part 6 · Frontier AI Model Comparisons: A First-Principles Introduction

fraction

P(A≻B)=11+e−(θA−θB).P(A \succ B) = \frac{1}{1 + e^{-(\theta_A - \theta_B)}}.
fraction

What this part means

Divide the expression above the line by the one below it.

Its job in the formula

The expression above the fraction bar is divided by the complete expression below it. The denominator must not be zero.

The passage around this formula

A different style of evaluation sidesteps fixed question sets altogether by asking humans which of two anonymized model outputs they prefer, on real conversational prompts, and aggregating millions of these paired judgments into a ranking. Chatbot Arena, the platform behind this approach, describes itself as using “a pairwise comparison approach” that “leverages input from a diverse user base through crowdsourcing,” with the underlying statistics resting on a standard paired-comparison model rather than a raw win count [ 6 ] . The relevant object is a Bradley–Terry model: each system i is assigned a latent strength θi\theta_i , and the estimated probability that system A ’s output is preferred…

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Learn the underlying idea

A fraction a/b means a divided by b. The top number is the numerator; the bottom number is the denominator, and it cannot be zero.

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Sources cited in the surrounding passage

These citations provide research context; check each source for the exact claim it supports.