Bradley-Terry Model Is Just A Logistic Function

$$p^{\ast }(y_1>y_2|x)=\frac{\exp (r^{\ast }(x,y_1))}{\exp (r^{\ast }(x,y_1))+\exp (r^{\ast }(x,y_2))}=\sigma (r^{\ast }(x,y_1)-r^{\ast }(x,y_2))$$

(Rafailov et al., 2024, pp. 3–4)

Reference

Rafailov, R., Sharma, A., Mitchell, E., Ermon, S., Manning, C. D., & Finn, C. (2024). Direct Preference Optimization: Your Language Model Is Secretly a Reward Model. arXiv. https://doi.org/10.48550/arXiv.2305.18290