Why logistic regression is equivalent to Bradley-Terry model?

Nobody has claimed this yet.

Dear maintainers,

Thank you for your valuable arena. I am currently researching the way of LLMs evaluation and got stack with a question about Bradley-Terry model.
As it stands, from multiple sources, BT is obtained through maximizing BT likelihood (as well as in your paper). However inside the code, logistic regression is fitted on some kind of "one-hot" matrix, where +1 is model_a and -1 is model_b, and target is 1 in case model_a wins and 0 if model_b wins. Lets neglect controlling length of answer for simplicity, but I can not understand why this is equivalent to BT model.

Could you please explain this or give me some sources where i could find the derivation?

```
def compute_elo_mle_with_tie(
df, SCALE=400, BASE=10, INIT_RATING=1000, sample_weight=None
):
from sklearn.linear_model import LogisticRegression

ptbl_a_win = pd.pivot_table(
df[df["winner"] == "model_a"],
index="model_a",
columns="model_b",
aggfunc="size",
fill_value=0,
)
ptbl_tie = pd.pivot_table(
df[df["winner"].isin(["tie", "tie (bothbad)"])],
index="model_a",
columns="model_b",
aggfunc="size",
fill_value=0,
)
ptbl_tie = ptbl_tie + ptbl_tie.T
ptbl_b_win = pd.pivot_table(
df[df["winner"] == "model_b"],
index="model_a",
columns="model_b",
aggfunc="size",
fill_value=0,
)
ptbl_win = ptbl_a_win * 2 + ptbl_b_win.T * 2 + ptbl_tie

models = pd.Series(np.arange(len(ptbl_win.index)), index=ptbl_win.index)

p = len(models)
X = np.zeros([p * (p - 1) * 2, p])
Y = np.zeros(p * (p - 1) * 2)

cur_row = 0
sample_weights = []
for m_a in ptbl_win.index:
for m_b in ptbl_win.columns:
if m_a == m_b:
continue
# if nan skip
if math.isnan(ptbl_win.loc[m_a, m_b]) or math.isnan(ptbl_win.loc[m_b, m_a]):
continue
X[cur_row, models[m_a]] = +math.log(BASE)
X[cur_row, models[m_b]] = -math.log(BASE)
Y[cur_row] = 1.0
sample_weights.append(ptbl_win.loc[m_a, m_b])

X[cur_row + 1, models[m_a]] = math.log(BASE)
X[cur_row + 1, models[m_b]] = -math.log(BASE)
Y[cur_row + 1] = 0.0
sample_weights.append(ptbl_win.loc[m_b, m_a])
cur_row += 2
X = X[:cur_row]
Y = Y[:cur_row]

lr = LogisticRegression(fit_intercept=False, penalty=None)
lr.fit(X, Y, sample_weight=sample_weights)
elo_scores = SCALE * lr.coef_[0] + INIT_RATING
if "mixtral-8x7b-instruct-v0.1" in models.index:
elo_scores += 1114 - elo_scores[models["mixtral-8x7b-instruct-v0.1"]]
return pd.Series(elo_scores, index=models.index).sort_values(ascending=False)
```

Start with the compute_elo_mle_with_tie entry point and trace how the X, Y, and sample_weights arrays are constructed before the sklearn LogisticRegression fit. Compare that construction with the Bradley-Terry likelihood, and document a derivation or authoritative references that explain the equivalence, including how ties are represented.】【。

Written by the indexing model from the issue text.