Add chess-style Elo ratings for players

Each player's rating starts at 1500 and updates transactionally with
every finished match: a team's rating is the mean of its two members
and the standard K=32 formula decides the zero-sum delta applied to
both members of a team. Ratings are per game type in a new
player_rating table; match_player records each match's elo_delta.

- GET /api/leaderboard exposes elo and sorts by it
- GET /api/me/matches includes per-player elo deltas
- new GET /api/me/ratings returns the caller's rating per game type
- frontend: Elo column on the leaderboard, per-match delta in the
  history page, current rating in the lobby
- python -m tavolo.backfill_elo recomputes all ratings from the
  recorded match history (one-off backfill for existing matches)

server/tests/test_elo.py

@@ -0,0 +1,73 @@
+"""Unit tests for the chess-style Elo math in :mod:`tavolo.elo`."""
+from __future__ import annotations
+
+import unittest
+
+from tavolo.elo import (
+    INITIAL_RATING,
+    K_FACTOR,
+    expected_score,
+    match_delta,
+    team_rating,
+)
+
+
+class ExpectedScoreTest(unittest.TestCase):
+    def test_equal_ratings_give_even_odds(self) -> None:
+        self.assertAlmostEqual(0.5, expected_score(1500, 1500))
+
+    def test_higher_rating_is_favoured(self) -> None:
+        self.assertGreater(expected_score(1700, 1500), 0.5)
+        self.assertLess(expected_score(1500, 1700), 0.5)
+
+    def test_scores_sum_to_one(self) -> None:
+        self.assertAlmostEqual(
+            1.0, expected_score(1600, 1400) + expected_score(1400, 1600)
+        )
+
+    def test_four_hundred_points_is_ten_to_one(self) -> None:
+        self.assertAlmostEqual(10 / 11, expected_score(1900, 1500))
+
+
+class TeamRatingTest(unittest.TestCase):
+    def test_mean_of_members(self) -> None:
+        self.assertEqual(1600, team_rating([1500, 1700]))
+
+    def test_empty_team_rejected(self) -> None:
+        with self.assertRaises(ValueError):
+            team_rating([])
+
+
+class MatchDeltaTest(unittest.TestCase):
+    def test_equal_teams_exchange_half_k(self) -> None:
+        delta = match_delta([1500, 1500], [1500, 1500], winner_team=0)
+        self.assertEqual(K_FACTOR // 2, delta)
+
+    def test_favourite_gains_less_than_underdog(self) -> None:
+        favourite = match_delta([1700, 1700], [1500, 1500], winner_team=0)
+        underdog = match_delta([1500, 1500], [1700, 1700], winner_team=0)
+        self.assertGreater(underdog, favourite)
+        self.assertGreater(favourite, 0)
+
+    def test_losing_side_loses_the_winners_gain(self) -> None:
+        # Zero-sum: the losers' delta is the negation of the winners'.
+        win = match_delta([1600, 1500], [1400, 1500], winner_team=0)
+        loss = match_delta([1600, 1500], [1400, 1500], winner_team=1)
+        self.assertEqual(-win, -abs(win))  # winner gains
+        # Losing the same pairing costs K * E, winning gains K * (1 - E);
+        # both are computed from the same expectation, so loss = win - K.
+        self.assertEqual(win - K_FACTOR, loss)
+
+    def test_team_average_decides_not_individual_ratings(self) -> None:
+        # [1700, 1300] averages 1500, same as [1500, 1500].
+        mixed = match_delta([1700, 1300], [1500, 1500], winner_team=0)
+        even = match_delta([1500, 1500], [1500, 1500], winner_team=0)
+        self.assertEqual(even, mixed)
+
+    def test_initial_rating_constant(self) -> None:
+        self.assertEqual(1500, INITIAL_RATING)
+        self.assertEqual(32, K_FACTOR)
+
+
+if __name__ == "__main__":
+    unittest.main()