gen-define-the-elo-rating-system-and-give-one-short-worked-examp-bdf8

Define the Elo rating system and give one short worked example.

# Elo Rating System The **Elo rating system** is a method for calculating the relative skill levels of players in competitive games. Originally developed for chess, it's now used in many sports and games. ## How It Works - Each player has a rating (typically starting around 1200-1500) - When two players compete, the system: 1. Predicts the expected outcome based on rating difference 2. Compares actual result to expected result 3. Adjusts both ratings accordingly ## Key Formula **Expected score for player A:** ``` E_A = 1 / (1 + 10^((R_B - R_A)/400)) ``` **New rating:** ``` R_A_new = R_A + K(S_A - E_A) ``` Where: - R_A, R_B = current ratings - S_A = actual score (1 for win, 0.5 for draw, 0 for loss) - K = maximum rating change per game (typically 16-32) ## Worked Example **Setup:** Alice (rating 1600) plays Bob (rating 1400). Alice wins. K = 32. **Step 1 - Expected scores:** - E_Alice = 1/(1 + 10^((1400-1600)/400)) = 1/(1 + 10^(-0.5)) = 1/1.316 = **0.76** - E_Bob = 1 - 0.76 = **0.24** **Step 2 - New ratings:** - Alice: 1600 + 32(1 - 0.76) = 1600 + 7.68 = **1608** - Bob: 1400 + 32(0 - 0.24) = 1400 - 7.68 = **1392** Alice was expected to win, so she gains only 8 points. Bob loses 8 points.