How do you turn a run of wins and losses into a single skill number? The ELO system, born in chess and now everywhere from ping-pong to matchmaking, says a rating GAP predicts a win probability by a logistic curve: 400 points of lead means about 91% to win. Beat someone stronger and your rating jumps; lose to someone weaker and it drops hard. Slide the gap and read the odds.
The Elo expected score is E = 1/(1 + 10^(−Δ/400)), a logistic function of the rating difference Δ: equal ratings give 0.5, +200 gives ~0.76, +400 gives ~0.91. After a game the update is R′ = R + K·(S − E) where S is the actual result (1/0.5/0) and K the sensitivity — so beating a higher-rated opponent (low E) gains the most. A fail-loud self-check throws unless Δ=0 gives 0.5 and Δ=400 gives ~0.909. ◆ real rating theory, node-verified.
Elo assumes a stable logistic skill model and independent games; real skill drifts, and the 400/base-10 scaling is a convention (Glicko/TrueSkill refine it). The number is a MODEL of relative strength, not an absolute measure.