Log5 Matchup Probability

Log5 answers the question a schedule constantly poses: two sides have played the same field and posted different records, so what happens when they play each other? Bill James published it as part of the same body of baseball work that produced Pythagorean expectation, and the name is a leftover from an earlier and different method — it involves no logarithms and no fives.

The formula is much prettier in odds than in probabilities. Convert each win rate to odds — a .600 side is 1.5 to 1, a .400 side is 0.667 to 1 — and the odds of the matchup are simply the ratio: 1.5 divided by 0.667 is 2.25, which is 0.692 as a probability. That is the whole content of the model, and in that form it is visibly the Bradley-Terry paired-comparison model of 1952: give each side a strength, and let the chance one beats another be its share of the pair's total. It is also, therefore, the same model as Elo, on a different scale — Elo's ratings are the logarithms of these strengths.

Two properties make it checkable without a calculator, and both are worth confirming against your intuition. Equal win rates always give exactly 0.500, however strong the pair — two .700 teams are a coin flip against each other, which is right, since their records say they are equal. And a .500 side is a perfect yardstick: against anyone, its chance is one minus the opponent's win rate. That is the sanity check to run whenever you suspect you have mistyped something.

The assumption doing all the work is that both records were earned against the same field. Two teams in different divisions, on different schedules, in different eras or against different opposition strength are not comparable inputs, and feeding them in anyway produces a confident number built on nothing. Strength-of-schedule adjustment has to happen before log5, not inside it.

And a small sample is not a win rate. A side that has started three-and-one has a .750 record and no meaningful strength estimate at all; log5 will happily tell you it beats a .500 team 75% of the time. The formula has no way to express uncertainty about its inputs — which is precisely the gap Glicko fills on the ratings side, and no equivalent exists here. Nor does log5 know anything about home advantage, rest, injuries, or the way one style of play matches up against another. It is a model of relative strength and nothing else, and it is a good one within those limits.