Deuce: what the loop is worth to the server

At deuce a game can only end when one player wins two points in a row. That rule is where tennis scoring first does something clever with a small edge: it squares it. The same loop comes back in a tiebreak and in a set played without one. From deuce the server wins with probability p² ÷ (p² + (1 − p)²), where p is the share of points they win on serve — and that is always further from a coin flip than p itself.

How often a game gets there

A game reaches deuce when both players win three of the first six points. Between equals that happens 31.25% of the time — about one game in three — and less often the stronger the server: 27.6% at 60%. Once there, a game between equals comes back to deuce once more on average before someone takes two points in a row, which is why deuce games feel longer than the arithmetic of a single game suggests. Between equals the whole game averages 6.75 points.

What no-ad takes away

No-ad scoring replaces the deuce loop with a single deciding point at 3-3. That point is won with probability p — no squaring — so the amplification disappears and the server gives some of the hold back. That holds at every serve strength above 50%: the loop always helps the server, so removing it always costs them. The cost peaks at a server who wins 65% of points, who holds 2.9 percentage points less often under no-ad. Weaker servers lose little because they barely have an edge to amplify; much stronger ones lose little because they seldom reach deuce at all.

Between two exactly equal players it makes no difference whatsoever — both scoring systems hold 50% — which is the point worth making to anyone who calls no-ad a lottery. It is not more random between equals. It is less kind to whoever is better, and games are shorter: 5.81 points on average between equals rather than 6.75.

What no-ad does to a whole match, alongside a match tiebreak, is on the formats page.

What the model assumes

Every figure assumes each point is an independent trial at a fixed strength for the server. That is an assumption, not a fact about tennis. A study of 86,298 points at Wimbledon from 1992 to 1995 (Klaassen and Magnus, "Are Points in Tennis Independent and Identically Distributed?", Journal of the American Statistical Association, 2001) found that real points are not quite like this, and that the divergence is small. Read these figures as exact for players who serve every point at the same strength.