Average rating of opponents: the measure that argues with Buchholz

It measures the same thing Buchholz does - how hard your path through the tournament was - but takes it from ratings known before round one rather than from results. Sometimes the two disagree, and then everything depends on which of them your regulations list first. That is exactly how the 2022 European champion was decided, and on Buchholz it was the other player who stood higher.

Formula and example

ARO = sum of opponents' ratings ÷ number of games played

It uses the ratings in force before the event began. Your own rating is excluded and so are the results: the measure answers one question only - how strong were the opponents the pairing handed you.

Five rounds, five opponents

23802205245519802310
  • Sum of ratings: 11330
  • ARO = 11330 ÷ 5 = 2266
  • AROC 1 - the same calculation without the weakest opponent (1980): (113301980) ÷ 4 = 2338

Cutting one opponent lifted the value by almost a hundred points. That is why regulations usually specify the cut version: a single chance pairing against a much weaker player stops deciding the outcome.

How it differs from its neighbours

Three quantities that get confused constantly, though they answer different questions.

Average rating of opponents (ARO)

Looks only at the ratings your opponents carried before the event. How you did against them changes nothing.

Measures the difficulty of the draw in its purest form: who got the hard road and who got the easy one.

Buchholz

Looks at what the opponents actually scored in this event. Their ratings are irrelevant; their results are everything.

Measures the same difficulty, but by what happened rather than by what was expected.

Tournament performance rating (TPR)

Takes the average rating of the opponents and adds an adjustment for the percentage you scored.

Answers not «who did you play» but «what level did you play at».

The measure based on opponents' actual scores is covered on the Buchholz page, and the full catalogue with formulas is in the tie-break reference.

European Championship 2022: when ARO outweighed Buchholz

Matthias Bluebaum and Gabriel Sargissian both finished the event in Brezice on 8½ out of 11. The regulations provided for no play-off games, so the title was decided on tie-break, and the first measure in the list was the average rating of opponents: 2594 against 2578, a margin of 16 points to Bluebaum.

The interesting part is that on Buchholz it was Sargissian who stood higher. With a different order of the same measures in the same regulations, the European champion would have been a different player with exactly the same games. Nothing improper happened: the list was published in advance, which is precisely why it is published in advance.

More cases where a tie-break decided the outcome are collected in the tie-break reference.

Where the measure stops working

Unrated players

The main limitation. In an amateur event where half the room is playing for the first time, there is no obvious number to substitute for an unrated opponent. The regulations have to state one, otherwise the calculation becomes ambiguous.

Stale ratings

The measure uses the figure from before the event. A fast-improving junior carrying last year's rating deflates the ARO of everyone who plays them, even though they are performing far above that number.

Round-robins

There everyone faces the same field, so the averages come out nearly identical and separate nobody. Round-robins use the Sonneborn-Berger score and the Koya system instead.

Ratings are stored with the event

To compute an average rating of opponents you need their ratings as of the start. In the Vibe Chess app every player carries a rating and the round history is kept in full, so ARO or any other measure can be reconstructed after the event without digging out paper protocols.

Questions about ARO

How is the average rating of opponents calculated?+

Add the ratings of every opponent you played and divide by the number of games. The value is rounded to a whole number. Your own rating is not part of it, and neither are the results: only the ratings count.

How does ARO differ from performance rating?+

Performance rating answers what level you played at: the average rating of your opponents plus an adjustment for your score percentage. ARO is only the first half, without the adjustment. Two players can therefore share an ARO and have wildly different performances.

What does AROC 1 mean in a crosstable?+

The same average with the single weakest opponent cut out of the calculation. It removes the effect of one chance pairing against a much lower-rated player, especially if that player then withdrew.

When is this measure used?+

In events where every participant has an official rating. In an amateur tournament full of first-timers it loses meaning, because there is no obvious number to substitute for an unrated opponent.

What do you substitute for an unrated opponent?+

Whatever the regulations say: usually zero or the minimum value adopted by the federation. Without such a clause the measure cannot be applied at all, since each reading would give different placings.

Can ARO contradict Buchholz?+

Yes, and not rarely. Buchholz uses the opponents' actual scores, ARO their pre-event ratings. Highly rated opponents can have a poor event, leaving a player with a high ARO and a low Buchholz. That is exactly how the 2022 European Championship was decided.

Are unplayed games counted?+

No. Rounds without a game do not enter the average, because there is no real opponent with a rating. A player who took a bye therefore has the measure computed over fewer games.

Where does it belong in the tie-break order?+

Usually after Buchholz, towards the end of the list. Major Swiss regulations often place the cut version, AROC 1, third or fourth - right before the direct encounter and a drawing of lots.

The order of tie-breaks decides more than it looks

The same tournament with the same games produces different winners under different tie-break orders. Publish the list before round one and let the app do the arithmetic.