Median Buchholz: Chess Tie-Break

Median Buchholz

Definition

Median Buchholz is a Swiss-system tie-break method that measures the strength of a player's opposition by summing the final scores of their opponents, then reducing the influence of extreme outliers by discarding the single highest and single lowest opponent scores before summing. In symbols:

• Buchholz: sum of all opponents' final points.
• Median Buchholz: sum of all opponents' final points, excluding the highest and the lowest opponent scores.

Variants you may see in regulations:

• Buchholz Cut 1 / Cut 2: Drop the single lowest (Cut 1) or the two lowest (Cut 2) opponent scores.
• Median-Buchholz-2: Drop the two highest and two lowest opponent scores.
• Solkoff (US): Same as Buchholz (no cuts).
• Modified Median (US): A family of rules related to Median Buchholz; specifics depend on the federation’s handbook.

How it is used in chess

Median Buchholz is employed to rank players who finish with the same number of points in Swiss tournaments (individual and team). It is especially popular because it mitigates distortions caused by facing one unusually strong (or unusually weak/withdrawing) opponent. Event regulations specify the exact order of tie-breakers (e.g., Direct Encounter, Buchholz Cut 1, Median Buchholz, Sonneborn-Berger, Rating Performance, etc.). Median Buchholz does not affect pairings; it is only used to order tied scores in the final (or interim) standings.

Why organizers use it (strategic and historical significance)

The classic Buchholz score can be skewed by a single opponent who either performs far above expectations or withdraws early and scores near zero. Median Buchholz trims these extremes, giving a more robust estimate of the average strength of your opposition. As Swiss tournaments grew in size through the mid-20th century and tie-breaks became more consequential, arbiters adopted cut-based Buchholz methods (including Median) to improve perceived fairness. FIDE lists Buchholz variants (including Median) among recommended tie-break options in Swiss events.

Strategically, players can’t directly “play for” a better Median Buchholz except by trying to score well against strong opposition; however, it’s a common adage to “root for your opponents” after you’ve played them—every point they score later tends to help your Buchholz-type tie-breaks.

How to calculate Median Buchholz

1. List the final scores (match points) of all your opponents.
2. Identify the single highest and single lowest of those scores.
3. Discard those two scores.
4. Sum the remaining opponent scores—the result is your Median Buchholz.

Practical notes:

• If too few rounds were played to discard both extremes (e.g., very short events), regulations may specify cutting fewer scores or reverting to plain Buchholz.
• Byes and forfeits are handled per event regulations (software may assign “virtual opponent” values or special handling); arbiters normally announce this before round 1.
• Team events often compute Buchholz/Median on match points (sum of opposing teams’ match scores) rather than game points.

Examples

Example 1: Two players tie on 5.5/7 in a Swiss. Their opponents’ final scores are:

• Player A opponents: 6.5, 6.0, 5.5, 5.0, 4.0, 3.0, 2.0 (sum = 32.0)
• Player B opponents: 6.0, 5.0, 5.0, 5.0, 4.0, 4.0, 3.0 (sum = 32.0)

Plain Buchholz ties them at 32.0. Median Buchholz:

• Player A: discard 6.5 and 2.0 → 6.0 + 5.5 + 5.0 + 4.0 + 3.0 = 23.5
• Player B: discard 6.0 and 3.0 → 5.0 + 5.0 + 5.0 + 4.0 + 4.0 = 23.0

Result: Player A ranks ahead on Median Buchholz 23.5 vs. 23.0.

Example 2: A “withdrawn opponent” effect. Suppose Player C’s seven opponents finish with scores:

• 7.0, 6.0, 5.5, 5.0, 4.5, 4.0, 0.0 (one opponent withdrew early)

Buchholz = 32.0; Median Buchholz discards 7.0 and 0.0 → 6.0 + 5.5 + 5.0 + 4.5 + 4.0 = 25.0. Median Buchholz prevents the 0.0 outlier from overly depressing the tie-break.

Related and contrasting tie-breaks

• Buchholz: Sum of opponents’ final scores; more sensitive to outliers.
• Buchholz Cut 1 / Cut 2: Discard the lowest one or two opponents’ scores only.
• Median-Buchholz-2: Discard the two highest and two lowest opponents’ scores.
• Sonneborn-Berger: Primarily used in round-robins; weights wins against higher-scoring opponents more.
• Direct Encounter (head-to-head): Ranks tied players based on their results against each other.
• Rating Performance/TPR: Orders ties by calculated performance rating; used in some events as a later tie-break.

Tips, pitfalls, and arbiter notes

• Always check the event’s tie-break order in advance; Median Buchholz might be first, second, or later in the sequence.
• Interim standings can shift late: since Buchholz-type tie-breaks use final opponent scores, your tie-break may change when other games finish.
• Half-point or full-point byes, forfeit wins, and unrated opponents have special handling that can affect Buchholz-type scores; rely on the event’s published rules or the pairing software’s documentation.
• For very small fields or short events, organizers often prefer head-to-head or playoffs over Buchholz-type methods.

Interesting facts and anecdotes

• “Rooting for your opponents” is rational: every point they score later generally boosts your Buchholz and Median Buchholz, sometimes deciding prize places among large ties.
• Many large opens specify multiple Buchholz variants in a row (e.g., Buchholz Cut 1, then Median) precisely to stabilize rankings when fields are huge and withdrawals common.
• The family of Buchholz tie-breaks is named after Bruno Buchholz, and has been standard in Swiss chess for decades alongside other classics like Sonneborn-Berger.