15 February 2016

Swiss Tournament Tie-Breaks

It is relatively easy to split a sum of money equitably between two or more recipients. However, cutting a trophy into halves or thirds only makes a mess. Consequently, chess tournaments use tie-breaks to determine the relative standings of chess players who finish with the same scores.

The Swiss System method for running chess tournaments accommodates a large number of competitors. It was first used in Zurich, Switzerland in 1895, where it was developed by Julius Müller, a founding member of the Swiss Chess Federation (see Edward Winter, Chess Notes 4118, 20 January 2006). Random pairings were used among winners and losers when first used, but the system has developed more sophistication since its first use.

Swiss System tournaments are non-elimination. The basic rules* are simple:

1) No one plays the same opponent twice.
2) In each round, players are paired as nearly as possible against others with the same score.
3) Each player should have approximately the same number of Blacks and Whites.

At the end of an event with a number of players that does not exceed 2 to the power of the number of rounds, there will be no more than one perfect score. It often happens, however, that there is no perfect score. Instead, a group is clustered at the top with one-half or one point below perfect. In such case, who wins first place? Tie-breaks are used to make this determination. Tie-breaks also sort players below first place for additional prizes.

Tie-breaks attempt to determine the first among equals by measuring the quality of each player's opponents. There are several methods for making this determination. Usually more than one is used by tournament directors. Below I describe tie-break methods and display how each sorts the participants in one particular tournament. As will be seen, different methods lead to different results.

In this event, three players shared first place, Lucas, Ernest, and Henry. Each of these three won four games and lost one. Each one's only loss was to another of the three. Lucas beat Ernest, the top rated player, but lost to Henry. Henry lost to Ernest.

Henry might prefer Cumulative, as that puts him on top. Lucas comes out on top with Average Opposition. Ernest prevails via Opposition Performance, Opposition Cumulative, and Solkoff.

U.S. Chess Federation rules prefer a system that employs Modified Median, Solkoff, Cumulative, and Opposition Cumulative in that order. In Washington State scholastic events, the standard tie-break system employed is Solkoff, Cumulative, and Opposition Cumulative.

Average Opposition

Average opposition employs the mean of the opponent's pre-tournament ratings. This system will tend to favor the higher rated player, but that did not occur in our sample tournament.

Standings. Average Opposition

#NameRtngRd 1Rd 2Rd 3Rd 4Rd 5TotTBrk[A]
1Lucas Beauchamp1418W12W5W2L3W44.01128
2Ernest Thayer1643W11W4L1W6W34.01123
3Henry Bird1448W14W10W8W1L24.01076
4Flannery O'Connor1069W7L2W11W8L13.01016
5David Duncanunr.W9L1L6W14W103.0950.5
6Gilbert Chesterton889L10W15W5L2W73.0888
7Robert Zimmerman419L4B---W13W9L63.0761
8Joanne Rowling809W15W13L3L4W143.0732.5
9Shaun Alexander934L5W12W10L7W153.0645.5
10Charles B. Brownunr.W6L3L9W11L52.01022.5
11Robert Frost793L2W14L4L10W122.0931
12E Dickinson432L1L9B---W13L112.0824.5
13Carl Sandburgunr.H---L8L7L12B---1.5553
14Rodney Serling588L3L11W15L5L81.0900.5
15Stephane Mallarmeunr.L8L6L14B---L91.0805

Average Opposition Performance

Rather than using pre-tournament ratings, opposition performance employs performance ratings. Performance rating is determined by opponent's rating + 400 for a win, opponent's rating - 400 for a loss, and opponent's rating for a draw.

Standings. Opposition Performance

#NameRtngRd 1Rd 2Rd 3Rd 4Rd 5TotTBrk[P]
1Ernest Thayer1643W11W4L2W5W34.01194
2Lucas Beauchamp1418W12W6W1L3W44.01182
3Henry Bird1448W14W10W7W2L14.01101
4Flannery O'Connor1069W9L1W11W7L23.01116.5
5Gilbert Chesterton889L10W15W6L1W93.0941
6David Duncanunr.W8L2L5W14W103.0934
7Joanne Rowling809W15W13L3L4W143.0766.5
8Shaun Alexander934L6W12W10L9W153.0744.5
9Robert Zimmerman419L4B---W13W8L53.0737
10Charles B. Brownunr.W5L3L8W11L62.01014
11Robert Frost793L1W14L4L10W122.0976.5
12E Dickinson432L2L8B---W13L112.0807
13Carl Sandburgunr.H---L7L9L12B---1.5743
14Rodney Serling588L3L11W15L6L71.0944
15Stephane Mallarmeunr.L7L5L14B---L81.0783


Cumulative sums the scores round by round. A player losing in early rounds will face weaker opponents than one winning in early rounds.

Standings. Cumulative

#NameRtngRd 1Rd 2Rd 3Rd 4Rd 5TotTBrk[C]
1Henry Bird1448W14W10W5W2L34.014
2Lucas Beauchamp1418W12W9W3L1W44.013
3Ernest Thayer1643W11W4L2W7W14.012
4Flannery O'Connor1069W8L3W11W5L23.010
5Joanne Rowling809W15W13L1L4W143.010
6Shaun Alexander934L9W12W10L8W153.08
7Gilbert Chesterton889L10W15W9L3W83.08
8Robert Zimmerman419L4B---W13W6L73.08
9David Duncanunr.W6L2L7W14W103.08
10Charles B. Brownunr.W7L1L6W11L92.07
11Robert Frost793L3W14L4L10W122.05
12E Dickinson432L2L6B---W13L112.04
13Carl Sandburgunr.H---L5L8L12B---1.52
14Rodney Serling588L1L11W15L9L51.03
15Stephane Mallarmeunr.L5L7L14B---L61.01

Opposition Cumulative

Opposition cumulative sums the cumulative scores of each player's opponents.

Standings. Opposition Cumulative

#NameRtngRd 1Rd 2Rd 3Rd 4Rd 5TotTBrk[O]
1Ernest Thayer1643W11W4L2W6W34.050
2Lucas Beauchamp1418W12W5W1L3W44.048
3Henry Bird1448W14W10W7W2L14.045
4Flannery O'Connor1069W8L1W11W7L23.048
5David Duncanunr.W9L2L6W14W103.039
6Gilbert Chesterton889L10W15W5L1W83.036
7Joanne Rowling809W15W13L3L4W143.030.5
8Robert Zimmerman419L4B---W13W9L63.028.5
9Shaun Alexander934L5W12W10L8W153.028
10Charles B. Brownunr.W6L3L9W11L52.043
11Robert Frost793L1W14L4L10W122.036
12E Dickinson432L2L9B---W13L112.028.5
13Carl Sandburgunr.H---L7L8L12B---1.522
14Rodney Serling588L3L11W15L5L71.038
15Stephane Mallarmeunr.L7L6L14B---L91.029


When two tied players have played each other, the winner of their game wins head-to-head. If they drew, however, or if more than two share the same score, head-to-head is useless. The scores below give each player 100, then subtract 1 for each loss to another player with the same score. As can be seen, it was effective for determining last place.

Standings. Head-to-Head

#NameRtngRd 1Rd 2Rd 3Rd 4Rd 5TotTBrk[H]
1Ernest Thayer1643W11W4L3W5W24.099
2Henry Bird1448W14W10W6W3L14.099
3Lucas Beauchamp1418W12W7W1L2W44.099
4Flannery O'Connor1069W9L1W11W6L33.0100
5Gilbert Chesterton889L10W15W7L1W93.0100
6Joanne Rowling809W15W13L2L4W143.099
7David Duncanunr.W8L3L5W14W103.099
8Shaun Alexander934L7W12W10L9W153.098
9Robert Zimmerman419L4B---W13W8L53.098
10Charles B. Brownunr.W5L2L8W11L72.0100
11Robert Frost793L1W14L4L10W122.099
12E Dickinson432L3L8B---W13L112.099
13Carl Sandburgunr.H---L6L9L12B---1.5100
14Rodney Serling588L2L11W15L7L61.0100
15Stephane Mallarmeunr.L6L5L14B---L81.099


The Kashdan system rewards aggressive play by awarding 4 points for a win, 2 points for a draw, 1 point for a loss, and 0 for an unplayed game.

Standings. Kashdan

#NameRtngRd 1Rd 2Rd 3Rd 4Rd 5TotTBrk[K]
1Ernest Thayer1643W10W4L3W6W24.017
2Henry Bird1448W14W11W7W3L14.017
3Lucas Beauchamp1418W12W8W1L2W44.017
4Flannery O'Connor1069W9L1W10W7L33.014
5Shaun Alexander934L8W12W11L9W153.014
6Gilbert Chesterton889L11W15W8L1W93.014
7Joanne Rowling809W15W13L2L4W143.014
8David Duncanunr.W5L3L6W14W113.014
9Robert Zimmerman419L4B---W13W5L63.010
10Robert Frost793L1W14L4L11W122.011
11Charles B. Brownunr.W6L2L5W10L82.011
12E Dickinson432L3L5B---W13L102.07
13Carl Sandburgunr.H---L7L9L12B---1.53
14Rodney Serling588L2L10W15L8L71.08
15Stephane Mallarmeunr.L7L6L14B---L51.04


The Median system sums the final scores of each player's opponents, discarding the highest and lowest. This system is also known as the Harkness system for its inventor, Kenneth Harkness.

Standings. Median

#NameRtngRd 1Rd 2Rd 3Rd 4Rd 5TotTBrk[B]
1Ernest Thayer1643W11W4L2W6W34.010
2Lucas Beauchamp1418W12W5W1L3W44.010
3Henry Bird1448W14W10W9W2L14.09
4Flannery O'Connor1069W7L1W11W9L23.09.5
5David Duncanunr.W8L2L6W14W103.08
6Gilbert Chesterton889L10W15W5L1W73.07.5
7Robert Zimmerman419L4B---W13W8L63.07
8Shaun Alexander934L5W12W10L7W153.06
9Joanne Rowling809W15W13L3L4W143.05
10Charles B. Brownunr.W6L3L8W11L52.09
11Robert Frost793L1W14L4L10W122.06.5
12E Dickinson432L2L8B---W13L112.06
13Carl Sandburgunr.H---L9L7L12B---1.54
14Rodney Serling588L3L11W15L5L91.08
15Stephane Mallarmeunr.L9L6L14B---L81.07

Modified Median

Modified median treats players with even scores (i.e. 2.5 in a five-round event) the same as the Median. Players with plus scores discard only the lowest scoring opponent's score. Players with minus scores discard the highest scoring opponent's score.

Standings. Modified Median

#NameRtngRd 1Rd 2Rd 3Rd 4Rd 5TotTBrk[M]
1Ernest Thayer1643W11W4L2W6W34.014
2Lucas Beauchamp1418W12W5W1L3W44.014
3Henry Bird1448W14W10W9W2L14.013
4Flannery O'Connor1069W7L1W11W9L23.013.5
5David Duncanunr.W8L2L6W14W103.012
6Gilbert Chesterton889L10W15W5L1W73.011.5
7Robert Zimmerman419L4B---W13W8L63.010
8Shaun Alexander934L5W12W10L7W153.09
9Joanne Rowling809W15W13L3L4W143.09
10Charles B. Brownunr.W6L3L8W11L52.011
11Robert Frost793L1W14L4L10W122.07.5
12E Dickinson432L2L8B---W13L112.06
13Carl Sandburgunr.H---L9L7L12B---1.54
14Rodney Serling588L3L11W15L5L91.08.5
15Stephane Mallarmeunr.L9L6L14B---L81.07


The Solkoff tie-break method sums the final scores of the opponents. No scores are discarded.

Standings. Solkoff

#NameRtngRd 1Rd 2Rd 3Rd 4Rd 5TotTBrk[S]
1Ernest Thayer1643W11W4L2W6W34.016
2Lucas Beauchamp1418W12W5W1L3W44.015.5
3Henry Bird1448W14W10W9W2L14.014
4Flannery O'Connor1069W7L1W11W9L23.015.5
5David Duncanunr.W8L2L6W14W103.013
6Gilbert Chesterton889L10W15W5L1W73.012
7Robert Zimmerman419L4B---W13W8L63.010
8Shaun Alexander934L5W12W10L7W153.09.5
9Joanne Rowling809W15W13L3L4W143.09.5
10Charles B. Brownunr.W6L3L8W11L52.015
11Robert Frost793L1W14L4L10W122.011.5
12E Dickinson432L2L8B---W13L112.010
13Carl Sandburgunr.H---L9L7L12B---1.57
14Rodney Serling588L3L11W15L5L91.012.5
15Stephane Mallarmeunr.L9L6L14B---L81.010

*The United States Chess Federation Official Rules of Chess, 6th ed (2014) list these three as priorities 1, 2, and 4, respectively. Number 3 is upper half versus lower half. The USCF rules also list a 5th priority: alternating colors.

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