Not surprisingly other sports and pastimes have well developed systems for resolving ties.  The ones which most closely fit croquet derive from chess.

Ties arise in non-knockout tournaments for a number of reasons, e.g.:

• when insufficient rounds are played in a Swiss tournament or within Block Play
• when players each have the same number of wins but A beat B, B beat C and C beat A.
• Ties can also arise when further events, e.g. the play-offs, or the final have to be cancelled due to bad weather.

The options available to a Manager to resolve the winner of a competition fall into the following categories:

• Number of wins
• Who beat whom
• Number of points
• Quality of opponent
• Subsidiary competitions

It is prudent to advertise which tie breaking systems will be implemented in which order before a competition is played.  This saves later threats against the Manager's life.

Number of wins

This is the obvious one - whoever has the most wins is declared the winner!

Who beat whom

If two people tie by other criteria then, if one of those players has beaten the other, they win.

Number of points

The number of points scored in all games by individual players is summed; the player with the most points wins. 

This is to my mind an unsatisfactory method.  In large events one player may have played against really strong competition whilst another may have had games against puppies. In a handicap competition good players will probably have to sacrifice loads of hoops whilst the bisques are consumed and hence will only win by small margins.  This therefore is not a good method.

Quality of opponent

There are a number of methods whereby an attempt is made to quantify the quality of the people who have been beaten. 

(a)    Buchholz (also known as Solkoff). Sum of oppositions' scores.
This attempts to value the quality of opponents by the magnitude of their wins.  Losses are included.  For each of the players who tied, their previous opponents' games against others in that competition are examined.  For each opponent the sum of the points they won by and lost by are summed.  Then all those summed points are summed to produce a 'quality of opponents'.  The tied player with higher 'quality of opponents' is the winner. 

(b)   Sonneborn-Berger.  Oppositions' weighted scores.
This is calculated as above by adding scores of opponents who have been beaten but losses are not included.

Subsidiary competitions

Some form of competition such as shooting at the peg, arm wrestling, duelling pistols...

Example

Consider the following results sheet.  It illustrates the techniques discussed above and indicates some of the problems.

How to read the table. Each row contains the results for a player, hence player B lost to A by 10 points, beat C by 10 points, beat D by 7 points, lost to E by 8 points and beat F by 9 points.  The number of wins is the number of positive (bold) values in the row.  The number of hoop points is the addition of all the points won and lost in that row; e.g.  for A: Hoop Points = +10 -10 +4 +5 -6 = +3.

Assume the decisons are made in the following order

1). Number of wins

A, B and C each have three wins hence this cannot be used to determine a winner.

2). Who beat whom

A beat B, B beat C but C beat A.  This is circular hence we cannot determine a winner by who beat whom.

3). Hoop points

B and C have both got the same number of hoop points (8); hence we cannot determine a winner by the maximum number of points

4). Quality of opponent

1) Buchholz system.

A played B, C, D, E, F; sum of their hoop points = +8 +8 +12 -10 -21 = -3

B played A, C, D, E, F; sum of their hoop points = +3 +8 +12 -10 -21 = -8

C played A, B, D, E, F; sum of their hoop points = +3 +8 +12 -10 -21 = -8

A beat B, D, E; sum of their hoop points = +8 +12 -10 = 10

B beat C, D, F; sum of their hoop points = +8 +12 -21 = -1

C beat A, D, E; sum of their hoop points = +3 +12 -10 = 5

'A' has beaten better quality opponents under the Sonneborg-Berger system than B or C.  'B' has beaten better opponents than C.

Fortunately in this example 'A' is the winner under both 'Quality of opponent' tests. It is not unusual however for one player to be selected by a Buchholz test and another under the Sonneborg-Berger.

For the above Example:

Player B would be declared the winner had the following criteria had been applied:

1. Number of wins
2. Who beat whom
3. Points (A is eliminated)
4. Sonneborg-Berger

Player A would be declared the winner under the following criteria:

1. Number of wins
2. Who beat whom
3. Sonneborg-Berger