Croquet Canada Gets A New Ranking System

by Louis Nel
Relative player strength is expressed by a Ranking Index, briefly Ranking, on a continuous scale of numbers, displayed to no more than two decimal digits.

Croquet Canada Gets A New Ranking System
From the Wicket Times March 1998 Vol.1 No.1


1. Description of the New System

Relative player strength is expressed by a Ranking Index, briefly Ranking, on a continuous scale of numbers, displayed to no more than two decimal digits. There is a nominal correspondence between Handicaps and Rankings, as shown in the following table:

Initial Rankings are determined by the Handicapping Committee. After that every game in a sanctioned tournament causes a Ranking Adjustment to be calculated. For example, if Mary (86.00) beats John (87.24) by a score of 20 to 10, Mary will receive (as explained below) an Adjustment of $0.13$ while John will get the negative of Mary's adjustment, namely -0.13. The size of the Adjustment depends on the scores obtained and on the difference between the winner's Ranking and the loser's Ranking. The Handicapping Committee will do the calculations, using computer software. Section 2 below describes how.

At the end of each tournament the Adjustments received by a member from games played are all added to the previous Ranking. The updated Ranking takes effect immediately after the tournament. For example, if Mary enters the tournament with Ranking 86.0 and picks up Adjustments of 0.13, -0.06, 0.11, -0.04 in her four games played, her Ranking immediately after the tournament will be 86.0 + 0.13 - 0.06 + 0.11 - 0.04 = 86.14.

The Adjustment formula will normally be the only means to change Rankings. The Ranking system will be the only one used for The Laws of Association Croquet (International Rules). For USCA rules, Rankings will be used in addition to Handicaps.


2. How Adjustments are Calculated.

The Winner's Adjustment is a product of two factors: a Score Factor S and an Assessment Factor A(D). The Score Factor is calculated as follows:

S = (Winner's Score)/(Winner's Score + Loser's Score).

For example, in a 20/10 victory we have S = 20/(20 + 10) = 20/30 = 0.67.
The Assessment Factor A(D) depends on the difference

D = Winner's Ranking - Loser's Ranking.

So for a winner Mary (86.0) and a loser John (87.24) that difference comes to:
D = (Winner's Ranking - Loser's Ranking) = 86.0 - 87.22 = -1.22, a negative quantity. Had John won, D would be +1.22. For a given D the value of A(D) can be found from the following table.

negative D A(D) positive D A(D)
-3.90 or less 0.25 0.01 ... 0.16 0.12
-3.89 ... -2.76 0.24 0.17 ... 0.32 0.11
-2.75 ... -2.20 0.23 0.33 ... 0.48 0.10
-2.19 ... -1.82 0.22 0.49 ... 0.66 0.09
-1.81 ... -1.52 0.21 0.67 ... 0.84 0.08
-1.51 ... -1.27 0.20 0.85 ... 1.04 0.07
-1.26 ... -1.05 0.19 1.05 ... 1.26 0.06
-1.04 ... -0.85 0.18 1.27 ... 1.51 0.05
-0.84 ... -0.67 0.17 1.52 ... 1.81 0.04
-0.66 ... -0.49 0.16 1.82 ... 2.19 0.03
-0.48 ... -0.33 0.15 2.20 ... 2.75 0.02
-0.32 ... -0.17 0.14 2.76 ... 3.89 0.01
-0.16 ... -0.00 0.13 3.90 or more 0.00
Notice that A(D) decreases steadily (without jumps) from a maximum of 0.25 to a minimum of 0.00 as D increases from below -3.9 to above 3.9. (For curious readers: the table is computed from the design formula A(D) = 0.25/(1 + exp(D)), where exp(D) (also written "e with superscript D") is the exponential expression widely used in science to model growth phenomena).


Mary (86.0) beats John (87.24) by a score of 20 to 10.
So D = 86.0 - 87.24 = -1.24.
In the table we find that -1.24 lies between -1.26 and -1.05, so it gives A(D) = 0.19.
The Score Factor, as we have seen, amounts to S=20/(20+10)=0.67.

So we have:

Mary's Winner Adjustment = (Score Factor) X (Assessment Factor) = S x A(D) = 0.67 x 0.19 = 0.13

John's Loser Adjustment = - (Winner Adjustment) = -0.13

All of these computations are rounded up to two decimals digits.

Remark (unimportant)

The World Chess Federation and the World Croquet Federation both use an expression of the form V/(1 + exp(PD)), where V and P are user selected constants. Neither organization uses a Score Factor (chess has no game scores). In our case, V = 0.25 and P = 1. The constant V controls volatility, while P controls the difference in skill level that corresponds to a unit difference in Rankings for accurately ranked players.


3. What does a Ranking mean?

If Sally is the best player, has the highest Ranking, and is still improving all the time, she may yet find her Ranking decreasing. This could happen for some of the following reasons: (a) Sally was initially ranked too high; (b) some competitors were initially ranked too low; (c) some competitors are improving faster than Sally is. For corresponding reasons, a player's Ranking may increase without improvement in his skill. Meanwhile, the sum of all Rankings in a fixed population remains constant.

These possibilities underline the fact that a Ranking is a relative measure, not an absolute one. It indicates approximate competitive position within the player population --- nothing more. Generally speaking, the more frequent the tournament participation, the more accurate the Ranking.

4. What's wrong with the Handicap system?

It is generally perceived to work well, but it has some unsatisfactory features. They are inconspicuous and are further concealed by Handicapping Committee interventions which come to the rescue before things get too bad. However, while these committees readily intervene in underrated cases (greasing squeaky wheels) they shy away from intervening in overrated cases. Our tests have shown that for both overrated and underrated players, automatic correction happens significantly faster in the Ranking system than in the Handicap system. For players with a fairly constant skill level, the two systems have similar volatility. The design parameters of the Ranking system were chosen with these effects in mind. Let us detail the mentioned unsatisfactory features in the Handicapping system.

Tracking points are hoarded.

Until they reach 28, tracking points remain without effect. A player staying around 1.0 -26 is taken as 1.0 in calculations while practically being a 1.5: an overrated player created by the system. An automatic system ought to correct aberrations, not create them. In the Ranking system the Adjustment points are immediately effective after the tournament.

Some overflow tracking points are lost.

While the system is already slow to keep up with a rapidly improving player, this slows it down even further. Information is wasted. Adjustment points are never discarded in the Ranking system.

Game scores are ignored.

This is a further waste of useful information. Isolated big wins are not significant (and will be canceled by isolated big losses). But a player who has significantly more big wins than big losses (e.g. a rapidly advancing player) deserves credit for this.

Instability and anomaly.

When Tracking points reach 28 or -28, a jump occurs in the relative positions of players. Jumps can cause instability and anomaly. The mentioned jump can cause John to gain 23 tracking points on an equally skilled rival Mary even while John is having a worse win/loss record. How? Suppose they are both at 5 -27 when Mary beats John in their last game of a tournament . So Mary becomes 5 -23 and John becomes 6 0. They get identical records in their following tournament, namely: lose to a 5, beat a 5, lose to a 5, beat a 5, lose to a 5, beat a 5. At that point, Mary is again 5 -23, while John has gone to 6 15. The very same win/loss record obtains in a further tournament. Mary stays at 5 -23 while John becomes 5 0. Had Mary lost that first game, she would have been 5 0 and John would have been 5 -23. Thus, sometimes, one game could cause a net difference of 46 tracking points --- in the unexpected direction. This kind of thing happens on smaller scale all the time.

Such instability and anomaly cannot arise in the Ranking system --- jumps do not occur. In the mentioned example, Mary and John would have had Rankings close together all the time, the better record just slightly ahead, as it ought to be.


Against a lower handicap, a player can gain up to 8 tracking points, but cannot lose any; the losing lower handicap can lose at most 4 points. So in every tournament, significantly more tracking points are gained overall than are lost. Consequently, for the population as a whole, handicaps decrease overall regardless of improvement in skill level.

For this reason we are introducing Rankings also for USCA rules play (in addition to Handicaps). The Rankings total for a fixed population remains constant. If we were to weight flights via Handicaps in USCA rules and via Rankings in International rules, the USCA flights would gradually become overweight relative to the International rules flights. To counter this obesity, both kinds of tournaments will be on a Rankings diet.

So now you know why Ranking Adjustments make my computer so happy that its terminal tends to get a little damp ...