Ranking Part 1: Scores for ARDF

Among other duties, the ARRL ARDF Committee is tasked with establishing a fair and equitable ranking system for use in team selection. It has been rightly pointed out that there are many considerations when selecting a team. Some considerations deal with subjective factors that have little or nothing to do with how well a candidate performed historically at ARDF events.

But competitive performance is one useful and objective factor that can be readily quantified and analyzed. And so competitive performance at ARRL-sanctioned ARDF events should be calculated and utilized as one component of a fair team selection process. It will also be helpful to competitors wanting to track their performance relative to their competition.

ARDF rules stipulate that the place (1st, 2nd, 3rd, etc.) of an individual competitor depends firstly on the number of transmitters found, and secondly on his or her running time. The rules provide no numerical formulae for deriving a single number to represent a competitor’s performance. But such a number would be helpful for facilitating a mathematically-rigorous ranking system, and would be the first of three steps to establishing such a system:

Establish a scoring system for ARDF competitions.
Define an approach for adjusting competition scores to make them comparable between courses with different difficulty levels.
Create a formula for combining adjusted scores to obtain a competitor’s rank relative to other competitors in his or her age/gender category.

The subject at hand is the first bullet above: how to represent a competitor’s ARDF performance at a single competition as a single numerical score. The other two bullets will be addressed in later postings.

Scores for ARDF

ARDF rules require that both the number of transmitters found, and the amount of time a competitor takes to reach the finish line, be used to determine a competitor’s place. And since a competitor’s score should accurately reflect their placement relative to other competitors, the score calculation must also utilize both the number of transmitters found, and a competitor’s total time, to derive their score.

Since the rules require the total transmitters found to be decisive in a competitor’s placement, time must never contribute more to a competitor’s score than does a single transmitter. In other words, no matter how fast a competitor runs a course, his/her score for that race must be less than the score of any competitor who found more transmitters. The rules demand it.

Most other properties of a score may be chosen arbitrarily. For instance, whether a score is proportional or inversely proportional to a competitor’s placement is totally a matter of convention. So, to keep things simple, the following scoring conventions will be used here:

A higher score reflects a higher (better) placement with a score of 100.0 being awarded to the first-place finisher(s) of an age/gender category.
All transmitters count the same amount toward the calculation of a competitor’s score.
Competitors’ times are scored relative to the fastest time of any competitor who found the same number of transmitters.

Item 3 is worthy of additional discussion. Note that all competitors within an age/gender category are always assigned the exact same course to run, and therefore the same number of transmitters to find. But the rules allow any competitor to choose, at their own discretion, to find only a subset of the assigned transmitters. A competitor might elect to find fewer transmitters if, for instance, finding more transmitters seems likely to result in their being classified Overtime (OVT). By choosing to find fewer than the assigned number of transmitters a competitor can avoid being classified OVT. But by choosing to find fewer transmitters a competitor accepts a placement below all the competitors who find more of the assigned transmitters than they. So, in effect, by choosing to find fewer transmitters they place themselves in a separate category of finisher. Since the rules effectively segregate finishers’ placements by the number of transmitters found, it is appropriate then that their times be likewise segregated and compared only to the times of others in their category who elected to find the same number of transmitters.

Score Calculations

Since all transmitters count the same toward a competitor’s score, and a competitor’s speed performance (time) must count no more than a single transmitter, each transmitter and the time component of the score is assigned a maximum score value of:

(1) $\begin{equation*}s_{N} = 100 / (N + 1)\end{equation*}$

Where
$s_{N}$ is the point value of each transmitter found (and maximum points awarded based on finish time),
$N$ is the total number of transmitters assigned to a competitor.

So, for example, if an M40 competitor is assigned five transmitters to locate, then each of the transmitters will have a score value of $100 / (5 + 1)$ or $16.\overline6$ points. The maximum number of points the M40 competitor can gain by having the fastest overall time among all M40 competitors on that course is also $16.\overline6$ points. So the total score for competitor “c” is given by:

(2) $\begin{equation*}S_{c} = (n_{c} + T_{1}/T_{c})\cdot s_{N}\end{equation*}$

Where
$S_{c}$ is the score for competitor $c$ ,
$n_{c}$ is the total number of transmitters found by competitor $c$ ,
$T_{1}$ is the overall time in seconds of the fastest of the competitors who found $n$ transmitters,
$T_{c}$ is the competitor’s overall time in seconds, and
$s_{N}$ is the value derived in equation (1)

Examples:

(Example 1) M40 competitor “A” is assigned to locate five transmitters and finds all five, and also has the best overall time (i.e., $T_{1} = T_{A}$ ) amongst all M40 competitors who found all five transmitters. He would have a score of:

$S_{A} = (5 + T_{1}/T_{1})\cdot 16.7 = 6 \cdot 16.\overline6 = 100.0$ points

(Example 2) M40 competitor “B” who found only four transmitters, and who had the best overall time amongst all M40 competitors who found four of the transmitters would have a score of:

$S_{B} = (4 + T_{1}/T_{1})\cdot 16.7 = 5 \cdot 16.\overline6 = 83.\bar3$ points

(Example 3) Let’s look at a very slow M40 competitor “C” who found all five transmitters but required 10 times as long to reach the finish line as did the competitor in Example 1:

$S_{C} = (5 + T_{1}/(10 \cdot T_{1}))\cdot 16.7 = 5.1 \cdot 16.\overline6 = 85$ points

Note in Example 3 that the slow finisher finding all five transmitters still received a score substantially higher than the best competitor who found only four transmitters in Example 2.

Other Observations

Note that equation (1) implies that the more transmitters assigned to competitors the smaller the effect of each individual transmitter and of time on the competitors’ scores. Consider: if there were only one transmitter to find, equation (1) says that finding the single transmitter would account for 50 points, and one’s finish time could account for up to an additional 50 points. But if we increase the number of transmitters to nine, then finding a single transmitter adds just 10 points to one’s score and time accounts for just 10 points as well. This can be difficult to reconcile when combining scores (e.g., for averaging, etc.) from competitions in which different numbers of transmitters were assigned.

To avoid the pitfalls of mixing or comparing scores derived from differing time and transmitter component values, the following approach is suggested:

Only scores for the same competition format will be combined or compared.
Only scores for competitors finding the same number of transmitters on the same course in the same age/gender category will be combined or compared.

Limitations of Scoring

The above approach for deriving a single score for competitors’ ARDF race results can be helpful for comparing performance among those competing within the same age/gender category and on the same course. But it would fall short if it were used to compare performances on courses of significantly different levels of difficulty. That subject will be addressed in Part 2.