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I am trying to rank participants based on sets of data that I have. The data used is for a competition. In this competition, you can participate in X amount of events, at the end of the event you participate in you get ranked in a Tier (eg from Tier 1 to Tier 6). Example:

Particpants  Tier 1 Tier 2 Tier 3 Total
John           2       3     2      7
Smith          1       2     4      7
Tom            3       1     1      5

In this example John was ranked in Tier 1 for 2 events out of the Total 7 he signed up for. The issue I'm having is that I am unsure what formula to use that would rank the participants, the ranking would take into consideration the amount of events you participated in (the higher the better) and the Tier you got per event (the more you have in Tier 1 the better etc).

Does such a formula exist? Please tell me if this seems to complex to calculate.

Taum
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  • It's not complex so much as broad. It's not too difficult to come up with a ranking system, but deciding which system is "right" depends on the situation is highly subjective. – Acccumulation Jun 22 '18 at 15:36

1 Answers1

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Briefly: it all depends on how you want to rank two players, one of them with more "Tier X or better" results, and the other with more "Tier Y or better" results. One choice might be better suited than another depending on the context; different choices produce different rankings -- in fact, essentially all "reasonable" rankings! See below for a more detailed explanation.

In general, a very natural representation of the "performance" of a player in scenarios like this one is a vector, with one component for each Tier, and the $i^{th}$ component being the number of events that player got into Tier $i$ or better (the lowest component then equals the total number of events the player participated in).

For example, with $4$ tiers (say, "winner", "podium but not winner", "good performance but no podium", "poor performance (but showed up!)") if a player got $0$ Tier $1$ results, $2$ Tier $2$, $1$ Tier $3$ and $5$ Tier $4$, the vector would be $|0 | 2 | 3 | 8|$: $0$ wins, $2$ podiums ($0$ wins $+2$ non-win podiums $=2$), $3$ good shows ($0$ wins $+2$ non-win podiums $+1$ non-podium good performance $= 3$), participating in a total of $8$ events ($0$ wins $+2$ non-win podiums, $+1$ non-podium good performance, $+5$ poor performances $=8$).

It seems obvious that if of two players $P_A$ and $P_B$, $P_A$ has the same or higher score in each component, he is at least as "good" as $P_B$ (we say that $P_A$ dominates $P_B$). E.g. $| 0 | 2 | 3 | 8 |$ dominates $| 0 | 1 | 2 | 8 |$. But if $P_A$ has some higher and some lower components than $P_B$, there is no obvious way to rank $P_A$ relative to $P_B$. It's entirely up to you to decide if two Tier $2$ results ($|0|2|2|$) are better or worse than a Tier $1$ and a Tier $3$ ($|1|1|2|$): in some situations one might be more appropriate, in others the other.

In fact, you can prove the following. Consider a set of players and choose an arbitrary ranking for the players you want to end up with, with the sole condition that no player that dominates another can be ranked worse. Then, for each tier $T$, you can find a (positive, integer) number of "points" $R_T$ to award to every Tier T result, so that if you rank your players according to the sum of points awarded to each, you get exactly the ranking you want!

Anonymous
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