Fairness and Recharge Rounds

Recharge rounds are the possible immediate rematch rounds where a match is replayed if the initial pairing results in equalizing the number of losses for the two teams. The most common (but not the only) use of recharge rounds is in unifying the lower and upper brackets in a double-elimination tourney.

Yesterday the propriety of including a recharge round was considered in terms of efficiency, participation, and spectacle, and I found that none of these factors made the case for the recharge round was not particularly strong. This post will discuss the other one of the FEPS goals: fairness.

The basic case for the recharge round is often stated something like this:

How can you call it a double-elimination tournament if someone can not win the tournament even though they’re lost only once?

This is a fairness (A) argument – an argument for honoring reasonable expectations. And it’s better, in my mind, than most fairness (A) arguments because the reasonable expectation is not just a matter of people remembering the features of some other format, but an expectation that’s grounded in the very language we use to describe the format.

If there were something else to call a “double elmination”, careful directors could avoid the name and (perhaps) avoid that expectation. But there is no good name for a tourney that’s double elimination except that it omits the recharge round.

To counter this, some who favor omitting the recharge round argue that fairness demands this accommodation because it’s so much more difficult to get to the final through the lower bracket. Here’s a snippet from a discussion on a site devoted to battling robots:

For the “as needed” game, isn’t being 9-1 sufficient to beat a 5-1 winners bracket winner? Having to fight 4 less fights to get to the finals should be enough of an advantage without also having to lose twice.

In all but fairly exotic bracket structures (including those from Joe Czapski), one needs more wins to get to the final through the lower bracket than it takes to get there from the upper bracket. Is dispensing with the recharge round a fair compensation for this?

I have no way to quantify the fairness (A) argument in favor of the recharge round. But it is possible to address the claim as to fairness (C). Here’s a plausibility argument that suggests that fairness (C) should be lower for tourneys with a recharge round than those without one:

To decide whether a recharge round enhances fairness, consider the situation immediately following the “X” round, if the lower-bracket survivor wins. That team has a record of, say, six wins and one loss. The upper bracket champion has a record of say, four wins and one loss – the loss having been suffered just prior, in the “X” round itself.

Now, it stands to reason that a team that’s 6-1 should be, on average, better than a team that’s 4-1. So, if you hold an “R” recharge round, you’re just giving a somewhat worst team another opportunity to get a lucky win against a better team, and so fairness (C) should suffer because the prize is more often awarded to a less skillful team.

For a time, when I was waiting for my new simulator to be ready to test this, I had convinced myself that this was true, and that it would put a further nail in the coffin of the recharge round when I finally was able to run the tests.

But it’s not true. Now that I’ve been able to actually run the experiment, it appears that my earlier conjecture, that adding a recharge round would have a small positive effect in the item of fairness (C).

Here are the fairness (C) averages from 100,000 runs of a basic 16-team double elimination. In addition to running the experiment with and without a recharge round, I’ve also run it with and without a bracket shift, and with high skill (luck = 1) and low skill (luck = 3):

	high skill	low skill
16AB|C|DXR	64.05	30.80
16AB|C|DX	61.01	30.27
16ABCD|XR	64.18	31.45
16ABCD|X	60.92	30.82

(The bracket notation is discussed here.)

In every case, the version of the tourney with a recharge round has a higher average fairness than the equivalent version without one.

Why is the plausibility argument invalid? I think you have to keep in mind that playing more games means that there’s more opportunity for superior play to win out. Yes, it’s true that the recharge may allow for a lesser team to edge in ahead of a better one, but it will happen more often the other way round.

(You might also note that, while the earlier finding that the shifted bracket is fairer than the unshifted bracket, this is not the case for the high-skill case, where there is more skill progression.)

In conclusion, I have to say that while in many cases I still believe that it’s better to dispense with a recharge round, considered only in terms of fairness (C), the recharge round is justified.

Recharge rounds can be used for reasons other than ensuing that a double elimination really does mean that every team except the champion loses twice – in the next post, I’ll explore another use for the recharge round.