# Recharge Redux

From time to time I look back at old posts. Recently I looked at my pair of posts on the issue of recharge rounds, particularly the ones that require the winner of a “losers bracket” to defeat the winner of the “winners bracket” twice in order to win the overall championship.

Here I considered recharge in terms of efficiency, participation, and spectacle, and concluded that none of those three considerations really supported the practice of including a recharge round. But here I did some simulations to see if recharges contributed in a positive way to fairness, especially fairness (C). I concluded, with some surprise, that fairness was enhanced by the use of a recharge round.

But in looking back at that post, I’m struck by the fact that I did all of the simulation work on 16 brackets. A 16 bracket is the smallest bracket that’s capable of being shifted. Perhaps the results would have been different if I’d used a larger bracket, where the number of additional rounds played by players coming up through the losers bracket is greater. So I re-did the experiment, recently, using 64 brackets rather than 16s. And, sure enough, the results are somewhat different.

As before, I considered eight cases, with all possible combinations of three binary factors: [recharge|no recharge], [shifted|unshifted], and [high luck|low luck]. Note, however, that the results are not directly comparable because this time I’m using my revised fairness metric.

 low luck (luck = 1) high luck (luck = 3) 64 AB|C|D|E|FX 23.06 91.04 64 AB|C|D|E|FXR 20.94 91.04 64 ABCD|EF|X 22.56 88.62 64 ABCD|EF|XR 20.68 88.84

The numbers indicate fairness (C), with a winner-take-all payout scheme. In each case, I ran 200,000 trials, except for the high-luck, shifted models, where I ran 500,000 trials in hope that a longer run would cause the small difference to become statistically significant. It didn’t – the 0.22 difference is not significant.

To get a significant result, I bumped the luck factor up to 4 – a level that some of those who’ve commented on earlier posts think is a better fit for backgammon. At that level, the 64 ABCD|EF|X beat the 64 ABCD|EF|XR, with f(C) = 113.00 and 113.95, respectively.

So, the earlier result is confirmed with respect to low-luck events. But it’s not confirmed for high-luck events, and the conclusion is reversed for very-high-luck events.

I take this as validation for the widespread practice, at least in backgammon tourneys, of omitting the recharge round in designs that are otherwise perfect double-eliminations.

There is still, of course, a fairness (A) objection to omitting the recharge round. ‘How can you call this a “double elimination” if the F1 winner is not the overall winner just because she lost once, in the X round?’ But the recharge round is usually so bad for other reasons that I think directors would do well to steel themselves against the fairness (A) objection.

One other result to note in passing: in each case, the shifted bracket’s results are fairer than the corresponding unshifted bracket, though the benefit of the shift is smaller for the low-luck cases. This is in line with previous results.