Three Match Guarantees

One of the perceived disadvantages of knockout tournaments is that they don’t offer much play to weaker (or unlucky) players. They’re very efficient, but that efficiency comes at the expense of participation – half the field in a knockout tourney get to play only one match before being eliminated.

Running a double elimination will give all teams at least two matches, as will running a consolation bracket. To offer each team at least three matches, you can run a triple elimination (which is quite rare and tends to raise thorny issues of bracket design), or add a last chance as a consolation-to-the-consolation bracket.

There is another way to guarantee each team a third match that is much less complicated than running a full triple elimination, and yet still offers a team a chance to win the overall title after losing the first two rounds. Here is such a bracket for a field of 16: 16DE3GG. It’s essentially a double-elimination bracket with an additional round in the lower. It this fair? How does it play?

The added round is the F round, which takes E drops, inserted between the A and B drops in the lower bracket: A.E.B.C.D.|.X.R. It’s still possible, and usually desirable, to shift the C and D drops forward, though it’s also possible to draw the bracket A.E.B.|.C.|D.X.R, which is how you’ll see the bracket drawn in other sources. Note that I’ve chosen to draw this bracket with a recharge round. Since it’s possible for the tourney to be won by a team with two losses, it would seem more-than-usually unfair (in the fairness (A) sense) to allow a team with only one loss to still lose the tourney. It’s also true, as we might suspect for any 16 bracket, that the recharge also helps fairness (C).

The natural design to use for comparative benchmarks is the A.B.C.D.|.X.R. How does the interpolated F round with E drops affect fairness? Practically not at all, it seems. For the A.E.B.C.D.|.X.R, fairness (C) is 17.53 and 68.08 for luck at one and three, and for the comparison design, we get 17.53 and 67.97, neither of which difference is statistically significant.

This does not speak well for the A.E.B.C.D.|.X.R design. We’ve added a full round, with four extra matches, so you might expect at least some small improvement in fairness (C). Using those extra four games, for example, to make the D and I rounds of the A.B.C.D.|.X.R. best two of three games would bring fairness (C) down to 14.38 and 62.12. But here, the extra resources have vanished without a trace.

Perhaps we’re not seeing anything because the payout we’re using, 65/35, puts the prize money too far away from the F round we’d like to know more about. To get a better look at what’s going on, let’s rerun the analyses with a payout schedule that’s specifically designed to shine the spotlight on the effect of the interpolated F round. Donald the Potholer, in the comments to the recent TGT fairness (C) post, suggests a diagnostic payout strategy designed to make all parts of a design visible, but here I’ll go to the extreme of pulling nearly all of the money down close to the critical round. Thus, for both A.B.C.D.|.X.R and A.E.B.C.D.|.X.R, I’ll use this payout: 12/8/8/8/8. The overall winner gets a small bonus, with a twelve percent payout. Eleven other players each get eight percent payout – that’s everyone except the four players who are the first to be eliminated: in the E round in the traditional design, or in the new F round in the three-game-guarantee bracket. These four players get nothing.

Now the effect of the new guarantee round is dragged front and center, and it’s not pretty. For A.B.C.D.|.X.R with the new payout, fairness (C) is 11.88 and 26.29. But for the A.E.B.C.D.|.X.R design it’s 15.92 and 43.25.

Another diagnostic tool for showing what’s happening is to track the round-by-round fairness (b:X) calculations. Here’s the full list of fairness (b:X) numbers for A.E.B.C.D.|.X.R, with the F round highlighted: {0.22, 0.14, 0.08, 0.08, 0.40, 55.36, 5.07, 0.26, 0.59, 0.34, 15.28, 3.54}. The two rounds that stand out from the rest are the two that pit teams with a different number of losses against each other. A bad number for the reconciliation round (round K) is, perhaps, unavoidable. But the F round is gratuitously bad.

From a fairness (C) perspective, the three game guarantee is something between a non-factor and a catastrophe, depending on your focus. Using a top-heavy payout scheme it matters little because it affects only teams that are very unlikely to cash anyway. But if you adjust the scheme to focus on the new F round, you see that it’s a very bad round indeed. In order to give the 0-2 teams another round, you’re making the E round essentially meaningless.

This is not to say that there’s no reason to adopt a three-game-guarantee design, it’s only to say that there’s no fairness reason to do so. There is, obviously, a participation reason. But if you want to offer a three game guarantee, and realize that to do so you need to make one round meaningless, it would seem to make more sense just to drop the 0-2 teams into a special consolation bracket, or even into just a consolation game. That’s the approach we saw with the Little League World Series.

A final shout out to another of my perceptive commentators, jamesbcrazy, who discovered an anomaly in some of my fairness (C) calculations, and helped me trace it to overly aggressive clipping of Gaussian skill distributions. For this post forward, all simulations (unless otherwise stated) will clip at -10 and 10, not at -4 and 4.