# Finding the Second and Third Best

Getting the Most from an Afternoon looked at the choice of bracket for a time-limited tourney and concluded that moving to the use of a shifted bracket was probably a bad idea because it might make the event run too long.

This post will consider another aspect of the design. How fair is it? And can the fairness be improved by making changes to the distribution of the prize fund?

I’ll begin this analysis by considering the performance of the two brackets in terms of how they tend to distribute the top three prizes. I’ll do this without regard to the way the prize fund is divided. In a subsequent post, I’ll bring the payout schedule back into the analysis.

One way to compare the performance is to look at the average skill levels of players in the money. For the unshifted bracket:

champion: 1.057; runner up: 0.492; consolation winner: 0.697,

Compare the paying places for the shifted bracket:

champion: 1.057; consolation winner: 0.802; consolation runner  up: 0.352.

As a point of reference, the mean Z scores for the three best players (of the 13):

best player: 1.660; next best: 1.164; third best: 0.850.

The choice of bracket doesn’t affect the way the overall winner is chosen, so it’s not surprising that the two designs have the same Z score for the champion. But the shifted bracket is considerably better at finding a good player for the second spot. On the other hand, the third-place winner with the shift is conspicuously weak.

One way to understand this is to look at the results when luck is set to zero, that is, where the better player always wins. In both brackets, the top prize always goes to the best player.

In the unshifted bracket, the second-best player wins the second prize only when they happen to be drawn into the half of the draw that doesn’t include the best player. When they are in the same half of the draw, they’ll drop into the consolation, where they’ll be assured of winning the third prize. So third prize always goes to either the second- or third-best players in the draw. But the second prize can be won by a player as weak as the eighth-best, which happens whenever the top seven are all drawn into the same half of the thirteen bracket. It goes to the second-best player about 54% of the time.

In contrast, with zero luck, the shifted bracket will always give the second prize to the second-best player. But the third prize goes to the third-best player only about 75% of the time, and can go to a player as weak as the fifth best.

Now, putting back in a generous luck factor suitable for a game like backgammon weakens these expectations considerably. But the traces of the zero-luck expectations can still be seen in the actual results for the two brackets.

The shifted bracket is better at finding a strong player for the second prize, but less good at giving the third prize to a good player. The unshifted bracket does the reverse: it’s less likely to bestow the second prize on a good player, but more likely to do so with the third prize. In fact, it tends to give the third prize to a better player, on average, than the one who receives the second prize.

This calls into question what I’ve been calling the second and third prizes for the past few paragraphs. If, in the unshifted bracket, the consolation winner is better, on average than the runner up, maybe it’s really the second prize, and the runner up is the third prize.

This doesn’t really matter, perhaps, for some purposes, but it does matter when it comes to distributing the prize money (or points, or whatever). In the next post, I’ll consider some options for how the prize fund should be allocated in the unshifted bracket given these results.