We’ll continue the FEPS analysis of the Swiss backgammon tourney run last weekend, turning our attention to fairness. The results are, as might be expected, somewhat different depending for the different aspects of fairness. Fairness is experienced differently by different players.
Fairness (A) will always favor established practice over an innovation. So, for backgammon, fairness (A) favors the bracketed consolation format over the Swiss because it’s more familiar. For chess, where the Swiss system is dominant, the reverse is true.
Because my tourney simulator cannot simulate Swiss pairings, I wouldn’t be able to results for fairness (C) directly but for the fortunate fact that we’re considering an event with exactly 32 entrants. For the Swiss, this means that there’s never any need to pair players from different score groups, so the tourney is nearly equivalent to one that could be run on a bracket. However, to run a faux-Swiss tourney in a bracket, it’s necessary (in order to avoid repeat pairings) to draw the bracket so that the drops always pair players whose round-by-round records are the same. Thus, while a true Swiss is happy to pair a player whose first two rounds were W-L against another whose record is L-W, and later to pair a W-W-L-L player against one who’s record is L-L-W-W, that doesn’t happen in my Swiss-ish bracket. This entails a small damaging effect on fairness (C), similar (if not quite so severe) to that caused by the failing to properly interleave drops in another bracket.
Comparing fairness (C) results also requires a common payout schedule. If only one place was paid, all but 31 matches of the 80-match Swiss would be irrelevant, and the fairness (C) results exactly equal to those calculated from the 31 matches in the top bracket of the consolation tourney. But Swiss I’m trying to approximate paid six places: 40%, 12%, 12%, 12%, 12%, 12%.
The consolation format would commonly pay four places: 45%, 22%, 22%, and 11%. But it can be made to pay 40%, 12%, 12%, 12%, 12%, 12% by equalizing the rewards for the upper bracket finalist, the consolation winner, the consolation finalist, and the two consolation semi-finalists. This is bound to compromise the fairness (C) because it gives equal rewards to players with a mean skill level of 1.301 (the consolation winner) and 0.658 (the consolation semi-finalists).
With these unnatural arrangements in place, the measured fairness (C) for the faux Swiss is 42.58, and for the consolation 42.41 (measured at luck = 3; 200,000 trials). The difference is barely significant in itself, but considering the extent to which disparate results are conflated in the consolation, I think it’s reasonable to conclude that fairness (C) favors the consolation. And, given that the consolation would ordinarily be played with somewhat longer matches in the upper bracket, the fairness (C) preference is clear.
If, however, you think differently about the payout schedules, there’s a very different result for fairness (C). One way of considering the benefit earned by players in bracketed tourneys is to think of participation itself as the benefit. In this view, the reward for winning a round in a knock-out tourney is simply the right to play another round.
Now, recall earlier attempts to derive optimal payout schedules through an iterative process failed because they almost always devolved into a schedule that rewards every entrant equally, at which point measured fairness (C) is optimal. That’s just what happens in a Swiss! Everyone gets to play the same number of rounds.
This tracks with the reactions I got to my backgammon Swiss. The most skillful players, who expected to play several rounds anyway, wanted some of those rounds to be longer, and perhaps also wanted a more nuanced payout schedule. From my simulation, the top four players, by skill, averaged more than five matches apiece. But the common run of players, who simply want to play some challenging backgammon and don’t necessarily expect to do well, loved the Swiss format.
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