If you have ever played in Hockey pools (or any other kind of pools) you know that if you do not get a good drawing rank, your chances of winning anything are greatly diminished. So, here’s how a typical pool works. There are pool players that will form “teams” with league players (usually real players from the real leagues, with their standardized scores) from a total of league players.
To form the teams, the players put their numbers in a hat, and the numbers are drawn one by one, determining in which order, in each round, pool players will get to choose their next pick in the remaining league players. That is, if the order drawn is, say, 5, 3, 4, 2, 1, then pool player number 5 gets to choose first, picking one league player, then goes pool player 3, and so on, until all pool players picked a league player, thus completing one round. There are of those rounds so that each pool player has his team of league players.
The league players’ strength tend to be distributed according to a vaguely zipfian law, meaning that the league player occupying rank has a strength proportional to . The consequence of this, is that if you get first draw, you’re really advantaged, because you always get to pick the strongest remaining league player. Conversely, if you do not get a good draw order, you’re pretty much done for.
So the usual way of drawing league players (assuming the best strategy is to pick the strongest available league player and that you cannot really pick 2- or 3-combos) is really favoring the pool players according to their draw order, even when the league player score isn’t zipfian. Even if they’re uniformly distributed, the first pool player to draw is disproportionately advantaged.
Let us consider the results of the usual greedy scheme for 100 experiments, each with 5 pool players, 25 league members with scores uniformly distributed on 0 to 100. The league players’ scores are different from experiment to experiment, while the draw order remains the same, namely 1, 2, 3, 4, 5, without loss of generality. The teams’ total scores (vertical axis) are distributed as:
The box plots show clearly that the advantage is proportional to draw rank. However, if we redetermine pick order at the beginning of each round (that is, we put back the numbers in the hat and draw them again at the beginning of each round), we can remove that bias. Consider now the same experiments (the same league player scores than in the first set of experiments) but with drawing order decided randomly at the beginning of each round. We now get teams with strenght distributed as:
where the bias is essentially removed.
It is open to debate whether team scores as equal as possible make a better Hockey pool game, but it certainly prevents the first pool player—the one that was chosen to pick first—from getting disproportionately advantaged, and the last one to be so disadvantaged as having to next to no chance of winning anything. A better, or at least more egalitarian, strategy is to draw the pick order at each round, thus removing the bias.
However, pools like that cannot be played overly strategically because your performance will be largely determined by the order in which you are allowed to pick your next league player. Of course, if you know that in real like, such-and-such trio performs really well and you already have two of the three players, maybe your pick needn’t be the strongest remaining league player but the one that completes your trio. Whether you use the greedy approach with the draw order decided at the start or the randomized approach, it will be difficult to get real-life duos or trios in your team.