Do Split Leagues Really Mean More No-Shows?
The Broomball Caucus conference is weighing whether to divide its teams into “softcore” and “hardcore” divisions. A common argument against such a split is that the softer league would concentrate the very teams most likely to forfeit — dragging down the whole season with no-shows. On the surface, that worry sounds reasonable. But the math tells a different story: splitting leagues along forfeit propensity actually reduces the overall forfeit rate for any given game.
Consider a league where half the teams forfeit with probability a and the other half with probability b, where a < b.
Case 1: Split leagues. Put all the a-type teams in one division and all the b-type teams in the other. A forfeit occurs when exactly one of the two teams fails to show. In the hardcore division, that chance is 2a(1-a); in the softcore division, it is 2b(1-b). Averaged over the whole league, the per-game forfeit probability is a(1-a) + b(1-b).
Case 2: One combined league. Every game draws from the full pool, so each team has the average forfeit probability (a+b)/2. The chance a given game is lost to a no-show is then (a+b)(1-(a+b)/2).
Note that a forfeit here means one team shows up and the other doesn’t. If both sides fail to appear, nobody really loses out on play.
Plotting the two expressions reveals what’s happening. In the split scenario, the forfeit probability falls off sharply toward each corner — if, say, no “softcore” teams ever show up but all “hardcore” ones do, there are zero problem forfeits. In the single-league case, the math is harsher: a few unreliable teams drag the forfeit probability up for everyone else, including the most dedicated squads. Superimposing the two curves, the split-league outcome is at worst equal to the single league, and strictly better everywhere else.
The conclusion holds under the assumption that forming separate leagues doesn’t itself change a team’s forfeit behavior. That assumption is debatable — it’s possible that softer divisions could make games feel less intimidating and actually increase attendance. But if forfeit rates are simply divided up rather than mixed together, split leagues are the clear winner for minimizing wasted games.
