Kickball was the one of the most fun and anxiety inducing parts of middle school. On one hand, the game was simple in its mechanics and enjoyable to play. On the other hand, the picking of teams by team captains reveals a ranked preference that often feels uncomfortable for those picked towards the end. Playground politics aside, here we’ll look at a few different picking rules and their effects on the fairness of the resulting game.

individual players

Not all people are equally skilled when it comes to kickball. Let’s go a step further and say that kickball skill follows a bell curve like the one shown below. Most people are close to the average skill level. Exceptionally good and bad people exist in smaller numbers in the tails of the distribution. Every individual is assigned a number representing their skill, pulled from a normal distribution.

picking rules

There are lots of common rules used for picking teams.

  • random - count off people in the group like “team 1, team 2, team 1, team 2, team 1, …” This is a favorite of gym teachers, though I’ve never seen kids elect to do this themselves.
  • 1-1 - Team 1 picks 1 person. Then, team 2 picks 1 person. This repeats until all the students belong to a team.
  • 1-2 - this system adds a twist to the 1-1 system. Team 1 begins by picking 1 person. Team 2 then picks 2 people. Team 1 picks 2 people, Team 2 picks 2 people and so on until Team 1 picks the last person. Such a system is seen as “more fair” because it compensates team 2 for not getting to pick 1st.

comparison

One way to see the differences between these 3 strategies is to simulate the picking of kickball teams under the rules and see what happens. In order to evaluate the outcomes, it’s useful to come up with a metric to track so that we can better compare the rules.

  • 1st pick advantage rate - how often does team 1 end up with the “more stacked” team? Here we can just sum up the skill levels of the members of each team to compare.

Let’s take a look at the simulation results!

This seems a bit odd. The simulation was calculated with a population sizes of 4, 6, 8, … all the way to 100. I was expecting that as the number of people increases, whatever unfairness might exist would disappear over time. Instead, we’re seeing oscillation, with alternating numbers of players being fair and unfair. More specifically, for population sizes divisible by 4 (4, 8, …, 100), picking with the 1-2 system ends up with fair teams, according to 1st pick advantage rate (as defined above). However, other even population sizes (6, 10, 14, …) leads to 1st pick advantage rates of around 83%. What gives?

Let’s try looking at two other metrics besides 1st pick advantage rate. It looks like for those even numbers not divisible by 4, 1st pick really does give an advantage. However, how bad is it really? We can use the following metric to figure out.

  • absolute skill spread - by how much do the teams differ in summed player skill? To calculate this, we look at the absolute value of the difference between the summed skill levels of players for each team.

So it appears there’s still some oscillation. Here’s the same graph as above, now splitting into separate curves for the divisible by 4 case and the other even numbers. For convenience, we’ll label the numbers divisible by 4 with “0 mod 4”, meaning that they have a remainder of 0 when divide by 4. The other even numbers are (2 )

From this, we see that there’s a decay as the population size increases. That’s a good sign– it shows us that as you pick from a larger pool, the advantage of going 1st diminishes. For an even group of people not divisible by 4, at 6 people you can expect the team picking first to be about 3% better skilled overall, while with 50 people that drops to about 0.3%. So if you find yourself picking second and your team losing, it’s not too unreasonable to blame the picking scheme if the total number of people is an even number not divisible by 4. However, it’s clear from the data that the fairer way to pick these games is to use the 1-2 system rather than the 1-1 that’s commonly used.