# Calculating the “most balanced” scenarios of a game?

Figured you math dudes (and dudettes) could knock this one out of the park!

I've got a website that stores users' game scenario results. Each scenario record can have one of three possible results: Side 1 wins, Draw, or Side 2 wins.

To this end, I'm currently representing balance by showing the simple percentage of total victories for each result. Example:

Result Totals
Side 1 - 5
Draw - 1
Side 2 - 4

Display
Side 1    Draw    Side 2
50%       10%     40%


but the astute reader will immediately note that using this methods means a scenario with 5 plays split 2-1-2 and a scenario with 100 plays split 40-20-40 will have the same apparent "balance".

What I would like to do is take into account the total plays for a scenario. Thereby, a scenario with a split of 41-20-39 will rank much higher than one splitting 2-1-2, even though the simple percentage is actually less balanced.

Does that make sense? Ideally, this would map to a scale of some sort, so scenarios could be given discrete balance ratings.

In fact, your measurement of balance with 2-1-2 and 40-20-40 are the same-each player has the same chance of winning. What is different is the assurance that the measurement is reasonably accurate. Under the usual (worrisome) statistical assumptions you can calculate a standard deviation on the number of player 1 wins as $\sqrt{Np(1-p)}$ where $N$ is the number of games and $p$ is the fraction won by player 1. So for 2-1-2 the standard deviation on the number of player 1 wins is $\sqrt{5\cdot0.4\cdot0.6}\approx 1$ So you would say player 1 wins $(40 \pm 20)\%$. In the case of 40-20-40 you would have a standard deviation of about 5, so you would say player 1 wins $(40 \pm 5)\%$. It's not that the second measurement shows more balance, but you have better reason to think it is close.