I have a small fun problem that I gave myself to represent mathematically, but I got stuck very quickly.
Suppose there is a group of people (named A, B, C, D, E, F and G). Every week they meet:
- every person speaks then nominates the next person to speak
- $A$ always begins the sessions
Someone in the group suspects that B, C and F always seem to prefer nominating each other. Data has been collected for a number of sessions to try and test this hypothesis:
| Week | A | B | C | D | E | F | G |
|---|---|---|---|---|---|---|---|
| 1 | 0 | 3 | 1 | 5 | 4 | 2 | 6 |
| 2 | 0 | 4 | 1 | x | 5 | 2 | 3 |
| 3 | 0 | 1 | 3 | 6 | 2 | 5 | 4 |
| 4 | 0 | 6 | 3 | 2 | 4 | 1 | 5 |
| 5 | 0 | 6 | 5 | 2 | 4 | 1 | 3 |
| 6 | 0 | 1 | 6 | 5 | 4 | 2 | 3 |
| 7 | 0 | 2 | 6 | 3 | 5 | 1 | 4 |
| 8 | 0 | 6 | 5 | 4 | 2 | 1 | 3 |
| 9 | 0 | 1 | 2 | x | x | 4 | 3 |
| 10 | 0 | 5 | 6 | 3 | 1 | 2 | 4 |
| 11 | 0 | 2 | 3 | x | x | 4 | 1 |
| 12 | 0 | 6 | 3 | 5 | 4 | 1 | 2 |
| 13 | 0 | x | 1 | 2 | 3 | x | 4 |
For each week's session (rows) the numbers represent the order in which each member spoke. Implicitly, a person speak in position $i$, nominated the person that spoke in position $i+1$. People who were absent on any particular session is marked as with an $x$.
In words I could start describing the problem as follow. (Please forgive my butchering of mathematical notations :) )
with $P(J)$ being the probability of $J$ being picked, and with $P(J|K)$ being the probability of $J$ being picked by $K$, we can state
$B$, $C$ and $F$ are bias if:
$P(C|B) + P(F|B) + P(B|C) + P(F|C) + P(B|F) + P(C|F) > P(B) + P(C) + P(F)$
However when going down this route, how do I represent:
- the absentees
- people going last on various weeks
- the forced "choice" of the person who goes second to last
- the fact that the choice of those outside the "bias group" to avoid nominating members of the bias group, will result in higher chance of the bias occurring (as B, C and F would end up being the last people to speak and have no choice but to nominate each other)
Then I started thinking whether there was a statistical threshold that defines the bias or whether it could be defined in a different way. So I started looking into graph theory and measures of modularity, but ended up confusing myself more.
Naturally, this is an example of a more generalised problem.
Given
- a set of people $G$, who take turns at an activity and then nominate the individual to go next,
- subset $B$, the bias subset, defined as a subset of $G$ where individuals are more likely to nominate from within $B$ than they are outside of $B$
How do we show that:
- a given subset $B_1$ is bias
- there exist a bias subset in $G$
Did I word a well known mathematical problem in the worst way possible (I did try searching for this online with not much success)?
Thank you very much.