Fantasy Football Expert Consensus Rankings

Question

Suppose there are n experts who rank m players. Let's assume there are no ties, a lower ranking is considered "better", and the rankings done by each expert are complete. For each player, we know:

• the best rank
• the worst rank
• the average rank
• the standard deviation

Without knowing how each expert voted, how would you determine what percent of experts would prefer a given player from a subset of all players?

I really only need these results to be accurate to the nearest percent or so, therefore reasonable assumptions to simplify the calculation are acceptable.

EDIT: Here is what I have tried so far (assuming a normal distribution):

Answer 1

In general, you won't be able to recover this from the given information. Let $Y_{ij}$ be the ranking given to player $i$ by expert $j$. There are $n\times m$ rankings that you want to infer.

• If we assume each expert gives rankings from the set $\{1,2,\cdots n\}$, then $n$ degrees of freedom are lost (because once a particular expert assigns $m-1$ rankings, the final ranking is fixed).
• Specification of the min, max, mean and sd for each player $j$ reduces the problem space by at least $4$ for each player - hence $4m$ degrees of freedom are lost.
  • The problem space may actually be reduced further with this information, since the rankings are from a finite set. This will be complicated to deal with in a general setting however.

This argument demonstrates that when $m\times n$ is bigger than $4m + n$, the inference is unlikely to be tractable.

Thats not to say the problem is hopeless. By making some assumptions, we may be able to get reasonable estimates for the quantity you are interested in. I'll think about this and add to this answer if I have more time.

Answer 2

If we assume a normal distribution, then we can calculate the percent of experts who ranked the player at or below the current pick using a Z Score Table.

Then we can weight the expert consensus based on the total percent of judges and the number of picks. This seems like it is accurate to the nearest whole number fo the data I am working with (which implies that our assumption of normal distribution was valid).