Ordering Furthest from a Given Permutation I made a competition draft with some friends of mine and I've been wondering some of the math behind it:
Suppose there are $n$ people in a competition, and a group of people pick what they think the results are going to be. At the end of the competition, the following metric is used to measure error:
$E(expected, actual) = \Sigma_{i=1}^n | expectedRanking_{contestant_i} - actualRanking_{contestant_i}|$
So for example, if someone placed 4th but I guessed them to be first, that would be 3 points added to my error.

*

*What is the max error possible? How would you find this case?


*Suppose the $k$th placing is a tie. For this case, let's use the actual ranking as whatever is closer to the expected, $k$ or $k-1$. So, if person A ties for 3rd (so no 4th place technically), and I expected them to get 1st, my error increases by 2, but if I expected them to get 5th, my error increases by 1. What is the max possible error now? And how would you find this case now?

This is known as deviation distance, and is maximized when the ordering is the reverse of the true ordering. For the proof see
More distance functions for order-based encodings
Ronald, S

1998 IEEE International Conference on Evolutionary Computation Proceedings. 
IEEE World Congress on Computational Intelligence (Cat. No.98TH8360), 1998, 
p.558-563

I don't understand the second question. It is obvious that the maximal distance occurs when there are no ties, in which case see above.
