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The best way of explaining what I'm interested in calculating is comparing it to a popular sports pool game called NFL Survivor. The aim of the game is to pick one (and only one) team to win during each round of the NFL... if you team loses, you are knocked out of the game... if your team wins, you can continue to play the game but the team you picked isn't available to pick again during the rest of the game.

What I'm interested in is the same except if you lose... you don't get knocked out, you continue to play the game and you still have a chance of winning if you make enough correct picks. On top of this, if you win that team wouldn't be available to pick (like the standard survivor) but if you lose... that team would still be available to pick.

So given the above (you don't get knocked out and a lose doesn't remove the team), I would like to know how to calculate a score (or weighting) for players that end up with the same amount of wins at the end of a season. Feel free to argue differently, but I think that players with more wins at the beginning of a season deserve a higher score/ranking.

Here's a few examples... where I'd want to rank players. They all use the following configuration:

  • 8 rounds in the season
  • 9 teams in the competitions (I think this is arbituary but for the sake of having a number)
  • ✓ Marks a win
  • ✗ Marks a loss

Players tied on 1 win

Round - 1 2 3 4 5 6 7 8
Player 1 - ✓ ✗ ✗ ✗ ✗ ✗ ✗ ✗
Player 2 - ✗ ✓ ✗ ✗ ✗ ✗ ✗ ✗
Player 3 - ✗ ✗ ✗ ✓ ✗ ✗ ✗ ✗
Player 4 - ✗ ✗ ✗ ✗ ✗ ✗ ✓ ✗
Player 5 - ✗ ✗ ✗ ✗ ✗ ✗ ✗ ✓

Players tied on 2 wins

Round - 1 2 3 4 5 6 7 8
Player 1 - ✓ ✗ ✗ ✗ ✓ ✗ ✗ ✗
Player 2 - ✗ ✓ ✓ ✗ ✗ ✗ ✗ ✗
Player 3 - ✗ ✓ ✗ ✗ ✓ ✗ ✗ ✗
Player 4 - ✓ ✗ ✗ ✗ ✗ ✗ ✗ ✓
Player 5 - ✗ ✓ ✗ ✗ ✗ ✗ ✓ ✗

Players tied on 4 wins

Note: This has an additional row to show the number of teams they had to pick from on any given round... its another way to think about the problem and might be helpful.

Round - 1 2 3 4 5 6 7 8
Player 1 - ✓ ✗ ✓ ✓ ✗ ✗ ✗ ✓
Player 1 - 9 8 8 7 6 6 6 6
Player 2 - ✗ ✓ ✓ ✓ ✓ ✗ ✗ ✗
Player 2 - 9 9 8 7 6 5 5 5
Player 3 - ✗ ✗ ✓ ✓ ✗ ✓ ✓ ✗
Player 3 - 9 9 9 8 7 7 6 5
Player 4 - ✓ ✓ ✗ ✗ ✗ ✗ ✓ ✓
Player 4 - 9 8 7 7 7 7 7 6
Player 5 - ✓ ✗ ✗ ✗ ✓ ✓ ✗ ✓
Player 5 - 9 8 8 8 8 7 6 6

I could go on, but I think you get the picture.

My initial thoughts were to use a simple addition of the round where the players got their wins to give the players a weighting. E.g. given the same example as above with players tied on two wins:

Round - 1 2 3 4 5 6 7 8
Player 1 - ✓ ✗ ✗ ✗ ✓ ✗ ✗ ✗
Player 2 - ✗ ✓ ✓ ✗ ✗ ✗ ✗ ✗
Player 3 - ✗ ✓ ✗ ✗ ✓ ✗ ✗ ✗
Player 4 - ✓ ✗ ✗ ✗ ✗ ✗ ✗ ✓
Player 5 - ✗ ✓ ✗ ✗ ✗ ✗ ✓ ✗

In order of ranking: (lower weight = better)

  • Player 2 - weight = 2 + 3 = 5
  • Player 1 - weight = 1 + 5 = 6
  • Player 3 - weight = 2 + 5 = 7
  • Player 4 - weight = 1 + 8 = 9
  • Player 5 - weight = 2 + 7 = 9

This kinda works and on a basic level does make sense... but it's a little too simple and something tells me this isn't a new problem and could be answered with some statistics. It also doesn't sit well that players can have equal weights but different looks to their winning... which might be fair, I don't know... but again, I think it could be answered statistically.

EDIT: More thoughts on difficult / ranking

As @Silverfish pointed out, how they should be ranked is really subjective... so I thought I'd share my thoughts on how / why I think they should be ranked with early wins being ranked higher.

Given a trivial example with two players tied on 1 win a piece (again say 9 teams in the season)

Round - 1 2 3 4
Player 1 - ✓ ✗ ✗ ✗
Player 1 - 9 8 8 8
Player 2 - ✗ ✗ ✗ ✓
Player 2 - 9 9 9 9

You can see that player 1 got his win in the first round and player 2 got their win in the last round.

My thoughts are that Player 1 performed better because he selected his first win in the first round when he had 9 teams to pick from. The remainder of his picks he only had 8 teams to pick from... which means the game got more difficult for him straight after the first round. Player 2 on the other hand basically got 4 bites at the apple with the full 9 teams to pick from before he got his first win. So the game was easier for him for a longer period of time and player 1 performed better because he didn't need as many rounds to get a win on the board.

But things get trickier when players are tied on multiple wins with their wins scattered all over the place... so I guess I'm really asking how do you calculate the perceived difficulty of the entire game/season for each player?

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  • $\begingroup$ Sorry if this question has been answered elsewhere on the site... I have no idea how to search for it! Also, I took a stab at the tags... those with enough rep, feel free to change. $\endgroup$ – Charlino Sep 17 '16 at 11:29
  • $\begingroup$ This is an interesting question, but "how they should be ranked" is a rather subjective issue, which I don't think would be amenable to an objective, calculated answer. Perhaps when you say "difficulty calculation" in the title, really you are after something like a calculation of the probability? $\endgroup$ – Silverfish Sep 17 '16 at 11:56
  • $\begingroup$ Hi @Silverfish, your right, it is subjective... I've made an edit to hopefully explain my thoughts a little better... I'm finding it hard to put them down in words TBH! Cheers. $\endgroup$ – Charlino Sep 18 '16 at 2:17

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