PING: Stats Gurus

[bryce]

I'm playing around with some 'power rankings' of sorts for a soccer league, and I'm trying to determine the odds of a tie given the relative strengths of the two teams. For example, say one team has a 63% chance of winning vs. 37% of the other team, per my calculation. But what happens when you throw the chance of a draw in there? How does one measure the odds of a tie given these two percentages?

I started thinking about it as follows - ok, if the two teams' strengths were 50/50, then the odds of a tie would be 100%. Or would it? Or would the chance of a tie actually be 33% (1/3 chance on each team, and 1/3 chance of a tie.)

And then I thought about a dreaded standard deviation, but only briefly, as that's when a severe case of tired-head set in. So I appeal to someone more versed in stats than I.

Thanks in advance.

[st.cronin]

Originally Posted by bryce

I'm playing around with some 'power rankings' of sorts for a soccer league, and I'm trying to determine the odds of a tie given the relative strengths of the two teams. For example, say one team has a 63% chance of winning vs. 37% of the other team, per my calculation. But what happens when you throw the chance of a draw in there? How does one measure the odds of a tie given these two percentages?

I started thinking about it as follows - ok, if the two teams' strengths were 50/50, then the odds of a tie would be 100%. Or would it? Or would the chance of a tie actually be 33% (1/3 chance on each team, and 1/3 chance of a tie.)

And then I thought about a dreaded standard deviation, but only briefly, as that's when a severe case of tired-head set in. So I appeal to someone more versed in stats than I.

Thanks in advance.

I am by no means a stats guru, but my first obvious thought is it depends on how much scoring takes place in your league.

[bryce]

Originally Posted by st.cronin

I am by no means a stats guru, but my first obvious thought is it depends on how much scoring takes place in your league.

Heh, not much, as I'm using data from one of the major professional European leagues to test out my theory...

[Pumpy Tudors]

I'm not a stats guru either, but here are my thoughts:

Regarding st.cronin's comment, I'm not sure why the amount of scoring is relevant. A 0-0 tie is the same as a 3-3 tie. If two team's "strength scores" (for lack of a better term) are equal, then there's still a pretty decent chance of a tie.

As far as the original question goes, I see 50/50 strength scores leading to 33.3% chance Team A wins, 33.3% Team B wins, 33.3% tie. I admit that there's practically no mathematical theory behind my reasoning. It just makes the most sense to me. If the teams are "equal", then each possible outcome has equal probability. So to take the 63/37 example, I guess I'd see it as follows: of the times that there's a winner, Team A wins 63% of the time and Team B wins 37% of the time. That part is obvious. I figured that I'd play with the idea of the probability of a tie being (underdog's win% * favorite's win%), so 37% * 63% == ~23%. That means that there's a winner 77% of the time, so is Team A wins 63% of those games, Team A actually wins about 48% of the time. Team B would win about 28% of the time, and there would be a tie about 23% of the time.

I can't think of any mathematical logic behind that, but it came up with a decent number, so I've decided to go with that for now.

[Mr. Wednesday]

Originally Posted by Pumpy Tudors

I'm not a stats guru either, but here are my thoughts:

Regarding st.cronin's comment, I'm not sure why the amount of scoring is relevant. A 0-0 tie is the same as a 3-3 tie. If two team's "strength scores" (for lack of a better term) are equal, then there's still a pretty decent chance of a tie.

My thinking would be that if you view a team's strength as leading to a probability distribution for number of goals scored, more scoring would stretch out the distribution and lead to more likely outcomes that did not involve both teams scoring the same number of goals.

[Pumpy Tudors]

Originally Posted by Mr. Wednesday

My thinking would be that if you view a team's strength as leading to a probability distribution for number of goals scored, more scoring would stretch out the distribution and lead to more likely outcomes that did not involve both teams scoring the same number of goals.

Noted and understood.

I'm tempted to load up Minitab right now and play with some numbers just to see what it says.

[Mr. Wednesday]

Originally Posted by bryce

I'm playing around with some 'power rankings' of sorts for a soccer league, and I'm trying to determine the odds of a tie given the relative strengths of the two teams. For example, say one team has a 63% chance of winning vs. 37% of the other team, per my calculation. But what happens when you throw the chance of a draw in there? How does one measure the odds of a tie given these two percentages?

To give an example from another sport where ties are a possibility (namely hockey) -- I'm reasonably familiar with one of the ratings systems that's used. In application to college hockey, it's known as KRACH, but as I understand the methodology is actually referred to as Bradley-Terry. As applied to hockey, the only consideration given to ties is that they go into the team's record as half a win. The predictive component only predicts probability of a win, and the team strengths are adjusted to make it so that the cumulative probable wins (summed without rounding) for each team matches their observed number of wins. You can use this to predict a probability of any team winning a future game, but nobody's addressed the likelihood of a tie (as yet).

[Mr. Wednesday]

Originally Posted by Pumpy Tudors

Noted and understood.

I'm tempted to load up Minitab right now and play with some numbers just to see what it says.

Having said as I did, I'm not sure if the numbers would work out that way in practice. I also have no idea what sort of distribution of goal scoring one might expect to see as a function of relative team strengths.

[QuikSand]

Well, think this through -- if there's lots and lots of scoring in any sport, it's less likely for the score to just end in a dead heat. Games like soccer or hockey where the scores are frequently 1, 2, or 3 are far more likely to nominally end in a tie than games where the scoring is much higher. Even on a soccer scale, a high scoring league with lots of 9-7 scores would be less likely to yield a dead heat than a league where a typical score is only 2-1.

Seems to me the lieklihood of a tie is at a peak (but obviously nowhere near 100% if the two sides are pretty evenly matched, and drops from there -- but not by all that much. I have no idea if there is a real function to set up for this, though -- that's just based on common sense.

[QuikSand]

Originally Posted by Mr. Wednesday

My thinking would be that if you view a team's strength as leading to a probability distribution for number of goals scored, more scoring would stretch out the distribution and lead to more likely outcomes that did not involve both teams scoring the same number of goals.

Yeah, that.

[Pumpy Tudors]

Another thought that just occured to me is that the impact of goal-scoring is limited by how much weight that statistic carries in the power ranking calculation. Since we don't know how the power ranking is calculated, I guess it becomes a little hard to determine how to alter the formula to account for ties.

[CraigSca]

This is one of those things where it probably makes more sense to do a quick simulation of this in VB or something and find your results. I doubt there's a function out there that can determine this.

[MIJB#19]

It might be smart to do some research on past results, for example by figuring out the average score or the W-T-L tables for a given team against a team being x number of places ranked lower than them at the end of the season.

[Pumpy Tudors]

Originally Posted by CraigSca

This is one of those things where it probably makes more sense to do a quick simulation of this in VB or something and find your results. I doubt there's a function out there that can determine this.

I think I'm just going to make a little Excel spreadsheet. I'll make up something like 20 teams and assign each one a random "power ranking" number on a 1-100 scale. Then I'll just use the formula I suggested earlier (which I admittedly pulled out of thin air) and see what it looks like. This'll just satisfy my own curiosity, and of course it doesn't take into account any of the very good comments involving distribution of goals scored.

[Huckleberry]

Seems like to truly predict ties you would need separate offensive and defensive power ratings for each team. This way a system could predict goals scored probabilities against the given opponent.

Let's take two evenly matched teams with the following distribution to the 3rd decimal place:

Team A Goals vs. Team B	Probability	Team B Goals vs. Team A	Probability
0	0.250	0	0.250
1	0.250	1	0.250
2	0.200	2	0.200
3	0.125	3	0.125
4	0.075	4	0.075
5	0.050	5	0.050
6	0.030	6	0.030
7	0.015	7	0.015
8	0.004	8	0.004
9	0.001	9	0.001
10	0.000	10	0.000

This works out to an 18.99% chance of a tie and a 40.505% chance for either team to win.

With a rating system based on each team having an overall power rating and having their performances assumed to be based on some sort of distribution, ties can't be calculated. You can only come up with the probability of each team performing better in that game, i.e. wins and losses.

[Huckleberry]

dola -

I did a ratings set for European club soccer for basically the 2004 calendar year. Data gathering was too cumbersome to continue. It's near the bottom of the link in my signature.

[Pumpy Tudors]

Originally Posted by Huckleberry

With a rating system based on each team having an overall power rating and having their performances assumed to be based on some sort of distribution, ties can't be calculated. You can only come up with the probability of each team performing better in that game, i.e. wins and losses.

This seems so obvious now, and I can't believe I missed it before. I had just finished my spreadsheet, too. Oh, well!

[Huckleberry]

About the only thing I could come up with would be based on how many goals the standard deviation is in the league. Then we say that the probability of a tie is the probability that the two teams performances in the game are within however many standard deviations 1/2 a goal works out to of each other.

[Mr. Wednesday]

I recall trying to check on tie frequency in my hockey data and not finding that it actually favored teams that were more nearly equal in rating. I should go back and check that, though... I've got a lot of seasons worth of data.

[Mr. Wednesday]

Going off on a tangent, one of the more interesting results I had in running hockey BT rankings was a season where for division 1 women, home ice was a disadvantage -- yes, you read that right, the road teams outperformed their predicted results! It was quite a surprise to me, considering the men typically run anywhere from 0.03 to 0.07 wpct higher than predicted for the home team.

[Huckleberry]

Originally Posted by Mr. Wednesday

I recall trying to check on tie frequency in my hockey data and not finding that it actually favored teams that were more nearly equal in rating. I should go back and check that, though... I've got a lot of seasons worth of data.

There will certainly still be a decent chance of a tie. Here's a new matchup:

Team A Goals vs. Team B	Probability	Team B Goals vs. Team A	Probability
0	0.050	0	0.500
1	0.100	1	0.300
2	0.250	2	0.100
3	0.300	3	0.075
4	0.150	4	0.025
5	0.100	5	0.000
6	0.030	6	0.000
7	0.015	7	0.000
8	0.004	8	0.000
9	0.001	9	0.000
10	0.000	10	0.000

Team B will win only 7.75% of the time, Team A will win 81.625% of the time. Even in a matchup where one team will win over 9 out of 10 times when there's a winner there's still a 10.625% chance of a tie.