Modeling long races mathematically...results may surprise you!

[Still_Learning]

The thread about whether SBV choked or not against Alex P. got me thinking.
I've done some statistical modeling of what can happen between two players of equal ability in a long race. The results may surprise some people.

I gave "Player A" and "Player B" each a 50% chance to win each game, then simulated races to 100. I looked at the maximum swing from A's biggest lead compared to A's biggest defecit in each simulation. Remember now, these results come from pure math--it's like coin flipping. Here are the results of 10 simulations, always from A's perspective:

1) "A" lost 98-100, biggest swing: 21 games (A was up by 11, then later down by 10, eventually lost by 2)
2) "A" lost 94-100, biggest swing: 17 games (up by 11, down by 6, lost by 6)
3) "A" won 100-98, biggest swing: 21 (down by 5, up by 16, won by 2)
4) "A" lost 93-100, biggest swing: 15 (up by 6, down by 9, lost by 7)
5) "A" won 100-90, biggest swing: 18 (down by 7, up by 11, won by 10)
6) "A" won 100-74, biggest swing: 34 (down by 3, up by 31, won by 26)
7) "A" won 100-88, biggest swing: 20 (down by 4, up by 16, won by 12)
8) "A" lost 92-100, biggest swing: 27 (up by 17, down by 10, lost by 8)
9) "A" won 100-88, biggest swing: 16 (down by 4, up by 12, won by 12)
10) "A" won 100-99, biggest swing: 20 (down by 11, up by 9, won by 1)

Now I remind everyone that these results came from, effectively, flipping coins.

I am *NOT SAYING* that psychological and physical factors like fatigue, heart, choking, and so forth don't come into play in long races. Of course they do! What I'm saying is, in a long race between evenly-matched players, you can expect one player to take the lead, and the other player to snatch it back, and once in a while, one player will seem to crush another player who's equally good. (Like in simulation #6, where A beat B 100-74.) More often, the race will come down the wire (like in simulations #1, #3, and #10). What causes this variance? All those psychological and physical factors, but also, luck--how balls act on the break, and the various rolls we all get or don?t get.

I'm also saying that if you want to predict the winner of the next race to 100 between players of equal ability...you might as well flip a coin.

 

[beetle]

Great analysis. Congrats to Alex and Ockham's Razor!

 

[smashmouth]

I'm also saying that if you want to predict the winner of the next race to 100 between players of equal ability...you might as well flip a coin.

so you've concluded that a match between two players of equal ability gives each player a 50% shot at winning/losing?

you'll forgive me if I don't ask you for Blackjack advice

 

[ScottW]

so you've concluded that a match between two players of equal ability gives each player a 50% shot at winning/losing?

Obviously, you did NOT read the entire post.

Epic fail.

 

[Andrew Manning]

I found your experiment extremely enlightening. I especially liked how in simulation #6, if these were pool players we'd be talking about how in the long race it becomes clear that A really is the dominant player. Also in #8, we'd all be talking about how player A choked, or alternatively about how player B has a ton of heart.

Of course pool is a game of skill, and this was an experiment of random chance, but I think this really says something about 9-ball and 10-ball when played at the highest level. The nature of the game, where each rack is short and winning a rack often comes down to who won the lag or who gets a more favorable result from a kick-safe, lends itself to a lot of lead changes. Even though no two players are of exactly equal skill, the score in racks will not necessarily reveal the differences in ability level, any more than they reveal the differences between heads and tails on a coin.

-Andrew

 

[ShootingArts]

thanks!

The 50-50 chance resulting in as wide of swings as it did was very interesting. As someone else said, most of us reading the score lines would come to totally erroneous conclusions.

I'm old enough to remember Alydar and Affirmed and after Alydar placing second to Affirmed in all three of the triple crown races I was still never positive which was the better horse. Alydar and his backers had the last laugh, he was the far more productive stud.

If you have time and it isn't asking a lot could you model a couple of races to 300 and/or 500? Looking for a number where the 50-50 odds even out and wondering if there is even such a thing.

Final thought: This should give everyone a reminder to not quit fighting until the last ball falls. Sometimes it is indeed luck that is making a big difference in a score. Of course if someone is too lucky too long . . . .

Rep coming your way!

Hu

 

[Fatboy]

I'm also saying that if you want to predict the winner of the next race to 100 between players of equal ability...you might as well flip a coin.

that's what I have been saying all along....tails anyone??

 

[mullyman]

Yeah, it's like that Strickland/Reyes race to 120 in Hong Kong. At that time there weren't any 2 players on this planet stronger than those 2. The final score really was a toss up. Day 2 had Earl running rack after rack and getting something like a 20 or 30 game lead. Day 3 came along and Reyes came back and took it. I think the final score was something like 120 to 117. There is no loser with a score like that. Someone got to 120 first. Had it been something like 120-75, yeah, the other guy got his teeth kicked in. But 120-117, nah, someone went home with a big paycheck and the other poor sap didn't get anything for his fantastic play.
MULLY

 

[av84fun]

Obviously, you did NOT read the entire post.

Epic fail.

And just as obviously, he doesn't seem to understand the rules of probabilities. OF COURSE the odds of EQUAL PLAYERS winning is 50/50.

Vegas understands that quite thoroughly...which is why there are no 50/50 bets in casinos!

They also understand that in spite of the fact that each "coin toss" is a 50/50 proposition, it is also true that in, say, a series of 6 tosses, there is only a 5/16 chance of a 3/3 outcome.

Excellent post. It demonstrates the fallacy of "long races = best player wins."

In fact, as Allison Fisher has proven, the best player, over time, will prevail.

Regards,
Jim

 

[av84fun]

The 50-50 chance resulting in as wide of swings as it did was very interesting. As someone else said, most of us reading the score lines would come to totally erroneous conclusions.

I'm old enough to remember Alydar and Affirmed and after Alydar placing second to Affirmed in all three of the triple crown races I was still never positive which was the better horse. Alydar and his backers had the last laugh, he was the far more productive stud.
If you have time and it isn't asking a lot could you model a couple of races to 300 and/or 500? Looking for a number where the 50-50 odds even out and wondering if there is even such a thing.

Final thought: This should give everyone a reminder to not quit fighting until the last ball falls. Sometimes it is indeed luck that is making a big difference in a score. Of course if someone is too lucky too long . . . .

Rep coming your way!

Hu

That suggests why I want to be reincarnated as a rodeo bull.

Hey, they work for only about 8 seconds a week, win most of the time and when they retire, they stud for a living!

(-:

 

[vagabond]

that's what I have been saying all along....tails anyone??

'Heads'-if you let me flip the coin of my choice.

 

[jsp]

The thread about whether SBV choked or not against Alex P. got me thinking.
I've done some statistical modeling of what can happen between two players of equal ability in a long race. The results may surprise some people.

I gave "Player A" and "Player B" each a 50% chance to win each game, then simulated races to 100. I looked at the maximum swing from A's biggest lead compared to A's biggest defecit in each simulation. Remember now, these results come from pure math--it's like coin flipping. Here are the results of 10 simulations, always from A's perspective:

1) "A" lost 98-100, biggest swing: 21 games (A was up by 11, then later down by 10, eventually lost by 2)
2) "A" lost 94-100, biggest swing: 17 games (up by 11, down by 6, lost by 6)
3) "A" won 100-98, biggest swing: 21 (down by 5, up by 16, won by 2)
4) "A" lost 93-100, biggest swing: 15 (up by 6, down by 9, lost by 7)
5) "A" won 100-90, biggest swing: 18 (down by 7, up by 11, won by 10)
6) "A" won 100-74, biggest swing: 34 (down by 3, up by 31, won by 26)
7) "A" won 100-88, biggest swing: 20 (down by 4, up by 16, won by 12)
8) "A" lost 92-100, biggest swing: 27 (up by 17, down by 10, lost by 8)
9) "A" won 100-88, biggest swing: 16 (down by 4, up by 12, won by 12)
10) "A" won 100-99, biggest swing: 20 (down by 11, up by 9, won by 1)

Now I remind everyone that these results came from, effectively, flipping coins.

I am *NOT SAYING* that psychological and physical factors like fatigue, heart, choking, and so forth don't come into play in long races. Of course they do! What I'm saying is, in a long race between evenly-matched players, you can expect one player to take the lead, and the other player to snatch it back, and once in a while, one player will seem to crush another player who's equally good. (Like in simulation #6, where A beat B 100-74.) More often, the race will come down the wire (like in simulations #1, #3, and #10). What causes this variance? All those psychological and physical factors, but also, luck--how balls act on the break, and the various rolls we all get or don?t get.

I'm also saying that if you want to predict the winner of the next race to 100 between players of equal ability...you might as well flip a coin.

Interesting experiment. How easy is it for you to simulate a race to 100 and record the maximum positive and negative swings? If it takes no time at all to simulate and record, I would suggest simulating more than only 10 races. If you can, try 100.

I would like to see how often races such as #6 and #8 come up with a much bigger sample size.

Race #8 closely resembles the Alex/Shane match. If being up 22 games at some point and eventually losing the match happens less than 5 times out of 100 races (5% of the time), then I think it's reasonable to suggest that Shane actually did "choke".

 

[s0lidz]

I believe the technical term for this is the law of averages. You just gave a perfect demonstration of the law.

I think Alex gave up the lead originally due to being tired from the tournament. If they would have started fresh Alex would have led the whole time. To me, Alex blew Shane out, the score just didn't show it.

 

[ironman]

I believe the technical term for this is the law of averages. You just gave a perfect demonstration of the law.

I think Alex gave up the lead originally due to being tired from the tournament. If they would have started fresh Alex would have led the whole time. To me, Alex blew Shane out, the score just didn't show it.

I never understood why so many had Shane the favorite in this match to begin with.

I like Shane and believe him to be a great player and exceptional young man. He is though just coming of age. Alex has been around the block far more turns than Shane meaning far more experience. Plus, the big factor IMO, he doesn't miss near the balls Shane misses. After all, that is usually the determining factor in most matches.

 

[mikepage]

Interesting experiment. How easy is it for you to simulate a race to 100 and record the maximum positive and negative swings? If it takes no time at all to simulate and record, I would suggest simulating more than only 10 races. If you can, try 100.

I would like to see how often races such as #6 and #8 come up with a much bigger sample size.

Race #8 closely resembles the Alex/Shane match. If being up 22 games at some point and eventually losing the match happens less than 5 times out of 100 races (5% of the time), then I think it's reasonable to suggest that Shane actually did "choke".

No need to simulate here. There is a simple formula for the chance/frequency of each score. I'm off to a meeting now, but I'll give it later.

 

[AuntyDan]

Here's an interesting article showing that, using a computer to randomly model baseball history there is almost no statistical difference between actual real world results like incredible hitting and winning streaks and the random computer results.

 

[Roy Steffensen]

I guess I am stupid.. I don't get this.

How did you come up with the results, etc?

 

[Bob Jewett]

No need to simulate here. There is a simple formula for the chance/frequency of each score. I'm off to a meeting now, but I'll give it later.

Simple is in the eye of the calculator .

One useful and fairly simple result is the expected difference in wins between two players after a certain number of games. If they are evenly matched, the average difference will be zero, of course, but if you take the square root of the number of games, that will give the "sigma" of the difference. If they play 100 games, a result of 55-45 is not surprising. A result of 65-35 (or worse) would be very surprising and has about a 1 in 200 chance to happen.

If the players are not evenly matched, the formula is a little more complicated:

average difference = N * p - N * q
sigma = 2 * sqrt (N * p * q )

where p is the chance for one player to win and q is for the other. This is an approximate formula that applies to reasonably long matches.

 

[av84fun]

I guess I am stupid.. I don't get this.

How did you come up with the results, etc?

Don't know if you're referring to my post but the odds on an equal outcome where there are 2 possibilities is:

x/y x 1/2 to 3rd power x 1/2 to 3rd power
where x and y are the two possibilities so, for example if you toss a coin 6 times the odds of an equal number of heads and tails is

6/3 x 1/2 to 3rd power x 1/2 to 3rd power = 20x1/8x1/8 = 5/16


Regards,
Jim

 

[jsp]

No need to simulate here. There is a simple formula for the chance/frequency of each score. I'm off to a meeting now, but I'll give it later.

I understand that the probability distribution of the losing score would look like half of a Gaussian bell curve, with the peak being at 99.

I'm more interested in finding out the probability of being up at least 22 games at some point in the match only to end up losing the race to 100. If there is a "simple" way to compute that probability, I'm all ears (but then again, I only did get a B in my college probability course. ). I just thought it would be easier (for someone else) to simulate. With a toddler and an infant in the house, I haven't had much opportunity to exercise my brain much.