Doing the math - What's the difference between a long race and several shorter ones?

[CreeDo]

I like to check out the statistics of various things (even though stats often don't tell the whole story).
Jay and CJ were discussing whether the Mosconi Cup's format
(multiple short races) is a good test that would even out luck enough for the best team to win.
And there's also been discussion of TAR's shorter races to 20-something vs. a single race to 100.

I'm an amateur at this stuff so anyone with better math, tell me if this is wrong.
I use this handy calculator which is sort of like a coin flip calculator, but you can "weight" the flip.
http://stattrek.com/online-calculator/binomial.aspx

So, we're interested in the odds that a worse player can still come out on top based on the race format.
Let's say the worse player is only 2% worse... in 100 games, he should win 48.

Odds that the worse player will win in single race to 5: ~45%
Odds that the worse player will win in a long race to 100: ~28.6%
So, race to 100 more fair than race to 5, duh.

But, what if we had a series of races to 5?
The Mosconi cup is race to 11 SETS, meaning whoever wins 11 races-to-5.

Well, we already figured out the worse player might still prevail 45% of the time, in a single race to 5.
What are the odds that will happen 11 (or more) times, out of 21 sets?
The math says 32.4%.

If you were to convert the mosconi cup into games, and every match went hill-hill, we're talking
a maximum of 9 games per set, out of 21 sets. That's 189 games.
So if the mosconi cup were a single long race, it would be race to 95 (just over half of 189 games).
What are the odds the same underdog can win in a race to 95?
The math says 29%.

The short version:

Racing to 5 gives a slight (2%) underdog about a 16% edge vs. racing to 100.
Not as big as I would have thought.

With only a 2% difference in skill, the underdog can still win over 1/4th of the time in a race to 100.
You'd need an absurdly long race to be 'pretty sure' the best player wins. You'd need to go to like 1,000
before the better player is ~90% to win the whole set, with such a small difference in skill.

21 races to 5 (Mosconi Cup format) gives the underdog about a 3.4% edge over a single long race.
Also not as big as I thought. Obviously the actual mosconi cup will have lots more complexity.
Some guys will be 2% underdogs, others will be 10%, and the underdogs are spread between both teams.
But the point is, a series of short races isn't much different from a single long race.

Applying this logic to TAR:

Single race to 100, we already know a 2% dog still has a 28.6% chance to win.
What if it's 3 races to 30? (the upcoming tar is 2 races to 30 and then a shorter race in 1p).
The math says the underdog wins each race ~37.9%.
So the odds of winning 2 out of 3 of these: ~32.2%

Best of 3 sets [race to 30] gives a slight (2%) underdog about a 3.6% edge vs. racing to 100.

So, is the mosconi cup a fair test of skill? I'd say it's pretty close to fair, if you think race to 100 as fair.
But even a race to 100 still allows a slight underdog to win between 1/4th and 1/3rd of the time.

I should mention the numbers change pretty drastically if you make the difference in skill go from
"slight" (2%) to not-quite-as-slight (5%).

The 2% dog wins 28.6% of his races to 100.
The 5% dog wins 7.8% of his races to 100.

The 2% dog gets a 3.4% edge by breaking up all the mosconi matches into races to 5.
The 5% dog gets closer to a 5% edge.

The 2% dog gets 3.6% edge switching up a race to 100 into 3 races to 30.
The 5% dog gets a nice 8.7% boost from this format.

So, the bigger the gap in skill, the more helpful it is to break long races up into multiple shorter ones.

 

[Black-Balled]

I think your figures sound reasonable...
I would play each of the top 25 euros a race to 3 and bet I would win thrice.

 

[bdorman]

Another way to look at it is that you're re-setting the odds every five games. The fact that team A lost a match to team B 5-1 doesn't carry over to the next match. In the next match both teams start at 0-0.

Let's use blackjack as an example. It is 99% statistically impossible to beat the house at blackjack over the long term (let's say 100 hands is long term). Your "luck" simply can't overcome the house's rules-advantage over the long term.

But if you play 3 sets of 5 hands each and refresh your bankroll with every new set (like starting the next match at 0-0) the chance of luck outperforming the rules-advantage rises significantly.

It would be interesting to know the game-count scores of previous Mosconi Cups.

 

[Johnnyt]

It's the same in horse or dog racing. None of the 3% that beat the dogs or horses play all races on the card. Almost on every card at every track there are 1 to 3 races you can beat with handicapping skills and how you build your bets. Most of the time I went to the dogs or horses to bet just one race. after that. most people go to the tracks to get that high from race after race. I went to make money. Johnnyt

 

[BasementDweller]

Interesting...

Couple of thoughts:

First, is a 2% dog really a dog in pool? Maybe the odds makers among us could think of a pairing that would have a 2% dog. I would think any matchup that close would just be a coin flip. I would think anything less than a 10 game difference in a race to 100 would be close to a coin flip. I think to make the numbers more realistic you would have to get into the 10 game underdog category. Using Shane as the example might screw everything up, but you have some numbers to work with using him.

Just thinking out loud here...

Shane vs. Hatch, 10-Ball Race to 100. I'm guessing Hatch would be a 20 game underdog (at least).

So what would the numbers be if Shane played Hatch short races to 5, 9, or 10??? I don't have my thinking cap on right now to figure them out.

The other thing that jumps out at me is the effect of the lucky rolls. In the Mosconi Cup, short race format, the lucky rolls can easily affect the outcome of a race. Some would say, well they play enough races that the rolls should equal out. I'm not sure if that is accurate or not. If you have 12 lucky rolls in the entire Mosconi Cup, the chances of those determining the overall match winner are greater than if you had 12 lucky rolls in the equivalent long race format. At least it seems like that in my head, but I don't have the numbers to back that up.

Then there's the psychological factor of the bad rolls or just a bad miss setting a player off because they know there just isn't enough time to make up for the bad rolls or a single mistake in this format. This added pressure makes it better for the viewer but tougher for the competitor.

 

[fast&loose designs]

I think breaking it up into several sets also allows for mental re-grouping. If I was doing a best of 5 sets, I would have more opportunities to mentally erase any deficits or losses in previous sets. That alone makes multiple sets more interesting, and is more prone to awesome comebacks.

 

[td873]

Math. Showing how longer races strongly favor better player (even a player that is only slightly better). As shown below, shorter races benefit weaker player, but really only when the skill level difference is not great.

When you have no chance of winning each game (85% or more chance of losing), you will have a VERY hard time no matter how short the set. When the skill levels are closer (say 1 or 2 balls difference, which might be 5% or 10% difference), your odds are better.

-td

 

[td873]

More race charts.

 

[td873]

Final set.

 

[Matt]

I think the numbers give some interesting results, but they are based on the assumption that a player performs the same on average in either situation. I think that there are a lot of players that can truly play above their average skill level for a short period of time (not just happen to get lucky for a bit), but fall off in a long race because they don't have the mental of physical stamina for it. I have also seen players that play well from behind once they get warmed up. With a long race, you also usually have the additional pressure of playing for more per set.

From an entertainment perspective, I think multiple short sets are the way to go. Someone can get blown out in one set and come back strong in the next set instead of grinding through 30 racks.

 

[Bavafongoul]

I know what I'm about to write doesn't answer your question directly but rather analogously. I still think there's a correlation to be made and so I wonder about the fairness of any race unless it's an extended match like with the way the U.S. Open Championship of years ago.

When pool started being competitively played, there was a large measure of luck involved. The match was played as straight pool is mostly followed except that the player with the break would slam the rack open trying to pocket any, as many balls as possible and then proceed to shoot the remainder of the the table calling each pocket and when the table got cleared, the balls were re-racked (all 15 balls), The player pocketing the last ball from the prior rack got to break the rack wide open hoping a ball would drop (the player does not call any pocket on the opening break) so the player could try running the table.

As you can see, there was certainly some measure of luck involved since if a ball doesn't get pocketed on the break, the player's inning is over and the opponent comes to the table as is. In the early 1900's, a Frenchman who had just won the U.S. Championship ( and more than once) came up with the idea of making the game more skillful and reduce the luck factor. He proposed leaving the last ball of the rack out and re-racking the 14 pocketed object balls. Hence, the name was adopted 14.1 or straight pool as we know came into existence.

Why is this even relevant to this thread? Great question and thanks for asking.......The U.S. Championship formerly was played in a round robin format not too dissimilar from today's format but the individual matches consisted on one game of 150 points. But the Championship.....the last two players standing with the best records......they didn't play a race to anything....they didn't play one game or a race to 10 games. These two players would stand toe to toe in an endurance match....typically playing two games of 150 points ( morning and afternoon/evening) and would travel to other cities in the United States and continue playing more matches until the first player reached 2500 points....not a race to 7 or 11 games......2500 points.

And if anyone really wants to appreciate how great Willie Mosconi truly was.......how he played at a level we've never seen since.........look at Willie's U.S. Open records......number of championships, consecutive championships, innings per match, average balls pocketed per inning, consecutive balls pocketed records, run-out wins.....I mean this man was a Pool God the likes of which we'll never see again.....like Bobby Jones was in golf.....way too much talent for one person.

Sorry.......I drifted a bit.......when the U.S. Championship (14.1) was played as a race to 2500 points between two players, there was no luck involved in the outcome.......None of these matches ended 2500 points -2382points..........there were so many games played that the momentum could switch and one player would become hot for a few games but there were so many games that momentum could easily reverse back and often did back and forth....except when you were faced with playing Willie.....he was like an avalanche and you can't ski in front of an avalanche as so many of his opponents learned..............you just wind up getting buried.

So in regard to the original question, a longer race favors the better player and the shorter race favors the weaker opponent, except in say 9 ball where anything can happen, more so than 10 ball or 8 ball.........I mean it worked in the U.S. Championship for decades....any player could get hot in a short race but a race to 2500 points "ain't no short race"..........don't ya think?

 

[voiceofreason]

Other factors have an increased bearing on a short race too, for example, breaking the first rack in alternate break race to five is more valuable than in a longer race.

 

[ivicafranic]

In time-consuming I think

 

[CreeDo]

Nice stats td873...
I may be a little lost.
It looks like each column represents a "skill bracket" like the winner is 95% likely to win, 90%, 85%, etc.
Down to a 50/50 coin flip.

Then you track the winner's odds of winning exactly [x] games, like 7 out of 13, 6 out of 13, and so on?
And totalling all those odds should end up with their cumulative odds of winning the set?

About "rolls": I think rolls are built into the numbers, somehow.
In other words, if I'm 5% better than my opponent, and we're temporarily ignoring
psychological factors... and we're ignoring 'bad nights' or 'good nights'...
then rolls are the reason I don't win every single race with my 5% edge.

Re: psychological edges... I think this is huge. So huge that it might overcome, say, even a 20% skill difference.
Momentum is a real thing even if it can't be quantified. Having breaks between sets have a real effect too.
And it's also true, nobody would try to handicap two players as 52/48.

That was as close as I could imagine two players being and you'd need them to play over 1,000 games to
come up with a figure that detailed.

I guess the main goal was to show that multiple short races is not HUGELY different from one long race,
in terms of making sure the best player comes out on top.

In other words, the US lost the mosconi cup because our players were not as good as their players, as a group.
And if someone wins 2 out of 3 tar sets, you can't really say "yeah but they wouldn't have won a single race to 100".
Odds are very good that it would have turned out the same either way.

 

[voiceofreason]

Nice stats td873...
I may be a little lost.
It looks like each column represents a "skill bracket" like the winner is 95% likely to win, 90%, 85%, etc.
Down to a 50/50 coin flip.

Then you track the winner's odds of winning exactly [x] games, like 7 out of 13, 6 out of 13, and so on?
And totalling all those odds should end up with their cumulative odds of winning the set?

About "rolls": I think rolls are built into the numbers, somehow.
In other words, if I'm 5% better than my opponent, and we're temporarily ignoring
psychological factors... and we're ignoring 'bad nights' or 'good nights'...
then rolls are the reason I don't win every single race with my 5% edge.

Re: psychological edges... I think this is huge. So huge that it might overcome, say, even a 20% skill difference.
Momentum is a real thing even if it can't be quantified. Having breaks between sets have a real effect too.
And it's also true, nobody would try to handicap two players as 52/48.

That was as close as I could imagine two players being and you'd need them to play over 1,000 games to
come up with a figure that detailed.

I guess the main goal was to show that multiple short races is not HUGELY different from one long race,
in terms of making sure the best player comes out on top.

In other words, the US lost the mosconi cup because our players were not as good as their players, as a group.
And if someone wins 2 out of 3 tar sets, you can't really say "yeah but they wouldn't have won a single race to 100".
Odds are very good that it would have turned out the same either way.

Some valid points there.confidence is a huge factor. That cross double is a different shot to win a match/set than it is if you lose a match/set by missing it.

I like the point re the misconi cup too..

 

[spartan]

It would be interesting to know the game-count scores of previous Mosconi Cups.

Racks and Games won of last 6 MC from 2007-2012. Of the 6 , Eu won 5.

EU USA EU USA
Racks Games
2007 93 81 11 8
2008 68 50 11 5
2009 78 88 7 11
2010 95 78 11 8
2011 90 70 11 7
2012 83 69 11 9
507 436 62 48
% 53.8%46.2% 56.4% 43.6%

 

[td873]

Nice stats td873...
I may be a little lost.
It looks like each column represents a "skill bracket" like the winner is 95% likely to win, 90%, 85%, etc.
Down to a 50/50 coin flip.

Then you track the winner's odds of winning exactly [x] games, like 7 out of 13, 6 out of 13, and so on?
And totalling all those odds should end up with their cumulative odds of winning the set?

Correct. But it also tells you the odds of having a particular score. For example, in the race to 13 chart, if the shooter is supposed to win 95% of the time, there is a 51.33% chance of winning 13-0, a 33.37% chance of winning 13 to 1 [losing just 1 game], and only ~3% chance of the other shooting winning 3 games. Totalling these gives you the entire odds of winning the set at each possible score combo (13-12, 13-11, 13-10, 13-9, etc etc).

If you take the time to look at the charts in great detail, the interesting thing is that the weaker player has a greater chance to win a few games. For example, at the 85% | 15% level, there is still a 99.99% chance of the the stronger shooter will win, but the weaker player has about an 83% chance of winning 1-4 games.

This might be instructive as to how many games on the wire can be given in a particular race. In the race to 13, a substantially stronger player (80% or higher in the race to 13 chart) could reasonably give 6 games on the wire and still have great odds at winning. Looking at the other end of the chart (60%, 55%), the slightly stronger player could still give one or two games on the wire and have 85% or greater chance at winning.

Taking another example, the Race to 5 chart, a player that should win 75% of the time has almost a 25% chance of winning 5-0. But the weaker player has over 50% chance of winning 2 games, and over 65% chance of winning 3 games. So giving 2 on the wire only gives the weaker shooter a 35% chance of winning the set (giving one game gives them 70% chance of winning the set).

This all plays into the way you match up, which is a skill by itself. The weaker player will want a game where they can get an advantage (shorter races and more games on the wire), and the stronger player will want longer races. The ultimate goal for most players and perhaps tournaments will be to get the races as close to 50|50 as possible.

Of course, these numbers are just mathematical illustrations and not real world application. However, over the course of time, the law of large numbers should bear these our pretty close.

-td