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Hickok
02-07-2006, 08:11 AM
My opponent offers me "3:2 on my money" playing 9-ball. I suggest "races to 3, 1 on the wire" instead.

Since I'm not the world's greatest mathematician I use Monte Carlo simulations to evaluate gambling propositions when possible. According to my simulation, the above two propositions are not equivalent.

Which is the better deal for me?
...and if someone can, WHY.

Egg McDogit
02-07-2006, 08:14 AM
it's better for you to just take 1 on the wire to 3 and jack it up.

My opponent offers me "3:2 on my money" playing 9-ball. I suggest "races to 3, 1 on the wire" instead.

Since I'm not the world's greatest mathematician I use Monte Carlo simulations to evaluate gambling propositions when possible. According to my simulation, the above two propositions are not equivalent.

Which is the better deal for me?
...and if someone can, WHY.

jsp
02-07-2006, 08:34 AM
My opponent offers me "3:2 on my money" playing 9-ball. I suggest "races to 3, 1 on the wire" instead.
I'm still new to the whole gambling lingo. What does your second sentence mean exactly?

I'm guessing it means that he's giving you one game to start with, no?

OldHasBeen
02-07-2006, 08:49 AM
3 to 2 on the \$ is much harder to give than a game on the wire going to 3.
That juice will catch up to him quickly.
And it is a lot easier for him to justify more single games than committing to other sets.

TY & GL, OHB

Andrew Manning
02-07-2006, 09:18 AM
So, what bet is best for you is a different calculation depending on what assumptions we make about your relative skill level. Let's first assume that the two of you are of equal skill, and so there is a .5 probability that you'll win any given game. The probability that you'll win a set to 3 with one game on the wire is 22/32 or .6875. I arrived at that number because there are 32 possible combinations of wins/losses in a 5-game series, and 22 of them involve you winning two before he wins three.

So if you bet all day, you'll win 68.75% of the time, which is 11:5 on your money. 11:5 = 4.4:2, which is much better than 3:2, so if you're of equal skill, races to three with one on the wire is much better for you. You should win about 47% more money with the wire game than with the odds.

In my next post I'll try the calculations for if he's 3:2 better than you; i.e. in a long string of games he wins 3 for every 2 you win.

-Andrew

jsp
02-07-2006, 09:29 AM
Which is the better deal for me?
...and if someone can, WHY.
If "one on the wire" means that he's giving you a one game headstart, then race to 3 with 1 on the wire is the better deal statistically. Here's why...

First assume you guys are dead even in skill. So 50% of the time you win and the other 50% he wins. So for \$1 gambled, how much money do you expect to have by the end of the day in both situations?

For 3:2 on your money, you would expect that you would have \$1.25 by the end of the day, if you win half of the time. 3:2 means that you win \$3 for every \$2 gambled, or equivalently you win \$1.5 every \$1 gambled. That means you net \$0.50 every win. If you win half the time, then you would expect \$0.25 return for every \$1 gambled.

How much money would you win in the other situation? For a race to 3 with 1 on the wire, you'd have to play at most 4 games to determine the winner. If you draw out all of the possible permutations of 4 games, you'll have 16 distinct outcomes (2^4). If you circle the instances where you win 2 games before you lose 3 games, you'll find that you'll win the set 11 out of the 16 outcomes. That's equivalent of having a winning percentage of 11/16, or 68.75% (assuming that each outcome is equally likely to happen, which is true if you still assume that you win 50% of the racks). So therefore, if you get double your money back, and you win 11/16 of the time, that means you'll have 2*(11/16), or 22/16, or 1 and 3/8 of the money you started with. So for every \$1, you'd expect to win \$0.375.

So statistically speaking, the second option is the better gamble. Anyone let me know if I screwed up my math.

EDIT: I guess Andrew beat me to the punch.

macguy
02-07-2006, 09:32 AM
My opponent offers me "3:2 on my money" playing 9-ball. I suggest "races to 3, 1 on the wire" instead.

Since I'm not the world's greatest mathematician I use Monte Carlo simulations to evaluate gambling propositions when possible. According to my simulation, the above two propositions are not equivalent.

Which is the better deal for me?
...and if someone can, WHY.

3 to 2 on the money per game, (not per set but per game) is a lot. He has to beat you 60% of the games you play just to stay even, and like 80% to really get into your pocket. If he is a much better player then you he will beat you but it may take a while and it could ware him down. If you should win several games in a row he begins to get buried pretty quickly. I played a world champion with 2 to 1 on the money and we see sawed around for a while, then I had a spurt and it was over, I won like 7 or 8 in a row and he quit.

jsp
02-07-2006, 09:35 AM
So, what bet is best for you is a different calculation depending on what assumptions we make about your relative skill level...
Hmm, i don't know if the relative skill level makes a difference. One bet should clearly better than the other bet, regardless of skill level. Either you win more money at a faster pace, or you lose more money in a slower pace. The second bet would always be better for you.

Andrew Manning
02-07-2006, 09:38 AM
So if we assume that he's better than you such that 3:2 would be a perfect spot and make an even game, than we have to assume the probability of him winning any given game over you is .6. That way in a long session, the two of you would break even if he's giving you 3:2 on your money.

However, if the probability of him winning is .6, then we can go back to the best-of-5 with one game on the wire scenarios and insert probabilities. As stated in my last post, there are 22 possible combinations of game outcomes that will result in you winning 2 before he wins 3, and there are 10 possible combinations with him winning 3 before you win 2. The probabilities of those 10 combinations, added up, total .4752, meaning he'll win 47.52% of the sets (and thus the money), and you'll win 52.48%.

So if he's better than you such that 3:2 on the money is an even game, you'll have the best of it in sets to three with a game on the wire, but only barely.

In either case, it turns out, you should take the game on the wire instaed of the odds. You'd have to redo all this math if he's a different amount better than you, though. Somewhere there's a turning point, where if he's enough better than you, you're better off taking 3:2, but anywhere below that level, you're better off with the game on the wire.

Can you tell I'm bored?

-Andrew

Andrew Manning
02-07-2006, 09:45 AM
JSP -

Relative skill level matters because it affects the expected return on your dollar differently for the two different bets. As I showed in my second post, if he's better than you, than the second bet is still better, but only extremely marginally. I expect there's a turning point, where if you're a heavy enough underdog, 3:2 on the money is better than one game on the wire.

It's good that the math turned out the same for us both using the equal skill level assumption though. Makes it more likely that we're right.

-Andrew

breakup
02-07-2006, 09:46 AM
And it is a lot easier for him to justify more single games than committing to other sets.

TY & GL, OHB

Ah ha! That is one reason why OHB only plays by the game (been looking for clues):cool:

jsp
02-07-2006, 01:09 PM
JSP -

Relative skill level matters because it affects the expected return on your dollar differently for the two different bets. As I showed in my second post, if he's better than you, than the second bet is still better, but only extremely marginally. I expect there's a turning point, where if you're a heavy enough underdog, 3:2 on the money is better than one game on the wire.

It's good that the math turned out the same for us both using the equal skill level assumption though. Makes it more likely that we're right.

-Andrew
Hey Andrew...you're correct, i'm wrong. Which bet you should pick certainly does depend on your skill level, or more precisely the probability of you winning a rack. I crunched out the math, and these are the equations I got for the expected value of your money per \$1 bet versus probability of winning each rack...

bet1 = 2x^2*(3x^2-8x+6)
bet2 = 2.5x

...where bet1 is the race to 3 w/ 1 on the side, bet2 is the 3:2 bet, and x is the probability of winning each rack. Below is the plot of each graph.

As you can see, there are 2 intersection points (3 if you don't count 0,0). So if your probability of winning is somewhere betwen 0.35 and 0.77, then you should go with bet1. Otherwise (if you're really good or really bad), you should go with bet2. Sorry, I just didn't feel like doing work right now.

EDIT: The blue line is bet1, pink line is bet2.

Snapshot9
02-07-2006, 01:36 PM
is the fact that the better player has already figured BEFORE offering the
bet is that he is an 80-20 winner, or he wins 4 games to 1 game for the
Opponent (you). Now, you may only figure the skills levels to be 70-30,
and therein lies the X factor, how far off from reality are you both off your
skill estimates. Say, the better player is closer to reality than you, then
reality could actually be (for arguments sake) 78-22. NOW, all you stat
experts go figure the odds of each bet with the skill level percentages
of 78-22 again, and see which is better.

Egg McDogit
02-07-2006, 01:40 PM
lol if the guy wants to play such short races, it's gotta be HUGE

is the fact that the better player has already figured BEFORE offering the
bet is that he is an 80-20 winner

jsp
02-07-2006, 01:55 PM
is the fact that the better player has already figured BEFORE offering the
bet is that he is an 80-20 winner, or he wins 4 games to 1 game for the
Opponent (you). Now, you may only figure the skills levels to be 70-30,
and therein lies the X factor, how far off from reality are you both off your
skill estimates. Say, the better player is closer to reality than you, then
reality could actually be (for arguments sake) 78-22. NOW, all you stat
experts go figure the odds of each bet with the skill level percentages
of 78-22 again, and see which is better.
If you look at my above graph, and trace where x=0.22, you can see that the 3:2 bet is better.

However, notice that the break even point for both curves is around x=0.4, or if your probability of winning each rack is around 40%. That means, if you're likely to win less than 4 racks for every 10 racks played, then you would be losing money in the longterm regardless of which bet you choose (although you would lose less money if you pick the 3:2 bet). So if this is the case, don't pick either bet...walk out the door! :)

Chris
02-07-2006, 02:00 PM
bet2 = 2.5x

Shouldn't Bet 2 = 3:2 = 3/2*5*x (per \$1 bet per game, five games)

Godfather
02-07-2006, 02:12 PM
The length of the race is a huge factor. A player with significantly greater skill can give ridiculous odds on the money in a very long race. Much tougher by the game. You all have seemed to cover the stats of this specific question in good depth, but the general principle applies that if a better player offers you a game and you don't know if it's good for you or not, it probably isn't. If you counter with a proposition that you don't know is any better, and he accepts, you probably have still booked a loser. If the guy's a great player, but a known go-off, then maybe you've got a winning proposition. The problem is, most great players have done it for a living, and it's tough to eat if you're a go-off, so a lot of them aren't, unless they've come in to money recently. This question is a little like asking, 'should I play a casino game where my odds are 49% or 48%' Either way, you lose.

jay helfert
02-07-2006, 10:27 PM
My opponent offers me "3:2 on my money" playing 9-ball. I suggest "races to 3, 1 on the wire" instead.

Since I'm not the world's greatest mathematician I use Monte Carlo simulations to evaluate gambling propositions when possible. According to my simulation, the above two propositions are not equivalent.

Which is the better deal for me?
...and if someone can, WHY.

First of all, if the two of you play about even, then either game is the NUTS! I have to assume he is the better player, because that is a lot of weight to give up either way.
Remember one thing, if he gives you a game on the wire in a race to three, he must beat you 3-1 in actual games played. He must continue to beat you three out of four games to win the money, whereas when getting 3 to 2 on the money he can stay even by winning three out of five games.
Winning three out of five games in the one game spot scenario will get him broke. So which game do you like now?