Understanding multi-line variance and bankrolls.
Let's say that I've determined that I can play $1 single line JoB and with the players club and other benefits I've determined that a good bankroll is exactly $20,000. Alternately, I can play JoB with 50-lines, 100-lines, or 9-line Spin Poker and get exactly the same $5 bet in per hand with the same benefits. Does my bankroll requirement change for these variants, and if it does change what would be a good ballpark figure for each variant? Moreover, how can I calculate the requisite bankroll an arbitrary multi-line game if I know the return and variance?
I'm trying to think about this from many angles and I get conflicting answers. For example, it seems to me that by playing many smaller lines my bankroll could be smaller because I am rarely getting absolutely nothing back on each play. Additionally, in an individual session it is quite easy to bust $100 playing $1 single line, but I typically get a lot of play out of the same amount on 50 or 100-line with the same $5 total bet.. However, it seems to me that the bankroll requirement could go up because the only time you "truly" hit a royal is when you get one dealt to you before the draw, and this makes your cycle go from 40k to 650k. This also makes sense to me considering the published variance numbers on the WoO site.
I'll also note at the bottom of the page here: https://wizardofodds.com/games/video-poker/appendix/1/ it says go to appendix 6 for risk of ruin in multi-play VP but the link just goes to a page about deuces wild.
Assuming two games have the same return, would you choose to play a standard 5-coin 1-line VP for a $X total bet per hand, or would you choose to play 5-coin 50-line VP for a $X total bet per hand -- mainly to reduce variance? I feel pretty clearly that the 50-line variant is the better choice, even though the true "cycle" hits only on a dealt natural royal.
Secondly, assuming that the 50-line VP is the better choice, would it make sense that due to the lower overall variance that the bankroll requirement of the 50-line game is half that of the single line game? That is to say, we could play $2X total per hand on the same bankroll?
Quote: abacabbAssuming two games have the same return, would you choose to play a standard 5-coin 1-line VP for a $X total bet per hand, or would you choose to play 5-coin 50-line VP for a $X total bet per hand -- mainly to reduce variance? I feel pretty clearly that the 50-line variant is the better choice, even though the true "cycle" hits only on a dealt natural royal.
Yes, if your goal is to reduce variance, then you would pick the 50-line at 1/50 the denom.
Quote:Secondly, assuming that the 50-line VP is the better choice, would it make sense that due to the lower overall variance that the bankroll requirement of the 50-line game is half that of the single line game? That is to say, we could play $2X total per hand on the same bankroll?
I am really not sure how those numbers work out. It would probably depend somewhat on the game (although probably not much).
I just recently finished writing a simulator to answer this exact question so if you'd like I can run some numbers for you and give you an exact answer.
But, if you are playing a negative expectation game, it doesn't really make sense to ask how much bankroll you need without more details. What's your goal here? Long-term you will lose everything playing a -EV game. If your goal is just to give the casino a certain amount of play (ie, $N in coin-in for some N) then this question can be answered.
Multi-line has a very strange skew to it -- variance does not tell the whole story. In short-medium length sessions, You have a few massive wins (from dealt royals) and lot of small-medium sized losses. You will lose more often than you do at single-line, but they will tend to be smaller losses.
Quote: AxiomOfChoiceYes, if your goal is to reduce variance, then you would pick the 50-line at 1/50 the denom.
Exactly. Note the 1/50th per hand. Do not bet the same X per hand or your swings will be crazy the total bet being the same for both is probably what you want.
I will play $5 single line games for a total bet of $25 or I will play the same game in 50 play at the 10 cent level for a total bet of $25.
Quote: AxiomOfChoiceI am really not sure how those numbers work out. It would probably depend somewhat on the game (although probably not much).
I just recently finished writing a simulator to answer this exact question so if you'd like I can run some numbers for you and give you an exact answer.
But, if you are playing a negative expectation game, it doesn't really make sense to ask how much bankroll you need without more details. What's your goal here? Long-term you will lose everything playing a -EV game. If your goal is just to give the casino a certain amount of play (ie, $N in coin-in for some N) then this question can be answered.
Multi-line has a very strange skew to it -- variance does not tell the whole story. In short-medium length sessions, You have a few massive wins (from dealt royals) and lot of small-medium sized losses. You will lose more often than you do at single-line, but they will tend to be smaller losses.
Sure, please do run a simulation. As for the game, assume it does not matter. The situation I'm talking about would include things like freeplay and such. For example NSUD returns 99.73% but this can be a +EV game when you factor in players club points (say +0.3% in freeplay), offers, drawing entries, etc.
So the scenario I envision is something like this. Let's say the game is 9/6 JoB with a 1% cashback. According to WoO's numbers, for a 1% ROR the bankroll needs to be 7256 bets. At the $1 single-line level for a total bet of $5 this would be a bankroll of $36280. Now if we choose to play 50-line 2c for a total bet of $5 the variance is reduced so the bankroll requirement should also be reduced. By how much I am curious.
Quote: abacabbSure, please do run a simulation. As for the game, assume it does not matter. The situation I'm talking about would include things like freeplay and such. For example NSUD returns 99.73% but this can be a +EV game when you factor in players club points (say +0.3% in freeplay), offers, drawing entries, etc.
So the scenario I envision is something like this. Let's say the game is 9/6 JoB with a 1% cashback. According to WoO's numbers, for a 1% ROR the bankroll needs to be 7256 bets. At the $1 single-line level for a total bet of $5 this would be a bankroll of $36280. Now if we choose to play 50-line 2c for a total bet of $5 the variance is reduced so the bankroll requirement should also be reduced. By how much I am curious.
I will run some numbers tonight. I've been looking to exercise my new program anyway.
Quote: AxiomOfChoiceI will run some numbers tonight. I've been looking to exercise my new program anyway.
Thanks!
Bad news: This data tells me that I have a bug. I'm only returning 99.47% on 9/6 JoB. I knew I should have written the rest of those unit tests.
Now I just want to make it faster. It takes me about 2 min 30 sec to get through 1 billion hands (4 threads, pinning all 4 virtual cpus at 100%). This is on a laptop.
Of the 2 min 30 sec, about 1 min 50 sec is spent picking random numbers to shuffle the deck, meaning I am only spending 40 sec doing everything else. I'm doing a minimal amount of shuffling (cutting off the shuffle after it has swapped the top cards in the deck -- not shuffling the part of the deck that I will never get to) I'm using mt19937 from <random>. Is there a faster random number generator that I can use? I tried a ranlux24_base and it was slightly slower.
I get that, with a bankroll of 2967 units, the optimal Kelly bet for single-line 9/6 JoB with 1% cash back is 1 unit.
I also get that, with a bankroll of 696 units, the optimal Kelly bet for 50-line 9/6 JoB with 1% cash back is 1 unit.
That means that, if you are risking the same amount per hand (eg, comparing 2c 50-line with $1 single-line), you need about 4.26x the bankroll for single-line as for 50-line.
This also means that you can play 4.26x the total stakes while playing 50-line as compared to single-line. In other words, if you have the bankroll for $1 single-line JoB with 1% cash back, you almost have the bankroll for 10c 50-line JoB with 1% cash back. Obviously the 50-line is better since you earn money much faster.
N-play var = var + (n-1)*cov
where
n = number of hands
examples:
full-pay Jacks-or-Better:
var = 19.5
cov = 1.97
1-play: 19.5 + (1-1)*1.97 = 19.5
9-play: 19.5 + (9-1)*1.97 = 35.26
50-play: 19.5 + (50-1)*1.97 = 116.03
100-play:19.5 + (100-1)*1.97 = 214.53
I'll leave it to you to calculate bankroll requirements.
Quote: HughJassI'll leave it to you to calculate bankroll requirements.
Unfortunately, variance is not a useful measure here because the results are not normally distributed.
Quote: AxiomOfChoiceUnfortunately, variance is not a useful measure here because the results are not normally distributed.
Right, running a sim will give a better result. Also even finding covariance values I have struggled with. Does anywhere other than jazbo's site (which has a limited listing of games) have them?
Quote: tringlomaneRight, running a sim will give a better result. Also even finding covariance values I have struggled with. Does anywhere other than jazbo's site (which has a limited listing of games) have them?
I suppose it is feasible to find an exact number through a brute-force analysis. I just don't see why it's useful.
Quote: AxiomOfChoiceI suppose it is feasible to find an exact number through a brute-force analysis. I just don't see why it's useful.
It's not that practical, no. More of a mathematical exercise than anything.
Quote: AxiomOfChoiceUnfortunately, variance is not a useful measure here because the results are not normally distributed.
I haven't played multihand before, but I'm going this weekend and planning on switching from single-line to 3-line, mostly because of the difficulty of finding decent payback at the strip casinos (MGM) where I'm staying. Back-of-the-envelope calculations suggest that one hour of $1 3-play yields a normal distribution of -34 ± 118, whereas three hours of 1-play yields -45 ± 187. But ... how far off normal is it? I only played one session on my last trip cuz I blew through $400 in under an hour (at $1 1-play) which normal distributions suggest is somewhere south of 3 S.D ... at the time I thought "oh, it must be common to lose 80 bets in an hour, I didn't bring the bankroll for this game," but further research suggested that this wasn't the case and I was just very unlucky (to the tune of 1:700).
For you VP regulars, what kind of swings do you see? What are bankrolls required for 1% and 5% ROR for (say) four hours of single-line VP? If the 50x bankroll is 1/4.x then I'd guess 10-line would be roughly half.
Note to OP: Appendix 3 covers some tidbits about multihand VP. Sadly, the variance formula is written as f(n,n) and I'd rather not disentangle the math.
Quote: ArchonyI haven't played multihand before, but I'm going this weekend and planning on switching from single-line to 3-line, mostly because of the difficulty of finding decent payback at the strip casinos (MGM) where I'm staying. Back-of-the-envelope calculations suggest that one hour of $1 3-play yields a normal distribution of -34 ± 118, whereas three hours of 1-play yields -45 ± 187. But ... how far off normal is it? I only played one session on my last trip cuz I blew through $400 in under an hour (at $1 1-play) which normal distributions suggest is somewhere south of 3 S.D ... at the time I thought "oh, it must be common to lose 80 bets in an hour, I didn't bring the bankroll for this game," but further research suggested that this wasn't the case and I was just very unlucky (to the tune of 1:700).
For you VP regulars, what kind of swings do you see? What are bankrolls required for 1% and 5% ROR for (say) four hours of single-line VP? If the 50x bankroll is 1/4.x then I'd guess 10-line would be roughly half.
Note to OP: Appendix 3 covers some tidbits about multihand VP. Sadly, the variance formula is written as f(n,n) and I'd rather not disentangle the math.
FYI, if you really don't want to play multiline but get some full pay paytables, you can still one line either Spin Poker, Super Times Pay games, or 25 play/50 play/100 play units (that don't have a min bet requirement) with a max bet per line. I know MGM grand has 9/6 JoB for Spin Poker for dollars, as well as 9/6 JoB and 9/6 DDB with Quick Quads for 50 play (different strategy needed for Quick Quads, fyi).
sounds fun and have funQuote: ArchonyI haven't played multihand before, but I'm going this weekend and planning on switching from single-line to 3-line, mostly because of the difficulty of finding decent payback at the strip casinos (MGM) where I'm staying.
if this is normal HERE .............................................................................................................................. abouts that far offQuote: ArchonyBack-of-the-envelope calculations suggest that one hour of $1 3-play yields a normal distribution of -34 ± 118, whereas three hours of 1-play yields -45 ± 187. But ... how far off normal is it?
imo, under 100,000 hands is way from normal
I would calculate it or just get Video Poker for Winners or something similar
one can also simulate this but calculate with a program is fast and accurate
this one works
http://www.beatingbonuses.com/simulator_java.htm
after fighting with java
I won this time
for 9/6jobQuote: ArchonyI only played one session on my last trip cuz I blew through $400 in under an hour (at $1 1-play) which normal distributions suggest is somewhere south of 3 S.D ... at the time I thought "oh, it must be common to lose 80 bets in an hour, I didn't bring the bankroll for this game," but further research suggested that this wasn't the case and I was just very unlucky (to the tune of 1:700).
for probability of ruin
300 hands
0.91% (1 in 110)
400 hands
3.61% (28)
500 hands
7.56% (13)
expect ruin
everyone will have an opinion or two or three+Quote: ArchonyFor you VP regulars, what kind of swings do you see?
What are bankrolls required for 1% and 5% ROR for (say) four hours of single-line VP?
If the 50x bankroll is 1/4.x then I'd guess 10-line would be roughly half.
imo,
get some software and prove it to yourself
of course, any one session, your mileage will vary a lot!
you know that
Sally
But, if you're playing and have to get the coin in, bring 1k units per 2k rounds.
If you're playing for fun, bring what you're OK/comfortable losing.
I guess in this post I'm just looking for a sanity check, not really specific math. I'll probably wind up simming it when I get back.
Quote: RSBut, if you're playing and have to get the coin in, bring 1k units per 2k rounds.
Yar. The idea was that it'd be a quick and cheap way to fill out the last XX few k TC I need to hit a mLife tier that I'm close to -- plus a way to possibly add something else to the set of games I'll play when I'm in Vegas. 99.54% is tempting as it's "eh, throw away a few hundred each trip, and maybe I get a big score, right?" It definitely racks up the TC but did not feel anywhere close to cheap.
A few background points:
- For roulette and craps, games with a (mostly) normal distribution, I bring a trip bankroll of 1EV+1SD. I'm ok with this RoR; it's what I've been doing for years.
- I'm wondering how to compute a trip bankroll for VP, since my last attempt didn't work too well
- Specifically, how much would a session bankroll be? How much cash do I stuff in my pockets as I walk to the other casino? Because I don't have a line of credit at the new place and I don't want to stuff $4000 into my pockets just to go play video poker for an hour (or three).
Video poker seems WAAAY swingier. Although I guess I don't really play slots, at 1 coin and 1 line, $20 lasts forever in a 25c slot machine.
So, yeah, thinking about it, normal looks like this:
. . . . ...,,:|:,,... . . . .
. . . . ...,,:|:,,... . . . . . . .
Which means $1 1-line VP is more like -$300/hr+-$100 ... with a chance to get an "unexpected bonus" from quads (+$125) or better. ie that I should plan to "survive" this 91% game. I only brought $400 last time, hence only a little bit of bad luck dropped me to -$400 in an hour.
So, two things:
1) If I want a 95-99% chance to not ruin on 1-line for three hours, I don't really need 3x this amount (that is, 3x the EV + 1.7x the 1SD, aka -$1070), since even with poor luck I should hit a quads or two. Maybe $800 will work. And if I don't get any quads in ~1800 games then I'd need a sanity break anyway. Yeah, I could sim this, but my schedule is kinda full for the next two days before my flight.
2) If I want to survive 3-line for 1 hour, I don't need 3x the 1-line amount even though EV is 3x worse since ... hmm. Since what? Would variance just be sqrt(3) times higher? Would this be exactly the same as question 1? I guess that's what I'm asking, sans a sim: how does the deal:draw covariance come out?
... and, after the end of this, i'm now thinking that the only benefit of multi-line is the time savings. That it packs more coin-in into the same hour without reducing or increasing variance per dollar wagered.
