We're Vulturing Wrong?

I recently had the same idea that many others have had I'm sure, but once an idea gets in my head (even if I know it doesn't work) I like to work the math out to prove it to myself. I remembered seeing the following quote from Crystal Math when I first started looking in to vulturing:

Quote: CrystalMath

Your post is full of great info. I just wanted to comment on the payback percentage. Even if you play a 10 play game and there is a single 2x multiplier on one hand, then your payback will be over 100%, just not much. If you find a mulitplier on a 3 play, then your lowest return on the worst paytable with only one 2x multiplier is 127%....

*for the record I have great great respect for CrystalMath and love the analysis he does, which is why I 100% blindly accepted this as fact upon first reading.

Well, my idea was: Why not play UltimateX and "vulture yourself." By this I mean play 10 coins, and as soon as you get a multiplier, then the next hand play it as though you were Vulturing it (5 coins). My initial idea was that the vulture opportunity was worth more than the expected loss of the originally played 10 coin hand. Well, let's do some initial math:

Playing single play dimes with a crappy ass pay table at my local casino:
House Edge (10 coin): .0422
Expected Return (10 coin): .9578

House Edge (5 coin): .0473
Expected Return (5 coin): .9527

I started to confuse myself with average multipliers, and a lot of numbers. So what I did was break it down step by step in english, and then quantify the English statements with math.

1) Play first hand at 10 coins
ER = B10 - (B10*HE10) = $1 - (1*.0422) = 1 - .04 = .96
ER = .96, LOSS of .04

2) Play Second Hand

A) Without Multiplier
ER = B10 - (B10*HE10) = $1 - (1*.0422) = 1 - .04 = .96
ER = .96, LOSS of .04

B) With Multiplier
This was my original idea... Spoiler: It's flawed, but follow along.

ER = [B5 - (B5*HE5)]*AvgMultiplier = [.5 - (.5*.0473)]*2 = [.5 - .02] * 2 = [.48]*2 = .96
ER = .96, WIN of .46

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Originally I thought this was correct, assuming the 'worst' that when you have a multiplier, on average it will be a 2x (I used the Wiz's video poker calculator to find the probabilities of each item - pair, 2 pair, etc - and weighted them with their resulting multiplier). Did I calculate the average multiplier wrong? Can anyone tell me what the average multiplier is for a DB game (2, 2, 2, 2, 2, 12, 10, 8, 4, 3, 2, 1) and how to get it? Remember this is for single play. So this didn't make sense. I started looking in to why it didn't feel right and I came upon the solution (I won't keep you in suspense anymore):

Even when you get a multiplier, you must still weight that against the odds you'll get a payable hand to actually use the multiplier. It doesn't matter if you have 12x, if you brick and don't even get a pair, you get paid 0 all the same. You can't just double your EV assuming you're getting paid. Thus, the real math should look like this:

ER = [B5 - (B5*HE5)]*(AvgMultiplier*PayableHandPercent) = [.5 - (.5*.0473)]*(2*.455)

This sums up to [.48]*(.91)... which if you can see will LESSEN your expected return since you're multiplying by LESS THAN 1... = .44

ER = .44, LOSS of .06

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Conclusion
So what does this tell us? Well, it tells us we need an average multiplier of 2.3 or greater in order to make the ER over our initial bet. This also means, We've Been Vulturing Wrong. If you see a 2x on a single line machine, it's not profitable even at a 5 coin vulture play because you must remember, it's your EV * 2(.455) weighted with the probability you'll get something to actually get paid on (.455).

.455 = SUM of probabilities of Royal Flush down to Pair.

Can anyone comment? I know it's been a fairly known idea that if you see 'any' multipliers you should vulture the machine, but according to my analysis above, that is incorrect.

I'm currently working on what multipliers you'd need to see on 3 play, 5 play, and 10 play, in order to actually vulture.

I welcome and encourage the math minds to contribute their thoughts, either confirming my analysis above, or denying it and showing me why you should play a 10 play machine with a single 2x multiplier.

But you can simply double the EV of any line with a 2x multiplier, because the chance that you get a no-pay hand is already accounted for in that EV figure. In your way of thinking you have essentially doubled the true effect of no-pay hands which, given their high frequency, has seriously distorted your results.

Here's two ways to think about it, using single-line games for simplicity.

1.) If there's a single-line 2x multiplier, work out the paytable for that hand as if it were standard VP (of course the paytable will just be double all the normal values), then plug it into a VP calculator. I effect, for that one hand you are playing with a much better paytable. Hopefully this clears us any confusion caused by thinking of it as a standard paytable with occasional doubles.

2.) Let's say that the base game game returns 95%. I assume we are in agreement that if you sit down and play a hand for 5 coins without any multipliers on a quarter game your EV is ($1.25*.95) - $1.25 = -$0.0625 But now realize that you are playing with a 1x multiplier in this situation, so according to your formula we should be able to analyze this as EV*1(.455). Obviously this can't be right, because it would mean that the EV of each game is approx. 1/2 of the EV for that game.

First, thanks for the response Ziggy. Your example with a 1x multiplier does make sense. Next, that was my original train of thought, but if that's the case, wouldn't that make "Vulturing Yourself" profitable?

Hand 1 (10 coin): ER = B10 - (B10*HE10) = 1 - (1*.04) = 1 - .04 = .96, NET LOSS .04

Hand 2:

A) 10 coin: ER = B10 - (B10*HE10) = 1 - (1*.04) = 1 - .04 = .96, NET LOSS .04

B) 5 coin: ER = [B5 - (B5*HE5)]*2 = [.5 - .-02]*2 = [.48]*2 = .96, NET GAIN .46


?

Essentially any hand without a multiplier you're losing 5% of your investment, but any hand with a multiplier you're gaining (a minimum with 2x) of 90% of your investment?

Thus, on a 10 play with only one 2x multiplier, you're gaining 90% but losing 5% on the other 9 hands, thus 90-45 you're still expecting a 145% return?

Let's take a simple example: you play two rounds (betting 10 coins on the first round, and 5 coins on the second round) of single-line DDB with optimal standard strategy (not UX strategy) with the following payoff and next-hand-multiplier schedule:

Outcome	Current Payoff	Next Multiplier
Royal Flush	4000	4x
Straight Flush	250	4x
Four Aces with any 2,3,4	2000	4x
Four 2s,3s,4s w/Ace,2,3,4	800	4x
Four Aces	800	4x
Four 2s,3s,4s	400	4x
Four 5s thru Ks	250	4x
Full House	45	12x
Flush	25	10x
Straight	20	7x
Three of a Kind	15	4x
Two Pair	5	3x
One Pair	5	2x
All Other	0	1x

This average return for this paytable (ignoring the multipliers for now) is 4.89365 coins.

If we take the probabilities of each outcome and multiply it by the multiplier it awards for the next round, and sum these results, we get the average multiplier for the following hand which is 1.98799.

So now we can simply do this:

4.89365 coins returned on the first round
+ (4.89365 * 1.98799) = 9.72853 coins returned on the second round
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= 14.62218 total coins returned

Since you bet a total of 15 coins between the two rounds, the return is 14.62218 / 15 = 97.4812%.

JB, I understand your example (for the most part), but there might be a caveat or two... For starters, I'm ONLY betting 5 coins when I have a multiplier, essentially vulturing the multiplier. We know from vulture math that one 2x multiplier on a single line will yield a 190% return.

Assuming my "coin" is .10, then on the 10 coin hand I'd expect back 9.527 coins:

ER = 1 - (1*.0473) = 1 - .0422 = .9578

You'll get a multiplier 45.5% of the time (that's the probabilities of getting a pair of jacks or better summed up). So it should be safe to say one in every 3 hands you'll get a multiplier (this is rounding of course, but also should prove the point).

Hand 1: 10 coin, ER = .9578, LOSE .0422 cents
Hand 2 (no mult): 10 coin, ER = .9578, LOSE .0422 cents
Hand 3 (mult): 5 coin, ER = [.5 - (.5*0.0473)]*2 = [.5-.024]*2 = [.476]*2 = .952, WIN .452 cents

SUM = (Wins) - (Losses) = .452 - (.0422*2) = .452 - .0844 = 0.3676, a net WIN of about 37 cents.


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On the first hand I bet 10 coins, but on UltX (with the potential multipliers) the ER = 0.9588. This means 10*.9588 coins are returned to me, or 9.588 coins.

On the 2nd hand I bet 5 coins (IF I HAVE A MULTIPLIER - this is why I think it should be *.455), w/ no potential future multipliers, the ER = (0.9527)*2 = 1.91. This means 5*1.91 coins are returned to me, or 9.55 coins are returned.

In this scenario I bet a total of 15 coins, and got back 19.138 coins, which would indicate 4.138 coins of profit, each coin being a dime, this would mean about 42 cents profit.

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Conclusions
I think through the responses I've come to the following conclusions:

1) Yes, it is worth vulturing a 10 play machine with even one 2x multiplier.
2) The reason it's unclear if you want to vulture yourself is you can't look at it in a single state/game. The whole idea is to vulture yourself ONLY WHEN you have a multiplier. Thus, you MUST take in to account the 2 scenarios for hand 2, one that you don't have a multiplier, which happens 54.5% of the time, and two where you do have a multiplier, which happens 45.5% of the time.

EV(Vulturuing Yourself) = EV1 + EV2(.545) + EV3(.455)

ER1 = First Hand 10 coins = Bet*HouseEdge = 1*.9588 = .9588, or let's call it 96 cents for a LOSS of 4 cents.

ER2 = 2nd Hand No Mult 10 coins (which happens 54.5% of the time) = Bet*HouseEdge = 1*.9588, or let's call it 96 cents for a LOSS of 4 cents, 54.5% of the time.

ER3 = 2nd Hand Mult 5 coins (which happens 45.5% of the time) = Bet*HouseEdge*AvgMult = .5*.9527*2 = .9527, or lets call it 96 cents for a GAIN of 46 cents.

EV(Vulturing Yourself) = EV1 + EV2(.545) + EV3(.455) = -.04 + (-.04*.545) + (.46*.455) = -.04 - .0218 + .2093 = +0.1475

Quote: Romes

Thus, on a 10 play with only one 2x multiplier, you're gaining 90% but losing 5% on the other 9 hands, thus 90-45 you're still expecting a 145% return?

That should be be (+90% x 0.1) - (5% x 0.9) because the 2x multiplier represents only 10% of the hands you're playing. Or 104.5%

On a 25-cent machine, the one hand with a multiplier is worth $1.25 x 2 x 0.95. The other nine hands are each worth $1.25 x 0.95. So for 10 lines for $12.50, the expected value is $13.06

Quote: TomG

That should be be (+90% x 0.1) - (5% x 0.9) because the 2x multiplier represents only 10% of the hands you're playing. Or 104.5%

On a 25-cent machine, the one hand with a multiplier is worth $1.25 x 2 x 0.95. The other nine hands are each worth $1.25 x 0.95. So for 10 lines for $12.50, the expected value is $13.06

You're exactly right with the 104.5%. I meant to edit my post for that, but with all the numbers swirling in my head I forgot to, thanks.

I think it comes down to the Average Multiplier. I don't believe I'm doing it correctly. At one point weighted all of the probabilities with their respective multiplier, at another point I said "well in this round we 'know' we have a multiplier, so let's use the bottom one of 2" but I don't have a concrete answer:

For the following game, what is the average multiplier??? More importantly, how do you calculate it?

Romes,

Your math is off because you are incorrectly thinking that the first hand with NO multipliers on board returns approximately 95% for that hand ONLY. That is not correct.

When no multipliers are on board, the hand is like playing standard VP but at double the price. So the EV for your first hand is about half of what a normal VP game would be (~47-48% for a crap paytable). You make most of that money back when you vulture yourself off.

Look at JB's post more closely. He's got it right. His results are for two hands the first at ten coins and the second at five coins. You earn 48.93% back for hand 1 since you're paying double and have no multipliers. You make most of that money back when you vulture it off.

TLDR;

Your EV comes from generating future multipliers.

You are (probably) trying to double count the value of the multipliers.




A very simple example is this:

I do $10,000 coin in on 9/6 JOB and I get $500 in FP. I give up about $50 doing the coin in (99.5% return on JOB), and I get $500 in FP. So the EV is ~ +$450.

But I can't come back next month with my $500 in FP and add in an extra $500 in EV.



Either add in the EV "up front", or add in the EV "at the end". But you can't add it in twice.

Quote: Romes

For the following game, what is the average multiplier??? More importantly, how do you calculate it?

Take all the 2.5 million starting hands, come up an optimal strategy for each one, then see how often each hand hits. Fortunately many other people have already done very similar work: https://wizardofodds.com/games/video-poker/tables/double-bonus/

Quote: tringlomane

Romes,

Your math is off because you are incorrectly thinking that the first hand with NO multipliers on board returns approximately 95% for that hand ONLY. That is not correct.

When no multipliers are on board, the hand is like playing standard VP but at double the price. So the EV for your first hand is about half of what a normal VP game would be (~47-48% for a crap paytable). You make most of that money back when you vulture yourself off.

Look at JB's post more closely. He's got it right. His results are for two hands the first at ten coins and the second at five coins. You earn 48.93% back for hand 1 since you're paying double and have no multipliers. You make most of that money back when you vulture it off.

This sounds accurate now. I was unaware the EV of the game/hand being play was based on generating future multipliers. Thus, you were absolutely correct in that I believed the first hand's EV was worth much more than it really is. Your example/idea of plugging the same numbers in to the Double Bonus calculator but at 10 coins instead of 5 shows the truth... 47.6% ER.

Hand 1 ER = 1 - (1*52.4) = 1 - 52.4 = 47.6, LOSS of 52.4 cents

Even adding the hand 2 A and B (mult / no mult respectively) this will not overcome this deficit.

I really appreciate everyone's help! I was starting to see numbers in my sleep while clearly overthinking this.