Hypothetical Gambling Situation

[DavidC]

Kinda embarassed to post this up, but my eyes have been opened to a whole different side of gambling recently, so hopefully some of the guys who have been around a while will find this neat. It's a gaming theory question:

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You have a net worth of $10k, when you get to $30k, you're going to buy a car. You care just slightly about money between $10k and $30k, but you care about $30k A LOT. You also don't particularly like or dislike gambling.

---

Your buddies are going to the casino and they like slots and you're hanging out with them. You sit down at a Video Poker machine near the slots so you can talk and stuff. Holy crap! You just hit a Royal Flush worth $5k. Your net worth is now $15k and you're $15k away from your goal of having the car.

Video Poker offers something called "Double Up", which is where you can choose to rewager your jackpot, at even money on whether the next card will be red or black. 1:1 odds (and a reshuffled deck) make this completely even money, and the house doesn't take a commission on the bet. You can "Double Up" as often as you please, and for as many iterations as you please. You can also "Quadruple Up" by betting on the next suit (again, 3:1 odds, this is even money).

Neither of these options risks any money other than what you just won at the machine (so you can't lose your initial $10k).

---

Firstly, what should your goal be? To get to $30k. For the purposes of this question, let's assume that you don't just take your $5k over to the poker tables. Now, here's the interesting part. What should you do, and why?

a) Double up twice. ($5->$10->$20) (25% chance)

or

b) Quadruple up once. ($5->$20) (25% chance)

There's two factors that I consider worth evaluating here, but one of them is much more important than the other. I feel quite strongly that there is only one correct answer.

Edit: Try not to read the replies before you reply yourself, this is kinda a neat question. Any observations that you want to make on the situation from a gaming theory context would be neat too.

I quadruple up once and take clubs....

running a simulation now to see....

WOOOO!!! 4c!! FOUR OF CLUBS! i won!

but wait... if i put this 20k on the quadruple up... i could win 90k!!!! thats enough for 3 cars!

this time i take spades, running simulation....

6c.... i lose. i knew i shoulda just stuck with my lucky suit =(

This should probably (heh) like be in the probability forum, but I'm curious anyhow.

This is a BR stats type of question (risk of ruin) that I basically have no clue about. I think the ROR is greater for the 2nd option (quadruple up) is higher because the SD is bigger. We would expect to go bust 3/4 times. Now if we do the former (double double), we expect to go bust 1/2 times on the first bet. When we don't go bust, we have the same shot at going bust 1/2. Bah, see, I really have no clue actually.....

If you are sure you are going to take the doubling up option twice, and not quit if you won the first time, then they are exactly the same. 1/4 of the time you will quadruple up with either option, the other 3/4 of the time you will bust out.

I'd like to hear another way of interpreting this that sticks to probability though. /images/graemlins/smile.gif

[DavidC]

In both cases you have the same probability of winning the 20k total (15k more). In both situations you have no bankroll ror because this is a one-shot thing and you're not going to be doing it infinitely (or even at all with your original $10k bank that you started with, you're just out on a night with the guys trying to blow as little cash as possible and still be in the slots parlour with them, but you got [censored]-hot lucky, and it looks like you have a chance at your car, keep in mind that it would be nice, but not really a big deal, if you came home tonight with $5k extra, but that you've got a chance at going for the car, and you've decided to do it... how do you do it?).

So, the EV and ROR is the same with both options, yet there's still one correct answer.

btw, eskimo, you rock! /images/graemlins/smile.gif

[Eeegah]

Going for the quadruple up saves me a button press /images/graemlins/cool.gif

Doubling up twice would decrease your standard deviation. If you come across this situation many times. I guess I still don't prefer one to the other for that reason. I see it as you're going all the way to 30k and it's a 25% chance. I go for the quadruple up and save the suspense. I could just imagine that damn "Who wants to be a millionaire" music and those lights as I was going for the second double up. Then I pick red and it's a SPADE! Wah wah waaahh. Whammies start running around and take all my cash. I'll take the quadruple up. Much easier to deal with the loss of $5k instead of $10k since just winning $5k is not an option.

Related Interesting Problem:
Is the Monte Hall Question already posted here? The one with the three doors (2 donkeys 1 car) and you pick Door 1 hoping there's a $30k car behind it(assumption: you don't want the donkey). He shows you door #2 with a donkey behind it, he then asks do you want to switch to door #3? Well do ya?

[Shillx]

What if a DeathDonkey is behind it though? /images/graemlins/smile.gif

Yeah of course you switch. Go from a 1/3 to a 50/50 shot. Or is it to a 2:1 favorite I dunno.

If you pick the car and switch you lose everytime. But you only pick the car 1/3 of the time. If you pick a donkey and switch, it will be the car. So yeah you go from a 2:1 dog to a 2:1 favorite.

Edit: DaveC, wtf is going on here. You smoking the good stuff again or is there really a correct answer here? It looks to me like no strategy even slightly dominates the other. And I've been doing game theory all damn day so I know /images/graemlins/smile.gif

[crovax4444]

why the hell not, lets answer this

First off, the door question, always switch. I vaguly remember the math, but I know it improves your odds.

As for the double/quadrouple, since mathmatically they are the same, I'd do what Eeegah did and save myself a button press.

Crovax

You take that 5K and stick it in your f---ing pocket!

[DavidC]

Shill, I ALWAYS smoke the good stuff. /images/graemlins/smile.gif

Eegah's right, but I considered time to be a minor factor in this case.

Badger has the correct answer, and the correct reason.

I made our "hero" have very small marginal utility of money, but positive marginal utility of money, between 15k and 30k, therefore it's best to quadruple up, because it sucks to lose the 10k, just like badger said. /images/graemlins/smile.gif

This saves you from that 25% of the time where you win the first one but lose the second one. /images/graemlins/smile.gif

So there are three factors:

1) Time (not really important)
2) Utility of Money (if it sucks worse to lose 10k than 5k, then quadruple rather than double if you have huge utility at 30k (+20k))

3) Something that I didn't consider before tonight, if you actually enjoy gambling, then it would be best for you to double twice. However, I'm not 100% sure, but I think that this is the type of person who would gamble at -EV recreationally in order to gain the utility of gambling. Anyways, I cut this possibility out by saying hero didn't particularly like to gamble.

[DavidC]

Cheater! /images/graemlins/smile.gif

[crovax4444]

cheating how? Remember, I have no friggen idea wtf your talking about mathmatically 1/2 the time. I simply picked the responce that I agreed with most :-p In essence, I plagerized Eeegah's responce.

Crovax

[AdamL]

What's the utility of 10k vs the utility of 5k here Dave? *cough*

(innocent, stupid look)

[DavidC]

[ QUOTE ]
What's the utility of 10k vs the utility of 5k here Dave? *cough*

(innocent, stupid look)

[/ QUOTE ]

[ QUOTE ]
You care just slightly about money between $10k and $30k, but you care about $30k A LOT.

[/ QUOTE ]

[DavidC]

[ QUOTE ]
You care just slightly about money between $10k and $30k, but you care about $30k A LOT.

[/ QUOTE ]

The numbers don't quite match up my personal circumstances, but it pretty much sums up what I'm going through right now (with the exception of the $5k royal /images/graemlins/cool.gif).

I've found that it APPEARS as though I have some utility of money, even though I don't really spend a ton of money or anything. I've called this marginal utility of money that I don't spend "utility of income". Meaning that on a daily/weekly basis I like to see a growth in my assets regardless of whether or not I have any spending goals, regardless of if I'm way over-rolled for my current limit and I've got the "Miller 3-months" behind me.

This is pretty weird, but seems to be general human nature.

Thus, your utility of 10k is higher than 5k, even if you aren't planning on spending the $10k or the $5k...

It was stated in the original question, though. Everything was there from the hero's perspective.

Anyways, enjoy your day, sir. I'm off to bed. /images/graemlins/smile.gif

[AdamL]

You knew this, because you ruled it out in your original post, but normally 5k in the hand is worth more than 50% of 10k *even* if you have a strangely shaped utility function (as you've described.) The reason of course is that the 5k has the ability to make money, even if nothing more than simple interest. You have two options which each have equal EV, but only one of them is guaranteed to be "playable".

Any surefire permanent addition to your roll is worth more than a 50% neutral EV shot at the same thing. There is no opportunity cost to keeping the 5k.

I hope that makes sense. "Well, if you had 10k you'd have twice as much opportunity to collect interest". True, but if you lose you'd have zero interest.

Given that interest and other bankroll-related activities are *compound* activities, a 100% 5000 is worth more than a 50% 10,000. The loss in compounded interest when you lose the double out-weighs the extra interest you'd have with 10k.

5k at 10% interest = 5500 ev, plus guaranteed compound interest.

50% x 10k at 10% interest = 5500 ev, but when you lose, you cost yourself in time-interset accumulation.


This is very similar to my suggestion that there are real costs to downswings, if they prevent you from doing things in real life which would make you money or prevent you from spending it on unprofitable outcomes.

If you rent for an extra 6 months due to a downswing in a booming real estate market, you lose a lot more than you would have if you had an upswing or stuck near EV. The "Sklansky Bucks" may be equal, but the real dollars won or lost are far from equal.

Likewise, if some individual were planning on buying a big money-pit, pain-in-the-ass, somewhat uneccessary, depreciating asset if they had an upswing, then they would actually prefer from a utility standpoint to NOT have an upswing. The upswing has "excess baggage" beyond the neutrality of the sklansky bucks that the individual shouldn't, from a strictly financial standpoint, wish to incur.

Of course, some people just want to have their toys. *COUGH*. /images/graemlins/laugh.gif /images/graemlins/laugh.gif /images/graemlins/wink.gif

I switch doors, I remember reading about it in my discrete math textbook. /images/graemlins/tongue.gif

I seem to recall that was in a Marilyn vos Savant column once. She was adamant about the answer despite many readers serious objections. I can't recall the dang details.

[mojobluesman]

I'm not sure which is correct in the theoretical world, but I can tell you which is correct in the real world because I've probably been gambling for longer than most of you are alive.

If getting to 30K is really important, then go for the quadruple immediately. Otherwise, after winning the first bet in a double up, the decision on whether or not to continue will be agonizing and a loss will put you on a 3 week drinking bender. /images/graemlins/blush.gif

If the 30K is not really that important, then go for the double up because the probability of winning is higher and it will give you a chance to re-evaluate your feelings and priorities after you are ahead the 10K.

she said to switch. something to do with the bayes theorem.

[gharp]

[ QUOTE ]
I seem to recall that was in a Marilyn vos Savant column once. She was adamant about the answer despite many readers serious objections. I can't recall the dang details.

[/ QUOTE ]
I think you're thinking of Cecil Adams. He got it wrong, but Marilyn was right (this time). (http://www.straightdope.com/classics/a3_189.html)

[DavidC]

[ QUOTE ]
If you rent for an extra 6 months due to a downswing in a booming real estate market, you lose a lot more than you would have if you had an upswing or stuck near EV. The "Sklansky Bucks" may be equal, but the real dollars won or lost are far from equal.

[/ QUOTE ]

Firstly, yes, the cost of keeping the 5k is basically the amount of time that you have to go without the car, which is significant due ot the shape of your utility curve. The interest doesn't compensate for this, in our current investment climate, though if you consider poker earnings "interest" (which you know you shouldn't), then keeping the 5k becomes much more attractive.

Regarding this point, I think that if you take on high variance with positive EV, you actually lose Sklansky Bucks when you've got something planned like taking out a mortgage... As you take on more variance you have a greater likelihood of renting, whereas the upside of that same variance wouldn't be as profitable to you as the downside would be unprofitable (you save a little on interest when you increase your downpayment, but you lose a lot when you don't have enough to pay it, particularly if real estate is in a downswing at the same time you are).

My response to that sort of situation, as you're aware, is to reduce your bankroll for all risk calculations, instead of your approach, which has been to say that you have baknroll X, but bet in a manner consistent with bankroll Y, where Y is less than X.

I think my approach is best, and it lends itself to evaluating all bets at baknroll Y, so that you avoid situations where betting at bankroll X provides a good but minimally acceptable hourly rate and then you're screwwed because you don't have your mortgage due to downswing. When you start mixing bankroll X and Y in your decisions, rather than using one or the other, you start taking on a real risk of ruin somewhere between your threshold at X and Y, which, if you really had your threshold at Y, means that you're taking on more risk than you desired.

[ QUOTE ]
Likewise, if some individual were planning on buying a big money-pit, pain-in-the-ass, somewhat uneccessary, depreciating asset

[/ QUOTE ]

You mean like a car, right? /images/graemlins/smile.gif

Let's just say that there are some utility factors involved here that supercede money. I intend to prove people wrong who say, "Man can not live on 'digs' alone."

---

Tubasteve, what's discrete math? I'm kinda curious. I think I used to call it Finite when I was in highschool, but I"m not sure.

[bozlax]

I've missed you, Dave. I haven't had a good headache in 6 weeks.

[Hashiell_Dammett]

Yes Marilyn vos Savant wrote about the Monte Hall Question and proved that switching doors will win you the prize twice as often as the goat. Apparantly, all sorts of academics and statisticians wrote to her to tell her she was wrong and that the odds were 50/50.

The Cecil Adams article is interesting because he asserts that Marilyn is wrong. Then he actually admits that Marilyn is right but manages to back pedal and say that she is not entirely right because you can not assume that Monte Hall will even open a door. I think he's reaching at straws there.

I first heard about it when my statistics professor used the Monte Hall Question to explain Bayes Theorum in his class (and since Bayes Theorum deals with givens), he proposed that it is a GIVEN that Monte Hall will ALWAYS show you an empty door.
It was pretty amazing how many people in that class still insisted that switching doors is a 50/50 proposition. I think Bayes Theorum is what confused them in the first place. Ultimately, it's a complicated way of explaining a pretty simple concept.

[crovax4444]

[ QUOTE ]
I've missed you, Dave. I haven't had a good headache in 6 weeks.

[/ QUOTE ]

holy crap, just got back from class and read the new stuff, I think I wanna return back to class now...less thinking involved...

Crovax

[Shillx]

[ QUOTE ]
I've missed you, Dave. I haven't had a good headache in 6 weeks.

[/ QUOTE ]

This is classic dude. NH. /images/graemlins/laugh.gif

[AdamL]

No, poker is not an interest garnering activity. The whole point is that the money can be used to create more money. There is a big difference in the dictionary but a small one on paper when we're talking about opportunity costs of not having money. I gather the point itself was clear.

The correct approach is to adjust bankroll based on your evaluation of what you'll need and how likely you are to need it. It just doesn't have to be 100%. "Calculated Risk" is the whole point, whether you are investing or gambling. The way poker players normally manage their bankroll is actually very conservative. (Which is fine, but not necessary for everyone.) The entire business world has been doing just fine using weighted probabilties of both likelihood and importance for their spending decisions with respect to their cash holdings. Maximizing utility often involves making probablistic estimates about what you're likely to need, when you'll need it, how likely you're going to need it, and how much (qualitatively, urgently) you'll need it if you do.

If you're doing that, your approach is best. It's not really proprietary - lol. /images/graemlins/smirk.gif

Certainly a car counts as "a big money-pit, pain-in-the-ass, somewhat uneccessary, depreciating asset." I know mine does. /images/graemlins/laugh.gif

See you shortly...

[Nomad84]

[ QUOTE ]
What should you do, and why?

a) Double up twice. ($5->$10->$20) (25% chance)

or

b) Quadruple up once. ($5->$20) (25% chance)

[/ QUOTE ]

*grunch*

At first I thought it was obvious that it was the same, but then it hit me. If the deck is not reshuffled on the second option, then you should obviously try option b and choose 2 red once and black once. This option actually has about a 2% player edge, if I'm understanding the problem correctly. The first choice is still 26/52, but the second is 26/51 if the first is correct.

[Nomad84]

Umm...apparently I misunderstood the question. And I don't know anything about utility. So, uh, nevermind.

[Eeegah]

Monte Hall is confusing, because it's only after one switch.

A better way to visualize it is if there were a million doors. We stick with door #1 while Monty eventually opens up every door, save one. Now which is more likely to have the car, the door you picked at random out of a million, or the one Monty has been avoiding for 999,998 door openings?

[AdamL]

[ QUOTE ]
Monte Hall is confusing, because it's only after one switch.

A better way to visualize it is if there were a million doors. We stick with door #1 while Monty eventually opens up every door, save one. Now which is more likely to have the car, the door you picked at random out of a million, or the one Monty has been avoiding for 999,998 door openings?

[/ QUOTE ]

Dude, you just made me hundreds of dollars in bar bets. /images/graemlins/laugh.gif

(The problem has always been convincing the losing party after the bet, that they are wrong. But this will make it pretty obvious.)

So, thanks. /images/graemlins/wink.gif