Giving up ev for reduced variance - from Gambling With an Edge

[Richard Munchkin]

Here is the situation. You have 4 envelopes, each envelope has a number 0 1 2 or 3. You arrange the envelopes making a 4 digit number. The lowest number would be 0123 in which case you would receive $123. The highest number would be 3210 in which case you will receive $3,210. (Please don't digress with ideas about gaming seeing through the envelopes.) To make things easy for you I will tell you that the average is $1,666. You have a choice of taking a sure $1,500, or making the 4-digit number. Do you

1. I'd take the sure $1,500
2. I'd take the variance.

 

[Richard Munchkin]

I should add that Bob and I will be talking about this on the show next week. This is from a promotion I believe at the Palms.

 

[tthree]

Normally I play it safe but I'd be happy with anything but the most significant number as 0. With that as only a 25% chance I would take the risk.

 

[Percy]

This type of scenario comes up the whole time in the game show 'Deal or No Deal'.

 

[Concept]

Normally I play it safe but I'd be happy with anything but the most significant number as 0. With that as only a 25% chance I would take the risk.

I don't think this is how the promotion works. I think there are 24 different combinations of values that you can get.

The answer to this is pretty straightforward. What you need to do is find the expectation and standard deviation. Then you use that information to calculate your Certainty Equivalent. Then if your CE is greater than $1,500 you take the variance otherwise you take $1,500. Here is how you calculate it:

Lets say your bankroll is 25,000 and you play at 25% Kelly.
Possible combinations are: 123, 132, 213, 231, 312, 321, 1023, 1032, 1203, 1230, 1302, 1320, 2013, 2031, 2103, 2130, 2301, 2310, 3012, 3021, 3102, 3120, 3201, 3210

Your expectation is the average = 1,667
Stdev = 1,105
CE = EV - ((EV^2 + Stdev ^2) / (2*Bankroll * Kelly %))
CE = 1,346.72

Since CE < $1,500 you take the risk free $1,500.

Now if your bankroll was 50,000 CE = 1,506.61 therefore you take the variance.

 

[tthree]

:whip:

I don't think this is how the promotion works. I think there are 24 different combinations of values that you can get.

The answer to this is pretty straightforward. What you need to do is find the expectation and standard deviation. Then you use that information to calculate your Certainty Equivalent. Then if your CE is greater than $1,500 you take the variance otherwise you take $1,500. Here is how you calculate it:

Lets say your bankroll is 25,000 and you play at 25% Kelly.
Possible combinations are: 123, 132, 213, 231, 312, 321, 1023, 1032, 1203, 1230, 1302, 1320, 2013, 2031, 2103, 2130, 2301, 2310, 3012, 3021, 3102, 3120, 3201, 3210

Your expectation is the average = 1,667
Stdev = 1,105
CE = EV - ((EV^2 + Stdev ^2) / (2*Bankroll * Kelly %))
CE = 1,346.72

Since CE < $1,500 you take the risk free $1,500.

Now if your bankroll was 50,000 CE = 1,506.61 therefore you take the variance.

If you count the 3 digit combinations there are 6 out of the 24 total combinations. 6/24 = .25 = 25%. Like I said a 25% chance of 0 being the most significant digit. Nobody said anything about placing a bet or having a bankroll. If that is the case I should take the sure thing. I am not sure that is what I would do though.

 

[Concept]

:whip:

If you count the 3 digit combinations there are 6 out of the 24 total combinations. 6/24 = .25 = 25%. Like I said a 25% chance of 0 being the most significant digit. Nobody said anything about placing a bet or having a bankroll. If that is the case I should take the sure thing. I am not sure that is what I would do though.

Sorry I dont follow how you get a 25% chance of 0. If you have 4 envelops labeled 0, 1, 2 and 3 and they are randomly ordered you will never get 0. This is a Permutation without repetition. So its n! / (n-r)! different combinations.

To make things easy for you I will tell you that the average is $1,666.

My math does match with this.

 

[tthree]

Most significant digit.

I don't think this is how the promotion works. I think there are 24 different combinations of values that you can get.

The answer to this is pretty straightforward. What you need to do is find the expectation and standard deviation. Then you use that information to calculate your Certainty Equivalent. Then if your CE is greater than $1,500 you take the variance otherwise you take $1,500. Here is how you calculate it:

Lets say your bankroll is 25,000 and you play at 25% Kelly.
Possible combinations are: 123, 132, 213, 231, 312, 321, 1023, 1032, 1203, 1230, 1302, 1320, 2013, 2031, 2103, 2130, 2301, 2310, 3012, 3021, 3102, 3120, 3201, 3210

Your expectation is the average = 1,667
Stdev = 1,105
CE = EV - ((EV^2 + Stdev ^2) / (2*Bankroll * Kelly %))
CE = 1,346.72

Since CE < $1,500 you take the risk free $1,500.

Now if your bankroll was 50,000 CE = 1,506.61 therefore you take the variance.

Here is how you get 0 for the most significant digit for your 4 digit permutation. Your 6 smallest numbers are 123, 132, 213, 231, 312, 321 or expressed more appropriately for the permutation 0123, 0132, 0213, 0231, 0312, 0321. This may go back to far into basic math for you to remember but in a 4 digit number the digit for thousands is the most significant digit. When the most significant digit is zero it doesn't need to be expressed in the number.

 

[AussiePlayer]

Does it cost anything, if not, there is nothing to lose.

 

[Concept]

Here is how you get 0 for the most significant digit for your 4 digit permutation. Your 6 smallest numbers are 123, 132, 213, 231, 312, 321 or expressed more appropriately for the permutation 0123, 0132, 0213, 0231, 0312, 0321. This may go back to far into basic math for you to remember but in a 4 digit number the digit for thousands is the most significant digit. When the most significant digit is zero it doesn't need to be expressed in the number.

Got it. I was just misunderstanding what you were trying to say. I thought you were somehow saying there was a 25% chance of getting $0.

Nobody said anything about placing a bet or having a bankroll.

You still need to treat this as a bet. Just because you aren't going to lose money doesn't mean that it shouldnt be treated as a bet. The amount of your bankroll is important here. To illustrate this lets take it to the extreme. Lets say a person that has a net worth of $1,000 has the option of taking $1,000,000 upfront or flip a coin for a chance to win $5,000,000. The expectation of taking the coin flip is $2.5mm. Even though taking the coin flip would maximize his EV he would be a fool not to take the $1mm upfront. On the opposite side if Warren Buffet were offered the same deal and he took the money upfront he would clearly be making a foolish decision.

 

[iCountNTrack]

Here is the situation. You have 4 envelopes, each envelope has a number 0 1 2 or 3. You arrange the envelopes making a 4 digit number. The lowest number would be 0123 in which case you would receive $123. The highest number would be 3210 in which case you will receive $3,210. (Please don't digress with ideas about gaming seeing through the envelopes.) To make things easy for you I will tell you that the average is $1,666. You have a choice of taking a sure $1,500, or making the 4-digit number. Do you

1. I'd take the sure $1,500
2. I'd take the variance.

It is not clear to me, how do you actually place the bet, like what amount do you bet?

 

[Richard Munchkin]

It is not clear to me, how do you actually place the bet, like what amount do you bet?

This is a free roll. It's a promotion at the Palms.

 

[pit15]

Does it cost anything, if not, there is nothing to lose.

You're totally missing the point here.

To answer the original question. I take the EV of 1666, because $1500 is very little $ to me.

Add 2 zeros to all the figures and I'd consider taking the 150K.

 

[blackriver]

i thought everyone would have said take the variance. some real nits in here

Got it. I was just misunderstanding what you were trying to say. I thought you were somehow saying there was a 25% chance of getting $0.


You still need to treat this as a bet. Just because you aren't going to lose money doesn't mean that it shouldnt be treated as a bet. The amount of your bankroll is important here. To illustrate this lets take it to the extreme. Lets say a person that has a net worth of $1,000 has the option of taking $1,000,000 upfront or flip a coin for a chance to win $5,000,000. The expectation of taking the coin flip is $2.5mm. Even though taking the coin flip would maximize his EV he would be a fool not to take the $1mm upfront. On the opposite side if Warren Buffet were offered the same deal and he took the money upfront he would clearly be making a foolish decision.

i like this one better as a fun question, how much would you need to be offered on a 50/50 to make you prefer it to a sure mil. i think the 5 was a good number. i cant make up my mind which i'd take (making me basically indifferent). what does kelly say about promo money thats bigger than your bankroll?

 

[Canceler]

i thought everyone would have said take the variance. some real nits in here

Nits, for sure. I was thinking that this question separates the gamblers from the non-gamblers. Even though I'm sure nearly everyone here would claim to be a non-gambler.

 

[WRX]

I haven't listened to the show yet (I was gambling at air time!), so haven't heard any spoiler. I believe Concept has the math right. Although I haven't combed through his calculations for accuracy, his approach is sound.

The key point to recognize is that if you go for the envelopes, in effect you're making a $1,500 bet with an expectation of a $166 win, or +11.067%, associated with fairly high variance. A lot of APs with adequate bankrolls would kill to have repeated opportunities to make such bets. Yet it's likely that a lot of people would give a kneejerk reaction that, since it's a one-time opportunity, they would instead just take the $1,500. Wrong. For an AP, or even for anyone who doesn't engage in advantage play on a regular basis but has substantial savings and has other investment opportunities, it's just part of the mix. The goal is not to ensure a positive return from the single bet, but to ensure a positive return from the collection of all the bets and investments one makes over a period of time.

So as long as you're well bankrolled, unless you really, really need the additional $1,500 right now, you probably want to take the envelope.

 

[zengrifter]

Here is the situation. You have 4 envelopes, each envelope has a number 0 1 2 or 3. You arrange the envelopes making a 4 digit number. The lowest number would be 0123 in which case you would receive $123. The highest number would be 3210 in which case you will receive $3,210. (Please don't digress with ideas about gaming seeing through the envelopes.) To make things easy for you I will tell you that the average is $1,666. You have a choice of taking a sure $1,500, or making the 4-digit number. Do you

1. I'd take the sure $1,500
2. I'd take the variance.

I think the proper decision is partially based on BR size, no? zg

 

[Nynefingers]

As Concept correctly pointed out, this is a simple application of certainty equivalent. If the certainty equivalent of taking the risk exceeds $1500 for you, you take the risk. If not, you take the $1500. If you'd asked me this 18 months ago, it would have been correct for me to take the $1500. Today, I'd take the risk for the higher EV (and more importantly, the higher CE).

 

[jaygruden]

I haven't listened to the show yet (I was gambling at air time!), so haven't heard any spoiler. I believe Concept has the math right. Although I haven't combed through his calculations for accuracy, his approach is sound.

The key point to recognize is that if you go for the envelopes, in effect you're making a $1,500 bet with an expectation of a $166 win, or +11.067%, associated with fairly high variance. A lot of APs with adequate bankrolls would kill to have repeated opportunities to make such bets. Yet it's likely that a lot of people would give a kneejerk reaction that, since it's a one-time opportunity, they would instead just take the $1,500. Wrong. For an AP, or even for anyone who doesn't engage in advantage play on a regular basis but has substantial savings and has other investment opportunities, it's just part of the mix. The goal is not to ensure a positive return from the single bet, but to ensure a positive return from the collection of all the bets and investments one makes over a period of time.

So as long as you're well bankrolled, unless you really, really need the additional $1,500 right now, you probably want to take the envelope.

Well said = +1

 

[Daggers]

Here is the situation. You have 4 envelopes, each envelope has a number 0 1 2 or 3. You arrange the envelopes making a 4 digit number. The lowest number would be 0123 in which case you would receive $123. The highest number would be 3210 in which case you will receive $3,210. (Please don't digress with ideas about gaming seeing through the envelopes.) To make things easy for you I will tell you that the average is $1,666. You have a choice of taking a sure $1,500, or making the 4-digit number. Do you

1. I'd take the sure $1,500
2. I'd take the variance.

well i look at it this way: think of it as if you grab a 0 or 1 you will make less than $1,500 and if you get a 2 or 3 then you will make more. 50% to make at least $613 more and 50% to make $1,320 and less at most. I would take the envelope because if i win more than offered, the smallest amount i would win would be further away from the offered price on the upside than the most amount i could win less than the offered amount. does this make sense or is it gambler's fallacy?