Problem: EV++ and risk tolerance in a multiple decision scenario.

[Top]

Here is a problem that I sometime think about as a fun theoretical excercise. Sometime I pose it to others to find out how they think about EV and risk tolerance:

A rich person offers you $1000. You can have this money with no conditions and walk away. This same person than says:

"I will give you a 5 in 6 chance (roll of a die for example) of doubling your money. If you roll 2,3,4,5,6 you double your money. If you roll a 1, you lose and I get my money back - all of it. There are other no other conditions."

Do you take the offer? Of course you do - right? EV+ !

Let's say you take the offer and win. He then gives you the same offer. 5 in 6 chances of doubling up the new sum($2,000 this time) or losing ALL the money. Do you take it this time? Of course.... right? EV+ !

He then offers his offer over and over again, to the extent of his full wealth (billions of dollars - theoretical question remember). Also keep in mind that this game is a one time deal. Once it's over - i.e. you lose all your money or walk away - there are no replays.

The dilemna of this problem is that calculating cumulative probability shows that you have less than 50% of getting to $8000 and less than 20% of getting to a million and less than 3% to getting to a billion. But at no point is the individual EV decision 'wrong' to go on.

This question is clearly a question of risk tolerance, but I find it interesting from the perspective that it is sometime CORRECT to turn down a very high EV+ situation. And in this game it will become correct to turn down a very high EV+ at some point.

[T50_Omaha8]

It has to do with how much impact the money will have on your life.

$1bn and $2bn will probably have a very similar impact on one's life, while the utility difference between $1k and $2k might be much more significant. I would probably quit at $4k myself.

Also, this has been discussed quite a lot lately. You probably don't even need to use the search function to find it.

[Top]

I am surprised your risk tolerance is so low. I didn't think anyone here would stop at $4000. I certainly wouldn't. The funny part is that by stopping at $4000 - you pass up an opportunity to have the best gambling odds you will most likely ever have in life on that kind of money. And yet you elect to stop. All those suckers betting in Vegas... with odds much worse than that...

[pococurante]

I would probably stop at $4k, possibly 8k. It's enough money to matter to me, and I'm just not comfortable with losing that much money at once.

Odds of being able to win 2k: 83%
4k: 69%
8k: 58%
16k: 48%
32k: 40%

While the odds are on your side, it's hardly "free money" to continue playing. At some point, getting great odds is less important than staying without your bankroll... for me that's about 5k.

[Top]

There are different ways of looking at this problem. Do we just pick an arbitrary stopping point because it 'feels' right? Or do we try to apply mathematical principles and real world scenarios?

One way one looking at this problem is to say that I only have 40% chance to get to 32K so I will stop at at let's say 8K - like pococurante did. So getting to 8K becomes a strategy to tackle this problem. That's how I used to approach this problem - with those type of 'strategies'.

But then I thought of it this way. If someone offered me 8K with a 1 to 5 odds of doubling it - and it was a one time deal - would I take it? And I can say that without reservation I would because in real life I would never get those type of odds AND because I am freerolling. Mathematically it's incorrect not to try and double up with such great odds when it is a one time deal and when I am freerolling. I then came to the conclusion that I would try and double up any arbitrary amount between $1 and $1billion if it is a freeroll with great odds and it was a one time deal.

So my question then became - why am I stopping at some arbitrary small number (like $8k) just because I am doing it multiple times in a row? Isn't each decision point independent? Is my strategy for this problem flawed? I have great odds with a exponential reward. I started looking at it this way:

If I stop at $8K I am then passing up a 28% chance of turning $8K into a million.

If I stop at $8K I am then passing up a 4.5% chance of turning $8K into a billion.

One billion is life changing to me, my family, all my friend, all my relatives, and all my favourite charities. $8K is is nothing in comparison. When in real life am I EVER gonna get such great investment odds? There is no stock, no bond, no real-estate deal that have potential for such great return for present dollars. I will never enter a $8k buy-in poker tourney in which I am 28% favourite to win a million. So why am I stopping? It's a freeroll!

Another way to think about this problem is that if you had two groups of 20 people playing this game and competing to get the most amount of money as a group. Each group then picked two strategies. One group would play each time until they got to $8K (or bust) and stop. The other played until they got to $1Mil (or bust) and then stop. The group with the $1mil strategy would crush the $8K strategy group majority of the time. So all repeating simulations point towards going deep in the game as the correct strategy.

And yet having said all that... by going for it big I am almost certain to walk away with nothing! So how can I justify that to myself or anyone else in my life who that would impact financially (if for example I busted out at the 500K or so level)?

The dilemna of this problem is that in order in to maximize your payout you have to play this game deep. And yet by playing this game deep you are almost certain to lose and get nothing.

[pzhon]

This has been discussed to death. That doesn't mean you will get reasonable answers by asking. According to the Kelly criterion, you should be willing to risk about 96.6% of your bankroll, which means you should keep doubling up until the part of your bankroll not on the table is less than about 1/28 of what you have on the table. While most advantage gamblers view the Kelly criterion (maximize the expected logarithm of your bankroll) as a bit too aggressive, it's a good start.

[pococurante]

Top: You are correct that there's no reason to stop just because it's being played multiple times in a row.

If someone offered me 8K with 1 to 5 odds of doubling it, I would probably not take the risk. While there are tons of people who would, to me $8,000 is too much to risk all at once. I'm just a nit.