LMAD: Money in the Bank

Today on Let's Make a Deal, a contestant played a game called Money in the Bank.

In the preliminary stage, she had to answer 4 simple questions valued at $500 each. Based on the way the questions were asked, players pretty much were guaranteed either $1000 or $2000, unless they were absolute morons. Today's contestant made $2000.

The good part:
If the contestant wanted to play on, she could 'invest' her money in one of five banks. Inside the 5 banks were multipliers or zonks. A zonk denotes lose everything. The breakdown was as follows:

Zonk - Zonk - 0.5x - 2x - 5x

She could keep her original amount, or invest. The question being, should she invest?

My argument:
On average, the player will win 1.5x (corrected from 3x), and should invest.

My wife's argument:
The player has a 3 in 5 chance of losing money, and should not invest.

Given that a player has one chance to play, whose argument holds more water? I can absolutely see my wife's point, but my mathematical mind says I have good point too. Do you invest, or not?

You've both got good arguments. But your math is wrong. You don't get 3x, you get $3,000 which is 1.5x - on average.

However, one variable is missing: Is $2,000 enough to make it as one of the contestants who will play in the Big Deal? If so, your wife's argument becomes stronger.

I haven't seen the modern version of the show, but on the classics, the 3 doors always had respecable prizes such that even if you picked the worst option, you'd most likely be trading up.

Quote: Dween

Today on Let's Make a Deal, a contestant played a game called Money in the Bank.

In the preliminary stage, she had to answer 4 simple questions value at $500 each. Based on the way the questions were asked, players pretty much were guaranteed either $1000 or $2000, unless they were absolute morons. Today's contestant made $2000.

The good part:
If the contestant wanted to play on, she could 'invest' her money in one of five banks. Inside the 5 banks were multipliers or zonks. A zonk denotes lose everything. The breakdown was as follows:
Zonk - Zonk - 0.5x - 2x - 5x

RTP of not playing is 100%.

Making the bet is a slot machine with pay table 0, 0.5, 2, 5 (these are all payouts that are x-for-1).

RTP of playing:

RTP = (0)*(2/5) + (0.5)*(1/5) + (+2)*(1/5) + (+5)*(1/5) = 150%

So you stand to win 0.5 units in profit by playing. Therefore, making the bonus wager is correct from a pure EV basis. Of course, if you need the money for rent, pizza or tennis shoes, the correct decision may be to take the original win.

I agree with Eliot, a 50% player advantage by "investing." Considering the risk, it is a good Kelly bet with wealth, not counting the $2000, of about $2,700 or more, which most of us have.

Quote: Wizard

I agree with Eliot, a 50% player advantage by "investing." Considering the risk, it is a good Kelly bet with wealth, not counting the $2000, of about $2,700 or more, which most of us have.

This is a great question. How do you (or do you at all) factor in DJ's comment regarding the amount won being a ticket to the next level? Let's estimate the "Big Deal" to be worth $50,000, and a $2,000 win was a 50/50 shot to advance. Do you still go for the investment?

You need to know the chance of advancing with $1000, $4000 and $10,000, and you need to know how likely you are to actually win the big deal prize.

Quote: dwheatley

You need to know the chance of advancing with $1000, $4000 and $10,000, and you need to know how likely you are to actually win the big deal prize.

Assume:
Chance of advancing with:
$1,000 = .25
$4,000 = .85
$10,000 = 1.0 (but would you risk it all for the chance to play for the $50.000?)

Probability of actually winning the big deal: .49

Here is a breakdown of the average "Big Deal" on the current incarnation of Let's Make a Deal.

As always, the top money/prize winner is given first shot, and refusal means they move to the next on the money ladder. In case of a tie (rare), they go to first chosen/first to win.

Those winning a car or a large cash prize almost always refuse giving up their prize for a shot at the big deal... but it has happened. On the flip side, on the old version (unsure about the current), it actually got down to a contestant giving up a Zonk to play.

The person that goes for the Big Deal will usually have a prize valued between $6000 and $8000. It is difficult to gauge the mindset of a contestant on the show, but it tends to be in that range.

The Big Deal will have 3 prizes: Small, Medium and Big.
Small is around $1500 to $2500.
Medium is around $6000 to $8000.
Big is around $22000 to $24000.

Big Deal in relation to Money in the Bank:
$5000 is not a guaranteed shot at the Big Deal by any means. Not only is there a 20% chance of getting $5000 in the first place, but there is the risk that a higher ranking contestant will not pass on a chance at the Big Deal, which is further combined with a 33% chance of winning the Big Deal if it even gets that far.

Thanks Dween. I thought the Big Deal might include the girls sitting on a burro as one of the three curtains. Is there always a prize?

Quote: Dween

Today on Let's Make a Deal, a contestant played a game called Money in the Bank.

In the preliminary stage, she had to answer 4 simple questions valued at $500 each. Based on the way the questions were asked, players pretty much were guaranteed either $1000 or $2000, unless they were absolute morons. Today's contestant made $2000.

The good part:
If the contestant wanted to play on, she could 'invest' her money in one of five banks. Inside the 5 banks were multipliers or zonks. A zonk denotes lose everything. The breakdown was as follows:

Zonk - Zonk - 0.5x - 2x - 5x

She could keep her original amount, or invest. The question being, should she invest?

My argument:
On average, the player will win 1.5x (corrected from 3x), and should invest.

My wife's argument:
The player has a 3 in 5 chance of losing money, and should not invest.

Given that a player has one chance to play, whose argument holds more water? I can absolutely see my wife's point, but my mathematical mind says I have good point too. Do you invest, or not?

You are looking at it through the lens of EV. Your wife through the lens of variance.

If you have $1000, the EV is $500, with a standard deviation of $2,121 and an absolute deviation of $1600. These numbers are greater than the starting sum, so in a one shot deal, I'd be less likely to take them.

Despite the "Monty Hall Paradox" being deacribed as three doors, two of which are zonks, 'doors' are only used on the Big Deal, and never hide a zonk. (At least that was true on the Monty Hall version of the show. Probably still true today.)

Quote: Dween

Today on Let's Make a Deal, a contestant played a game called Money in the Bank.

In the preliminary stage, she had to answer 4 simple questions valued at $500 each. Based on the way the questions were asked, players pretty much were guaranteed either $1000 or $2000, unless they were absolute morons. Today's contestant made $2000.

The good part:
If the contestant wanted to play on, she could 'invest' her money in one of five banks. Inside the 5 banks were multipliers or zonks. A zonk denotes lose everything. The breakdown was as follows:

Zonk - Zonk - 0.5x - 2x - 5x

She could keep her original amount, or invest. The question being, should she invest?

My argument:
On average, the player will win 1.5x (corrected from 3x), and should invest.

My wife's argument:
The player has a 3 in 5 chance of losing money, and should not invest.

Given that a player has one chance to play, whose argument holds more water? I can absolutely see my wife's point, but my mathematical mind says I have good point too. Do you invest, or not?

Me? I bet. But I don't care that much about $2000, so I take the highest EV available to me.

It's not just about whether you have an edge. It's also about whether you can afford to make the bet or not. On the other side of the spectrum, if you owe a mob boss $2000 and will get killed if you don't pay tonight, and you have no other source of cash, you'd be an idiot to bet.

For people closer to the middle of the spectrum, use the Kelly Criterion. The edge is 50%. (The player doesn't actually win 1.5x; they only win 0.5x. The "1" is just getting your money back.) The variance is 3.6. Kelly says that the max you should bet with a 50% edge and variance of 3.6 is 50%/3.6 = 1/7.2 of your bankroll. So, if your bankroll is over $14,400, you should definitely make the bet.

Now, someone (I want to say Stanford Wong, but I don't know if he was the first) pointed out that, while Kelly betting is optimal, anything less than 2x Kelly is still long-term profitable (anything higher than this actually lowers your long-term expected bankroll size). So, really, you should make the bet if your bankroll is bigger than $7200. This seems like a lot to bet from such a small bankroll, but, hey, it's a 50% edge. Deals like that don't come along every day unless your mississippi stud dealer is sloppy :)

So, if you're not willing to take an additional $5200 of your own money and set it aside to gamble with (note that investing money in the stock market, or doing anything else that has a variable return, counts as gambling), you should probably just take the $2000. If you already have more than that much money invested somewhere, then you should bet.

All this assumes that your goal is to die with as much money as possible. Which seems kind of silly, but I don't know any other way to mathematically analyze it.

Quote: AxiomOfChoice

For people closer to the middle of the spectrum, use the Kelly Criterion. The edge is 50%. (The player doesn't actually win 1.5x; they only win 0.5x. The "1" is just getting your money back.) The variance is 3.6. Kelly says that the max you should bet with a 50% edge and variance of 3.6 is 50%/3.6 = 1/7.2 of your bankroll. So, if your bankroll is over $14,400, you should definitely make the bet.

But the Kelly criterion is a statistical criterion, maximizing a rate of growth over a sufficiently long sequence of bets. "Statistical" here meaning that the large number of trials reduces your variance.
If you are playing only once, this criterion doesn't make sense. Variance is still high.

The original question is not answerable unless you know the objective of the player. Or, if you like better, his/her utility function. Dween maximizes EV (which is a questionable attitude for everybody except mathematicians, economists and bankers). His wife maximizes probability of winning something. Others maximize probabbility of going to the next stage of the game. Others still the EV of the overall game. Each time a different optimum choice!

Quote: kubikulann

But the Kelly criterion is a statistical criterion, maximizing a rate of growth over a sufficiently long sequence of bets. "Statistical" here meaning that the large number of trials reduces your variance.
If you are playing only once, this criterion doesn't make sense. Variance is still high.

I think I took this into account with my reply. Note that it's not just about making this particular bet again, but about making more bets in the future, where "bets" is loosely defined to include things like investing in stocks, bonds, real estate, etc. Basically, anything where you are given the chance to invest money at +EV. It's theorized that Warren Buffet used Kelly criterion to determine how much to invest in various profitable opportunities (I think I read that on Wikipedia)

My point is, if you have some money that you are going to continually invest in situations that you feel are profitable, then it makes sense to call that money a "bankroll" and use Kelly for each such opportunity (remember that all Kelly criterion is really saying is that, for each decision, you should maximize the expectation of the log of your bankroll, so it can be used to find things like optimal asset allocations, etc)