Expected Value: Deal or No Deal

You must have seen this new game on t.v. Its called "Deal or No Deal" and its hosted by Howie Mandel. The premise of the game is simple and copied from http://en.wikipedia.org/wiki/Deal_or_No_Deal:

Deal or No Deal involves a contestant, a host/presenter, a banker, and a number of briefcases (or boxes), with each having a different (and initially unknown) value. Each game starts with the contestant selecting one of the cases—this first case's value is not known until the game ends. During the rest of the game, the contestant opens the rest of the cases, one at a time, revealing its value. Each time after a specified number of cases are opened, the banker offers the contestant a certain amount of money to end the game. If the contestant takes an offer, the game ends, otherwise the contestant ends up with the money from the first case.


The first time I watched the game, I realized that there was an amazing application of mathematics here in the form of expected value. Expected value comes up in many situations, especially in poker which I play sometimes in my spare time. I think that with the suspense of the briefcases and the fact that people can earn quite a bit of money through such a simple concept has made this game quite popular.

So, as the wikipedia entry says, the banker offers the contestant a certain amount of money to end the game. This can be illustrated as follows, though not the exact numbers.

Say the contestant has 4 briefcases left: $1, $2, $750,000, $100,000. Now, the banker usually offers the expected value or the probability of picking each briefcase X the value of the briefcase. To illustrate in a simpler example, if I flipped a coin and gave you $2 every time it came heads, the expected value would be: (0) x (0.5) + ( $2 ) (0.5) = $1. So you would expect to gain $1 every throw. This concept is more important in poker where these situations occur more frequently, but what isn't so intuitive is that every throw, even the throws where you didn't earn any money (the coin came up tails), you would win $1. In fact, even when the coin comes up heads, you actually win $1 and not the $2 you get.

A bit of a sidetrack, but when two people with 1000 chips each have KK and AA in no-limit holdem and get it allin preflop, AA expects to win about 80% of the time. This means that of the 2000 chips in the pot, AA wins 1600 chips every time it is up against KK. When KK goes up against AA, it wins 20% of the 2000 chips or 400 chips, every single time.

Back to Deal or No Deal now, we want to see what the expected value of the briefcases are at that point because we can compare this value to the value that the banker offers. If the banker offers more money, we should stop because by playing on, we do not expect to get more money. If the banker offers less money, we should (usually) go on, because we expect more money by playing on. So, with the 4 briefcases above, we can do a simple calculation to see what kind of money we expect if we choose a briefcase.

Basically, we want to choose the briefcases that hold either the $1 or $2. There is a 50% chance in choosing one of these briefcases. So, with the 4 briefcases above, we have an expected value of:

(1/4) x ($1) + (1/4) x ($2) + (1/4 ) x ($750,000) + (1 /4) x ( $1,000,000) = $437,500

We ignore the small $1 and $2 terms (and most of the time we can 'ignore' briefcases in our calculations like $1000 and $5000 compared to the larger $300,000 and $500,000 briefcases; we're really just interested in seeing how much expected value the larger briefcases give us)

So if at that moment, the banker offers us (and in the late stages of the game, he will sometimes offer more money than expected) $450,000 then the math tells us to accept the banker's offer. The offer is more than what we expect if we went on so we should stop right now. This is analogous to the choice of $3 every time I didn't flip a coin and $4 for every time I flipped a coin and it showed heads. Hopefully everyone would choose to get the$3 off of me because in the long run the expected value of this situation (100%) x ($3) > ( 50%) x ( $4) or 3 > 2. So back to the example, we have $450,000 compared to $437,500 and so we'd make more money in the long run. Of course, this in the long run. If millions of people got to this same situation, they would all make money by accepting the banker's offer of $450,000.

Of course, a lot of people when they get to that point want to keep going because they may be greedy, or have some sort of intuition that they chose the $1,000,000 briefcase in the beginning. I think this is what makes the game exciting - the thrill of gambling. But if you always make the right decision, you'll earn money. Think about it this way, if you decided to, even once, choose the coin flip for 4$ instead of getting the sure $3 every time, you would theoretically lose $1. It is possible that you do gain the extra dollar for winning that one time, but remember, when you take that coin flip, you expect $2. Since you don't play the game over and over though, you really just have this one shot at the money and maybe thats why people do gamble, because right now, they only have one chance at getting more money.

So if you're ever on Deal or No Deal, whats your expected value?