Naive Investor Chapter 7 Interlude: Deal Or No Deal

Even with something as simple as Cash, people make the wrong decisions for subtle yet difficult reasons.

The best place to see this in morbidly fascinating action is Channel 7's Deal Or No Deal, which I understand is syndicated from the US.

So there are 26 cases, containing (usually) unique dollar amounts a contestant can win ranging from .50c (which for some reason is called the 'monkey') to the grand prize of $200,000.

A randomly selected contestant can select one of the 26 cases as 'their' case. This case will remain with them till the end of the game. They must then pick from the remaining 25 cases whose contents are revealed one by one. Periodically the 'banker' offers you a 'deal' which is a cash payment to stop playing.

You have two choices at these times 1) To 'Deal' or 2) To yell 'No-Deal' and keep playing.

Now just as in real life the banker has you by the pubic hairs, because ultimately you will have to play right through the game and get back to the case.

But there are two things about cash to illustrate here:

1. We are terrible at calculating odds.

So the first thing you do is pick your own case. Most people can wrap their heads around the fact that the bigger the cash prize you pick, over the course of the game the higher your offers will be. So if you pick the $200,000 initially then it isn't going to be in any of the cases you choose to reveal. So even if you eliminate all the good prizes early, you just have to keep on picking till there are two cases left.

Sounds easy right? Except you don't know what is in the case you picked. It could be the $200,000 it could be the 50c. If you open 24 of the remaining 25 cases, and end up with only two options left 50c or $200,000 how do you know what to do? To deal or not to deal?

You deal.

Here's why, when you pick your case you have a 1/26 chance of picking the best prize ($200). Now by simmitry there is a 25/26 chance that you didn't.

So in the first round you choose 4 cases to open, you don't reveal the $200k. There-to-fore the odds now are that you have a 1/22 chance you picked the $200k case, and a 21/22 chance you didn't? Wrong. The odds are still 1/26 you picked it, 25/26 you didn't.

The initial odds remain, as concerns your first choice, you get additional information, however it is still FAR MORE LIKELY that you didn't pick the best prize than you did. 25 times more likely.

Suppose then that the next case you open has the $200k in it? Well a) it proves how unlikely you were to pick it and b) you would just shift your focus to the next highest prize value. The odds that you picked the $100k prize for example are still 1/26, it is true of any prize value. you also had a 25/26 chance of not picking it.

Suppose you didn't reveal the grand prize though, once you get down to two cases left, the one you did pick and the last remaining of the 25 you didn't, are the odds 50:50?

No, the odds are still 1/26 you got the big bucks, 25/26 the last remaining person did. The bank only has to worry about what you had though, since the other turd holding a case only stands to win $500 by guessing correctly, that's not even 1 adspot for channel 7. The bank typically offers a value just a tad under the difference between the big prize and the last remaining amount, so for example if it was $200k and 50c the value would be somewhere around the $100k mark.

That's a really good deal, because your expected value is given by (200,000*(1/26)) so that is the outcome of winning multiplied by the odds of winning which I can tell you is: $7829

$10,000 would be a good deal. $100,000 is a great deal.

But crucially when there are 2 cases, and the $200,000 is in one of them, people think that the odds are 50/50.

Even I when first confronted with this fact of probability was convinced it was some kind of magic. But jsut think about it. You can accept that when you pick your case you have a 1/26 chance of winning the big prize. If everyone then immediately opened up their cases, not only would it be very short viewing, but you would expect the person to win the big prize money 1/26 episodes. So would the outcome be any different if they just opened the cases in order of lowest value to highest? You would simply expect the person who picks the case first to open the case last 1/26 episodes. Chances are, 25/26 times somebody else will open theirs last.

If you want to test it out, do what I did. Write the numbers 1, 2 & 3 on the top of a page. Then below them write in your best resemblance of randomness combinations of 2 x 'L' (L for Lose) and 1 x 'W' (for Win). Like this:

1 2 3
W L L
L W L
L W L
L L W
W L L
W L L
L L W
L W L

Then get your friend to pick a number between 1 and 3. Once they pick it scratch out one of the remaining 'L' option for each pick they do. So for example I pick 1 (W) in the first row, then you say 'Okay I'm going to reveal that 2 is a loser. Do you want to stick with one or switch to 3?' I switch to 3 and lose. But you should always switch to 3, just as you should always take the deal. Because if for the second round I pick '3' and you scratch '1' (the only remaining loser) by switching I win. You will find that by switching you win 2/3 times. By sitting put you will only win 1/3 times.

Who would have thought that understanding odds was so fucking difficult and counterintuitive?

2. We always forget Opportunity Cost.

This one is much simpler to illustrate. I'm walking down the street and I am approached by a wealthy banker type and he says to me: 'tohm, how would you like to win $200,000?' and I would say 'I suppose I could find a use for it.' and he would say (being a scrupulously honest banker) 'You have a 1/26 chance of winning the $200,000 how would you like to pay $100,000 for that chance?'

I would laugh and keep walking as, hopefully, so would you.

Even if you believed the odds to be better (namely 50:50 or 1/2) almost nobody would pay $100,000 for a chance to win $200,000. The potential loss would hurt far more than the gain.

Yet watching Deal or No Deal, people do exactly this all the time.

Why? Because they don't understand that the bank is offering them their money. They have won it, it is theirs, all they have to say is 'Deal', but they say 'No Deal' not just to huge sums like $100,000 but to amounts like $40,000 or 80% of their annual wages, sometimes that 200% of their annual wages.

Now if the show put $40,000 in your hands and said, 'It will cost you $40,000 to keep playing.' it would be a very different show, even though on paper, they are exactly the same.

The bankers deal doesn't feel like their money, so they have no trouble 'spending it' to play on for a pipe dream. Husbands and wives encourage each other to play on in the heat of the moment. If they were buying a share portfolio they hoped to earn them a 200% return on a principle of $40,000 you could bet they would spend more than 30 seconds thinking/talking about it.

You could also bet that few couples agree that a horse paying $26 is a good bet and they should put $40,000 down on it at cup time.

Conclusion:

Now you know the two basic 'rules' of Deal Or No Deal, it goes from being a pretty tedious gameshow hosted by quasi-talented Andrew O'Keefe to a captivating viewing experience. What other show demonstrates otherwise intelligent people making such routinely bad decisions with real money?