In the online gane I had 500,000 and 750,000 left and the bank offered 473,000. OK, let me think about that one. I “no dealed” and won 750K.
Yep. Apparently there’s some correlation between latex gloves being placed on one’s head, and severe, permanent hair loss. Parents, warn your kids!
It tells you nothing about your own door. Before he shows the dud door to you, your door has a 1/3 chance of being the right one. After he shows it to you, your door has a 1/3 chace of being right.
It’s the third unopened, unpicked by you door that you learn a lot about. Before he shows the dud door, it has a 1/3 chance of being right. After, it becomes 2/3, for reasons outlined in your post and elsewhere.
Wow, my first time out I settle for $58k. Turned out I picked the one cent case. I suck at picking briefcases. 
You can watch the UK version on YouTube. It’s interesting because it doesn’t have models holding the briefcases. Instead, each one is held by a potential contestant. When they pick one to play, they already have a box. So even less choice!
Not as sexy though.
Depends on the contestants, surely.
If you want to call “knowing I have a one-third choice of being right but with two doors left instead of three” nothing, then I agree.
And kinda cruel for the guy not playing who might have to open the case with the highest value in it!
We had a thread on this a while back when the American version of the show first came out and someone had a link to an online version of the game that was a heck of a lot faster, which is good because (by far) the worst part of this newer version as well as the show itself is that it’s so pointlessly fucking slow! Spare me the shiny-object graphics and “zshwooosh!” sound effects, please, I have an I.Q. over 70.
That online version’s math is so off I can’t imagine ever wanting to take the deal, here or in real life.
For instance, here I was down to two suitcases: 100K and 200K. Simple math shows my expected value to be 150K. The bank offered me 106! Online or in real life I’m willing to gamble 6000 for a 50/50 shot at 100,000.
Another example:
I had 8 suitcases left:
750,000
300,000
400,000
200,000
100,000
10,000
500
10
The bank was offering me around 130,000 here. Now if you do the “complicated” math of just adding the numbers up, you’d see the bank SHOULD be offering me around 220K. I’d lose nearly 100K in equity taking the deal here.
But if even that’s too difficult for you, count up the number of suitcases over 100,000. There are 5 of them. So 5/8ths of the time I’m walking out almost as great or a significantly greater winner just by turning the deal down right here.
I’ve never seen the show. Are the offers there just as bad?
First of all, I count 4 suitxases over $100,000, not five.
But here is the thing; if you turn down the deal, you open two more suitcases. And since you have very few low values left on the board, it is 50% likely at least one of the cases above your last offer will come off the board, with a 37.5% likelihood that it will be be $300,000 more, and 25% likely that TWO cases above your last offer will come off.
The thing to remember is that you are NOT choosing between the current offer and an even chance of getting one of the remaining cases. No, no, no, no; that’s where most of the contestants on the show screw up. You are, at any decision point before having one case left, choosing betweem this deal and the NEXT deal. And the magic of it is; the better your luck has been up to this point, the worse your odds are for your next round of selections, because you’re likelier to knock off the big ticket values.
To use your example, your values give you an expected average outcome, assuming you go all the way and never take a deal, of $220,063, so the offer of $130,000 is only about 60% of your expected value. But if you turn it down, you open two more cases. I haven’t worked out all the math yet but in your case the expected end outcome is likely to go down, not up, given the cases you have showing.
Also, keep in mind that the show’s goal is not necessarily to get away with giving you the least possible amount of money - if it were, they’d just not bother doing a show. They also want to drag it out as long as possible to increase the “drama”. If they give you a good offer and you take it, then that’s the end of the “fun”. So they lowball you a little bit.
The Wikipedia entry for Deal Or No Deal is excellent, outlining the mathematical strategy involved and comparison to the Monty Hall Problem.
Deal Or No Deal is interesting to watch because it’s combines pure statistics with the classic psychological ‘risk vs reward’ element in a fairly straightforward manner, although I agree that they over-dramatise it with all the fuss over box selection, stretching the show out with the tension a little too much.
I’ve witnessed the episode (UK version) with the first, and only so far, I believe, penny winner. This poor chap played until the bitter end, had two boxes left, the 1p and £100 (which is by no means fantastic but would have bought a slap up meal for him and his significant other) and decided to swap his own box for the last remaining one in the selection, earning a penny in the process.
Interestingly, a study (listed in the Wikipedia entry under the heading ‘Analyzing decision making under risk’) has shown that when contestants stumble, such as seeing their offers fall after taking out a strong box, they then tend to go for broke and start taking big risks.
Rickjay,
Sorry, I meant to say $100,000 or greater, not greater than $100,000. I included 100K into my consideration despite getting an offer 30K higher than that because $100,000 is still a nice payday regardless and I’d be willing to gamble it up on that potentially lost $30,000 to win upwards of $750,000.
As for the second part, let’s look at the worst case scenario here: I lose the two biggest boxes. It’ll happen, I think, about once every 28 times. In that case, my total prize pool is reduced to $600,000 with six boxes to go so I’d guess the banker would offer me around 70-80K.
The best choice I can make (which will also happen 1/28 times) nets me a pool of $1.8 million, an equity of 300K, and a banker’s offer of let’s say $250,000.
I’m really just fiddling around with numbers. In all honestly, I wasn’t even sure what point I was trying to make while I played around with them. I just found the math interesting. But as I looked it over again, I finally came up with a real point (hooray!)
The real, probable, worst case scenario, is to pick the 750K. This is going to happen 25% of the time. It doesn’t matter so much what your other choice is, you lose that and you’re probably losing over 30K off your banker’s offer regardless of your other pick.
So it comes down to this, based on the assumptions I’ve made:
~4% of the time you lose 60K by the next offer
~20% of the time you lose at least 30K by the next offer
~70% of the time your offer at least stays the same and could go up by as much as 120K.
I think it seems worth it to take the risk and choose 2 more. Obviously, with a powerful enough spreadsheet you can determine the full set of probabilities here. It’s really not that complicated…just tedious.
There’s no normal human who could figure this out in real time. I don’t think they’d let you take your laptop up on stage.
The part of the game that I find interesting, which is the real genius of it, is the self-correcting nature of the average prize value. If your initial 6-case selection is unusually unlucky - say you kill of 4 big ticket prizes - that’s bad, but it makes the decision to open more cases more inviting, because now low values are likelier to be picked. If the player goes on a bad streak and gets unlucky on his second and third rounds, it becomes increasingly profitable to keep playing because the board gets more and more favourable to picking cases.
On the other hand, if your initial selections are really lucky, it rapidly becomes stupid to keep picking. That’s why the initial offers are lowballs - they want you to play at least a little while - but in practice, I think most of the contestants would keep on going anyway, because they are, in poker terms, “on tilt.” You can see it in their eyes; Dude, I’m hot! I’m great at this game! Let’s open more!
Howie Mandel is good at egging 'em on too; he always emphasizes the positives, never the negatives. He’s brilliantly, subtly dishonest, always referring back to the Schroedinger’s Certified Check in the case. The Wikipedia article sums it up perfectly:
Either way, the game tends to make the player keep playing.
I took it down to five suitcases before accepting a deal for $108,989. The million was still in play but every other remaining suitcase was $25,000 or less. I “figured” that the odds of having the big winner was against me and this offer was better than every other remaining prize.
As it turned out, I had in fact picked the million dollar suitcase.
I think it would be possible for someone with a good memory and a head for figures. You’re only dealing with 26 numbers and a fixed amount of suitcases at each stage. It would probably be easier than counting cards at blackjack.
I wouldn’t go that far. Counting cards is a matter of adding and subtracing 1 to a running total. You don’t have to add or subtract 750k, 30k, 100, and 10. The differing orders of magnitude would make it more mentally taxing, IMO.
But card counting requires a much more accurate result. For purposes of evaluating a bank offer in this game, you could round things off to the nearest thousand dollars.
“Let’s see, I’ve got 1, 5, 25, 75, 100, 300, 1000, 10000, 50000, 300000, 400000, and 1000000 left. That’s about $1,761,000 divided by twelve. That’s around $147,000. They’re offering me $35,607. No deal.”
Just played another online game and did much worse this time. The million went out early and so did most of the other big prizes. I got out at $17,086. But I played on and got down to the last two - 5 and 25. They offered me $15.