Deal or No Deal: Two Cases

[GreyICE]

In my experience of watching the show the banker's offer is typically between 60% and 70% of E so it is almost never the best course of action to accept the banker's offer.

using this formula if you are left with two cases and have the $1,000,000.00 and the $0.01 left then E = $500,000.00. If the banker offers you less you have a better chance for opening your case. If the banker offers more, you should take the offer.

What makes the show so interesting is that it is real money on the line and it is psychologically difficult for most people to pass up a sure $350,000.00 for a 50-50 shot at $1,000,000.00

Wow, didn't I do a thread about this a while back?

Risk of Ruin.

Or even if your expected value is INFINITY, it's sometimes better to take the safe choice.

 

[Blackadder]

What makes the show so interesting is that it is real money on the line and it is psychologically difficult for most people to pass up a sure $350,000.00 for a 50-50 shot at $1,000,000.00

for good reasons!

This is not a hand of poker, where you make choices to maximize EV, knowing you will play many hands, so when this current hand gives you a big 0 , your strategy in the long run will pay off.

This is a unique opportunity to get a very significant amount of money. Unless the contestant is very rich already, it's wiser to chose the lower 100% certain prize.

 

[ZirconBlue]

I've watched the UK version a few times (top prize £250k) and it is clear that the Banker offers about 2/3 of the expected value of the box, with a significant amount of specific fiddling with the offered amount to take into account particular circumstances of the current contestant- sometimes, for instance, the Banker offers sums where the digits are 'neat' in some way.

It seems to me, although I haven't done the formal calculations, that the early offers are significantly lower than the Expected Value, while later offers are closer to the EV. Which, if true, would tend to keep contestants playing longer.

This is my impression, based on watching a handful of episodes of the US version.

ETA: I just realized that I've probably spent more time reading discussions like these about Deal or No Deal, than actually watching the show.

 

[69dodge]

I've never seen anyone 'deal' early, but if the offer significantly exceeds £15,000 they probably should, because with a lot of boxes left, the expected value of what remains is likely to show regression to the mean and fall back somewhere near £15,000.

That doesn't sound right. Can you give a specific example? That is, which boxes have been opened already and which are still closed?

 

[ponderingturtle]

1. A player chooses 1 case from 26. Each case has a different monetary value, with the lowest being $0.01 and the highest being $1,000,000. The player is not allowed to look in his or her case.

2. The player then eliminates case by case, which are opened to reveal their values.

3. The question is, if the player luckily manages to keep the top prize on the board until there are two cases left (the one on the board plus the one he or she holds), what are the odds he or she holds the winning case?

I say it is 50/50 but was tricked earlier by recent analysis of Monty Hall puzzles and similar situations.

Think of them this way. He now has the choice of the two. It is an equal probability either way because he has a fresh choice.

The illusion is that choosing not to change is not just as much of a choice as choosing to change.

He is still making a choice between A or B, so it will always be 50/50.

 

[rwp]

Think of them this way. He now has the choice of the two. It is an equal probability either way because he has a fresh choice.

The illusion is that choosing not to change is not just as much of a choice as choosing to change.

He is still making a choice between A or B, so it will always be 50/50.

Knowing how he got there matters. If an all-knowing host would have eliminated all of the other cases except one (purposefully not revealing the top prize), the probability would be 25/26 that the last case remaining would be the top prize, and 1/26 that you chose it originally.

The main reason the real situation is a 1/2 probability is that the contestant randomly eliminates cases to get to that point, and all of the information gathered along the way eliminates all other possibilities except for 1/2 in favor of winning and 1/2 in favor of losing.

 

[Badly Shaved Monkey]

That doesn't sound right. Can you give a specific example? That is, which boxes have been opened already and which are still closed?

If a lot of low value boxes go early on, then the offer often exceeds the initial EV.

However, the high value boxes, whose continued presence lifted that offer, have a low probability of being among the last few boxes. If the player stubbornly persists in carrying on to the later rounds having been made a high offer, the greater likelihood is that the high value box will get opened and the offer will get busted down to a much lower level.

I've just looked at Series 1 in the UK.

(By the way, I'd misremembered the actual EV, it's £25,712.12. I think I had the £15,000 figure in mind because the Banker always seems to play to about 2/3 of the actual EV)

Out of 66 games, there were 94 offers or final boxes over £15,000 (77 offers, 17 final boxes). On 36 occasions was a subsequent offer better than that.

If the criterion is set at £25,000 then the respective figures were, 52 (38, 14) and 19.

That means that it's roughly 50:50 (36/77, 19/38) whether an offer over £15k or £25k amount will be bettered by a later offer.

Now my head hurts and I want to go to bed.

Is there evidence of 'regression to the mean' in that? I think I've lost myself in the arithmetic!

 

[ShowerComic]

It seems to me, although I haven't done the formal calculations, that the early offers are significantly lower than the Expected Value, while later offers are closer to the EV. Which, if true, would tend to keep contestants playing longer.

What's interesting to me, is at the end of the game, when the contestant selects 'Deal' and takes the money offered by the Banker. Usually there are only a few other cases left unopened, and the host has the player go through the remaining cases. For each case, the banker quickly gives a value.

To me this means that whatever value the banker is giving, it is automatically calculated based on the hidden cases, the ones on stage, and at the player's side. Suspense, human nature, and commercial interruptions are a large part of the show. -- As with all game shows.

 

[Badly Shaved Monkey]

Here's another piece of information.

For those 66 games, mean maximum offer = £33,774, Mean box content £21,548.

I've not done any stats on it, but note that the mean box content is lower than the EV, probably because the EV is so heavily weighted by the £250,000 and it only turned up once in 66 games. Clearly the error bars on these numbers would probably be quite big.

Nonetheless, it does suggest that if you could take the Banker's maximum offer you are likely to beat the box. Of course, you only know what the maximum offer was in retrospect and as I have already found, a 'high' offer is followed by at least one higher offer about half of the time.

In the end, and has already been alluded to, the mainspring of the game is the willingness of a contestant to blow an almost guaranteed payout of several £000's for a small chance of a really high payout.

Max offer; count
>100,000;7
>=50,000; 13
>=25,000; 23
>=15,000; 41
>=10,000; 50
>=5,000; 62

 

[69dodge]

The whole thing is still not totally clear to me, but I'll try to explain what I had in mind when I said it didn't sound right.

1) I don't think regression to the mean applies here. The population is not infinite, nor even very large. And the sampling is kind of backwards: items are removed from a shrinking sample, not added to a growing one.

2) Sometimes, the average of the unopened boxes will go up after opening another box; sometimes it will go down. On average, it will stay the same. ("On average" means that the amount by which it might go up or down is considered along with the probability of it changing by that amount.)

3) The box amounts are more spread out, the larger they are: the next higher amount after $1 is only $1 more, but the next higher amount after $1000 is $1000 more. (Or something like that.) So, in any group of unopened boxes (or maybe, in most such groups?), more boxes are below the group average than are above it. So if you pick a box at random, it's more likely than not to be below average, and therefore to raise the average of the rest when it's opened and removed from the group. (But, consistently with (2), the less-likely decreases of the average tend to be larger than the more-likely increases.)