Previously I discussed some of the mechanics of Deal or No Deal and how it relates to Schrödinger’s Cat as an exercise in indeterminism, as a quick refresher here are a couple of examples for you to think about what you’d do if paying deal or no deal.
- The are 2 boxes in play, £100 and £10,000. The banker offers £3,800. Do you deal?
- There are five boxes still in play, the amounts left are £10, £100, £1000, £10,000, £100,000. The banker has offered you £12,500. Do you deal?
In the first scenario, you have the choice between taking the money or effectively gambling for all or nothing after £100 is only 1% of what you could win potentially win. The bankers offer is less than half the statistical winnings of £5,050. So statistically speaking, gambling is the right choice as over time you would win more – Of course you are not gambling over time and only have the one chance, so you could lose it all. From a more personal viewpoint, the amounts are unlikely to be radically life changing and you aren’t going to lose them. It’s just that you might not win them, which again suggests that going for the gamble is the best option for the contestant and it’s not merely being greedy.
The second case is even less clear cut, even though the Banker’s offer is £10,000 below the statistical winnings which at this point in the game would be £22,222. At first glance I thought I would deal with almost no hesitation. Firstly the statistical winnings only take account of one average, the mean.
Whilst useful as a starting point it can be skewed by excessively large or small values and here the only reason the statistical winning is £22,222 is because one of the prizes is over 9 times greater than the sum of the other four prizes put together. The bankers offer seems low compared to the statistical winnings, but note that it is still higher than 4 out of the 5 potential prizes. This means that if you didn’t deal and played to completion (ignoring for the moment that you would get another offer when down to two boxes) that you have an 80% probability of walking away with less and so cutting your losses at this point seems like a sensible course of action. You might only win £12,500, but if you play on it is a lot more likely that you will win less.
However can the odds be improved? Technically answer is no, but if you rephrase the question it continuing to play doesn’t seem like such a bad idea. In the example the bankers offer is reasonable close to one of the prizes and if one treats them to be the same the question changes slightly. Think about it, what is the difference between £10,000 and £12,500 in real terms to you? A few more shiny things perhaps, but the difference is negligible – Or at least it is for mathematicians who are used to treating horses as infinitely small points traveling at fixed velocities. Then the odds become:
- 20% Win more
- 20% Win the same
- 60% Win less
So playing to completion as it stand means the odds are only slightly worse than even that you are going to worsen your situation significantly, but that you have a 1 in 5 chance of winning big.
It’s still more likely that you will win less overall, but this decision is weighted to take into account the values in relation to each other and so isn’t quite as all or nothing and if playing to completion only I’d take those odds and gamble on my box.
But that is no how the game is played, we know that the banker will make another offer when we get down to just the two boxes and after that we have a chance to swap boxes, so it’s not just a case of playing to completion or dropping out now. At this point of the game however it’s still in the future and although indeterminate we can look at the probabilities and see if it’s worth going further.
Now although one box is fixed and you can’t open it to find out the contents, you can open three of the other boxes to remove possibilities. Taking advantage of the undefined nature of the box contents we can treat all contents as random. After 3 of the boxes are opened we are left with 20 possible combinations for contents of the boxes, despite there only being 4 possible combinations of boxes.
Allow me to demonstrate, I have box number 12 and the boxes in play are numbered 3. 8. 19, 20. (The numbers on the boxes don’t matter, I am just using them as examples.) After revealing three of the boxes I am left with the following 4 combinations:
12, 3
12, 8
12, 19
12, 20
I would also know the potential contents of the 2 boxes and then be able to work out how best to proceed. Realistically at this point if you have been lucky and the £100,000 is still in play you should get a banker offer of between £35,000 and £55,000 and if it was me I would take the money and run at that point and not risk the gamble. However if I had ridden my luck too hard and removed the £100,000 from the equation I might as well gamble on the contents of a box and hope for the top remaining prize as the amounts don’t really matter too much at that point. Unless it a choice between £10 and £10,000 for example with a banker offer of £5,000 – Although as at this point the game is considered lost I will not model the decisions further.
However with the five prizes still in play, you don’t know the exact situation you will find yourself in once you have got rid of 3 of the remaining boxes. Assigning the letters A, B, C, D and E to the contents of the boxes, the possible outcomes are:
AB BA CA DA EA
AC BC CB DB EB
AD BD CD DC EC
AE BE CE DE ED
However as the contents of the boxes are not known, the order is not important. AB is the same as BA. After all £100,000 in a box and £10 in the another box is the same as £10 in one box and £100,000 – Until you open one of the boxes they both contain £10 and £100,000, the cat is both alive and dead until you look at it. What this means is that the 20 combinations become 10 as they pair up with each other, this leaves:
AB BC CD DE
AC BD CE
AD BE
AE
Now if we assign a letter to each prize so that A=£10, B=£100, C=£1000, D=£10,000 and E=£100,00, we can see that the chance of leaving the £100,000 in play (and indeed any specific prize) is 40%. The chance of leaving the £10,000 in play is also 40%, however as there is a chance that both the £100,000 and £10,000 are left in play the chance of £10,000 representing winnings is actually 30%. The remaining 10% represents the £10,000 being ‘losings’.
So at the moment we have:
- 40% chance that top prize is £100,000.
- 30% chance that top prize is £10,000.
- 30 % chance you’ve flogged the horse of chance to death.
Assuming that you haven’t ridden your luck too far and are still in the game, what this boils down to is:
- 40% chance the banker offers approximately £50,000
- 30% chance the banker offers approximately £5,000
- 30% chance the banker offers a negligible amount
The statistical winnings using this model are (0.4 x 50000) + (0.3 x 5000) + 0 or as a numeric value, £21,500. This number is still heavily biased due to the large value of one prize compared to the others, but it allows you to weigh up the odds.
So contestant you have the choice walk away with £12,500 or gamble to the next stage potentially losing most of it. Obviously one part of the decision relies upon how important £12,500 is to you. Whilst probably not a life changing amount, if used wisely it could alter ones lifestyle over time or give a chance to have some experiences that otherwise might remained unfulfilled or in these days of credit meltdown might offer a fresh start. If the last reason is the case you’d be a fool to gamble further, for me however the choice is between buying shiny things or using it judiciously – Either way things I currently get by without.
In short I’d gamble as I have a 40% chance of increasing my winnings by a factor of 4 and those odds are good, especially with a potential fall-back position where I have a chance of still walking away with the second place prize of £5,000 if things don’t pan out and only a 30% chance of effectively winning nothing.
In fact with Deal or No Deal, statistically speaking you should lamost always gamble, over time you would win more than the banker will ever make as an offer, as the offer should always be less than the average (mean) winnings – As with gambling, the house always wins. However using the median is also worth contemplating especially if one of the prizes is disproportionately small or large and if all prizes except one are less than the banker’s offer you probably won’t be walking away with the biggest prize and might want to consider not being greedy and taking the guaranteed amount unless you wind up with a case like the one above which does offer a fallback position of sorts.
However I still have got to the question that started this whole train of thought off – Should you swap your box at the end of the game? I’ll get to that next time.