Beat The Banker / Deal or No Deal Odds Calculation Help...

Hello.

I'm trying to work out how the banker on the UK T.V. show "Deal or No Deal" Works out the Math of what Prize to offer a contestant

If you haven't seen the show General description follows

Contestant Chooses A Random Unknown Prize Between 1p to £100..


Prizes Increment As

1
5
10
50
100
200
500
750
1000
1500
2000
2500
3000
3500
4000
5000
6000
7000
7500
10000


(These are My Variables I wish to program), Depending on what Prize Boxes are left after the contestant has randomly chosen 1 of 20 Prize Box
Contestant Has Taken One of These Values Away Then Takes 5 of the values out of the " Pool " after those 5 are taken away a Sum is offered for the contestant to walk away with that sum or Gamble and take away Another 5 boxes Then a new offer is made for the Contestant to walk away.. etc etc until all boxes are opened or until Contestant Walks.

If at the end of this the Contestant Remains and The only Prize boxes Left Have the values 1 and 10000 The Offer would be 5000 That is as my mathematics skills Will allow me to Imagine..

I Need the formula to work out the value to Offer as the Contestant Plays the game

Hope that Makes Sense Cheers..

I'm afraid your going to have to roll your own algorithm. This forum discussion includes an excellent post on Deal or No Deal, including a few actual examples from the American version of the show. If I were approaching this problem, I would start with the following calculations of the remaining cases:
1. Mean (aka the expected value), median, and some sort of calculation that groups suitcases into groups of values (because there should be a difference between three suitcases of $1, $2, and $1M versus $1, $500K, and $500K, even though their expected values are equal. Use these values to compute some sort of "TRUE VALUE" amount.
2. The purpose of the show is drama, so the goal is to keep a contestant from actually taking a deal until well into the show. Thus, during the first few rounds you need to make low-ball offers so that the contestant declines the deal. As the game progresses, your offer should approach, but never equal the TRUE VALUE amount.

As to other tidbits. There is probably some psychology at play based on the behavior of the contestant. For example, a boring contestant may get better offers so that they get off the stage quicker. A risk-adverse contestant may get worse offers because the banker can get away with low-balling them. These are all things that you probably can't program into your game. Also, this TRUE VALUE should rarely, if ever, equal the expected value because expected value is only designed for repetitive events, and Deal or No Deal is definitely not repetitive... contestants get only one shot, which tends to make them naturally risk adverse (the same principle applies to other big money games, such as Who Wants to be a Millionaire, where it took over a decade before somebody actually answered the million dollar question incorrectly).

This is simple probablity maths. A "fair" offer is the mean value of all remaining closed boxes, including the one the contestant has. Of course, the banker never ever makes a fair offer, they always offer low. The host then wraps it all up in some mumbo-jumbo which never has any bearing on reality and the contestant inevitably panics and takes a stupid deal that anyone with the most basic understanding of probability maths would have laughed at.

Take this scenario:-
The remaining values are £1, £5, £20 and £100,000. A fair offer would be £25,006.5 (the mean). The banker offers £15,000. The contestant flaps. "Oh lordy, lordy, just one bad choice and I could lose the lot, I'll take the deal. I want to do up the spare room". They play through the rest of the game (to see what could have happened) and lo and behold the next box they open is the £100,000. Everyone cheers, slaps them on the back and whitters on about how clever they were and how they beat the banker.

Well, no they weren't and no they didn't... because if you ran that same simulation 1000 times they'd have been holding the £100,000 box themselve 250 times. Their average winnings would have been £25K (give or take) and they just stupidly walked away with £15K. From the information that was available to them at the time they actually took a £10K loss against the most likely outcome.

Can you tell that I think Deal or No Deal is a stupid game played by morons?

So if you really want an algorithm: take the mean, knock off some amount randomised between 10 and 40 percent. Everything else is irrelevant.

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Originally Posted by FunkyDexter

if you ran that same simulation 1000 times

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That's the key... you CAN'T play the (actual) game 1000 times. You only get to play once, and at least in the US version, you're generally dealing with life-altering quantities of money. This is why there is a huge difference for the player between a $400,000 offer when the two cases left are $1 and $1,000,000, as opposed to a $400,000 offer when the two cases contain $250,000 and $750,000. Their expected values are only 50 cents apart, but I would advise the former to always make that deal, knowing he has been low-balled by $100,000, while the latter has much better reason to press his luck.

It sounds like in the UK and Australian versions the cases contain less money relative to the typical contestants income (it makes sense, assuming the audience sizes are proportional to the national populations). Is £50,000 a life-altering amount for the typical Brit? If not, then we might as well be talking about different games entirely.

This is kind of similar to prisoner dilemma scenarios. Tit-for-tat is almost always the best strategy when there is repetition, but for the actual prisoner himself, he only gets one shot, so tit-for-tat doesn't even make sense for him.

That said, your suggestion to the OP is a good one, with the caveat that the game goes longer, the banker's offers should approach the expected value (since that is what happens in the actual game).

And for what it's worth, I don't like this show either, but mostly because of the over-dramatization. I find the mechanics pretty interesting, but am not willing to burn an hour for just 10 minutes of actual entertainment.

Hi,

Here is an idea for an algorithm based on an extension of what FunkyDexter has already posted. This does not take into account the psychology issue discussed by Lenggries, but I will touch on that later:-

1) Calculate the AverageBoxVlaue of the Boxes left to pick.
2) Calculate the Difference between the Lowest and Highest value box's.
3) Based on the Difference calculated in point 2, determine the number of box's defined as a LOW value box (i.e. <=Point 2) and defined as a HIGH value box (i.e. >Point2)
4) Then compute this calculation:-

(AverageBoxValue*(TotalNoOfHighValueBoxes/TotalNoOfLowValueBoxes))+LowestValueBox

As an example, consider these boxes left:-

Quote:

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5, 10, 15, 20, 100,000

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The Average is 20,010
The difference between Lowest value and the Highest value is 49,779.5
The number of LOW value box's is 4
The number of HIGH value box's is 1
The Result of the calculation is 12,504.38, which, as an offer seems about right?

To then add some sort of psychology into the mix you could then create an Enumeration such as:-

Code:

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Public Enum Confidence
  Mouse
  Timid
  Typical
  Butch
  Lion
End Enum

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And then manipulate the calculation to take into account some sort of psychological condition of the player.

Hope that helps,

Cheers,

Ian

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you CAN'T play the (actual) game 1000 times. You only get to play once

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We're dealing with probablity maths so that's irrelevent. Logically you should alwasy make the decision that has the best prospective outcome. I understand what you're saying but you're essentially advising the player to make their decision based on emotion rather than logic. $400,000 dollars may be a life changing amount of money but $1,000,000 is two and a half times as life changing. The right decision in life is to always, ALWAYS play the odds. You may lose out today but you'll win back tomorrow. (Which is the same reason you're better off not buying insurance if you can help it, collectivise your own risk, as part of a wider collective if necessary - it's cheaper).

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It sounds like in the UK and Australian versions the cases contain less money

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From memory I think they run from £1 to £100,000 in the UK but I don't watch it enough to be sure. It's certainly enough to be life changing if they win the jackpot though I don't think you'd retire on it. Realistically, most people win somewhere less than £20,000 which is a deposit on a flat. Nice, but not life changing.

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This is kind of similar to prisoner dilemma scenarios. Tit-for-tat is almost always the best strategy

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I don't know anything about that but it sounds quite interesting. Can you expand on it?

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mostly because of the over-dramatization

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Yeah, I blame Chris Tarrant. No quiz show since Who Wants to be a Millionaire has been able to go to a commercial break without a 5 minute suspense build up at the end of which the host declares "and we'll find out after this..." As a formula it's getting pretty old at this point. I watched a great episode of Who Wants to be a Millionaire once where the contestant had already won the Australian and US versions. He was utterly sure of himself in every answer he gave. Tarrant kept trying to do teh "Are you sure? it could be A." thing and this guy just shut him down cold every time. It was brilliant TV for all the wrong reasons.

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Originally Posted by FunkyDexter

We're dealing with probability maths so that's irrelevant. Logically you should always make the decision that has the best prospective outcome. I understand what you're saying but you're essentially advising the player to make their decision based on emotion rather than logic. $400,000 dollars may be a life changing amount of money but $1,000,000 is two and a half times as life changing. The right decision in life is to always, ALWAYS play the odds. You may lose out today but you'll win back tomorrow. (Which is the same reason you're better off not buying insurance if you can help it, collectivise your own risk, as part of a wider collective if necessary - it's cheaper).

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I agree with your statement that "Logically you should always make the decision that has the best prospective outcome", but disagree on your assumption that the "best prospective outcome" is measured only in quantity of money received. Since this game is presumably a once-in-a-lifetime opportunity, there is potential for tremendous regret, to the point of depression and suicide, if somebody blows it. In the US, you see this all the time with athletics. In baseball, for example, a player becomes a free agent after about 6 years with a team. Hitters typically reach their physical peak at ages 27-29, so let's consider Bryce Harper. He entered the league at age 19, which means he could become a free agent at 25, just before he is expected to peak. So if he is as good as everyone expects him to be, he could in theory sign a contract in 2018 worth over a third of a billion dollars. However, there is a good chance that his team, the Washington Nationals, will sign him to an extension this year or next worth much less than that. Why? Because Harper also carries significant risk: he could get hurt or even killed (presumably not on the field); he could fail to develop as expected; the US economy could collapse; etc. So in that sense Harper might be willing to except a guaranteed (and very much life-altering) $200M dollars now rather than a chance at $400M in 2018.

Deal or no deal is not altogether that different. $400K is life-altering to most Americans. $500K is also life-altering, but not much more so than $400K, but the point is that to a typical low-to-middle income American, $400K and $500K aren't much different. The difference between $400K and a million dollars to the typical American is MUCH less than the difference between $400K and $1, which is why accepting the $400K is still a very good deal for most people (for someone who is already wealthy or someone who will play this repeatedly, the definition of "best prospective outcome" obviously changes).

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Originally Posted by FunkyDexter

I don't know anything about [tit-for-tat] but it sounds quite interesting. Can you expand on it?

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Tit-for-tat is a strategy developed for a game theory tournament of a simulation known as the prisoner's dilemma. Much to everybody's shock, it proved to be the most effective strategy, while also being the simplest, of all, and has since come under significant study for its real world implications and effectiveness (including instances as wide-ranging fields as international business, communications, evolution and trench warfare). However, tit-for-tat only works if nobody knows how many iterations remain, which is why tit-for-tat cannot work for any actual individual prisoner in the prisoner's dilemma, since the current situation is presumed to be their first and last encounter with the dilemma (this creates an interesting paradox once lawyers get involved, since two defense attorneys could reasonably expect to find themselves in such situations again, and so they might advise their clients to act in the best interest of the lawyer rather than the prisoner).

Thanks for the input. Its more of a Pub style Game i was planning to do. The values I'd written are in Pence so 1p to 100 pounds. Possibly adding it to A Bingo ball Generator I Wrote some years ago for Pub Entertainment..

However i certainly have some Ideas to Mulch over. Thanks again.