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Deal or No Deal - Many a mickle maks a muckle

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December 24th, 2005


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03:38 am - Deal or No Deal
"Deal Or No Deal" is the most interesting new game show we've had for a while in the UK. It's a format that has been used in countries on at least four continents, as lovingly compared and contrasted by brigbother. After months of speculation, Britain finally has its own version, stripped across the Channel 4 schedules from Monday to Saturday at 4:15pm. It also represents Noel Edmonds' return to TV, after the wheels fell off Noel's House Party years ago. I half-wrote this in early November and then let it go for weeks; as the US is having their version at the moment, this strikes me as the appropriate inspiration to finish the job off while it's at all relevant.

It's a fairly dumb show; at heart, it's Let's Make A Deal writ large. (UK readers, Take Your Pick writ large is even more accurate. Someone made that comparison before me and I can't remember who.) Nevertheless, there's something to it; it poses a few interesting questions about people's attitudes and expectations pertaining to winning. In short, a contestant is selected and wins a sum of money (in general). They have no control over how much; they have very little idea as to how much at the beginning, but during the show, they get more information. They are given six opportunities to swap their unknown prize for a known sum of cash, based on what the information is; by the end, it's heads-or-tails as to whether the sum of money is one known value or another known value. If they accept a known sum of cash, they win that much; if they reject all six offers, they win their original sum. Er, that's it.

It's the Crazy Frog of game shows.

There are still some interesting questions it raises, though - ironic for a show which, at least in the UK, has no questions. (Some international versions do have an elimination quiz to determine the contestant to play the game; others do not.) The main driver powering the show is utility theory; man can only live in one house, drive one car, pay off one set of his credit card bills and so on. Accordingly, most people would prefer a guaranteed £50,000 to an even chance of winning £100,000 or nothing. (Not everyone, though, or the casino industry would not exist.) Deal or No Deal plays on this. If you had £45,000 for sure or a half chance at £100,000, which would you prefer? How about £40,000? £35,000? If the choice were not between zero and £100,000 but between (say) £20,000 and £75,000, would it make matters different? This show demonstrates the current exchange rate between birds in hand and birds in bush.

Here are some actual examples from the first couple of weeks of the show. Which of each pair would you prefer: either a guaranteed amount or an amount which is equally likely to be any of the two (or five) numbers listed? If you haven't looked at exchange rates for a while, £1 is worth about US$1.75 or so and xe.net has the other exchange rates.

Poll #639482 Deal or no deal?

Which would you prefer?

one of £5,000 or £15,000
22(84.6%)
£8,000 guaranteed
4(15.4%)

Which would you prefer?

one of £500 or £20,000
0(0.0%)
£9,900 guaranteed
26(100.0%)

Which would you prefer?

one of £10 or £35,000
2(8.0%)
£14,000 guaranteed
23(92.0%)

Which would you prefer?

one of £0.01, £0.10, £1,000, £20,000 or £35,000
7(26.9%)
£7,000 guaranteed
19(73.1%)

Which would you prefer?

one of £0.50, £50, £3,000, £75,000 or £250,000
5(19.2%)
£35,000 guaranteed
21(80.8%)


It'll be interesting to see which people prefer, but the dilemma posed is whether (for instance, taking the third example) the pain of losing almost all of £14,000 you could have had is more extreme than the joy of turning the £14,000 you could have had into £35,000. It's like Who Wants To Be A Millionaire? without those awkward questions. Then again, when Millionaire? gets interesting, almost all the viewers don't know the answers to the questions and the interest comes in following the contestants in their decisions as to whether to gamble or not. Both shows frequently feature life-changing sums of money as prizes, though obviously it varies from contestant to contestant as to what a life-changing sum of money is.

When there are five possible sums of money that might be the result, it's rather harder to work out what the appropriate levels of joy and pain from the various outcomes are. There is a branch of economics called utility theory which attempts to quantify abstract amounts of joy and pain to work out why people take the decisions they do, and to help people work out how to make better decisions for themselves. When trying to choose between a definite value and an indefinite amount, you are effectively trying to work out how good and bad the possible outcomes are, which is a process of assigning utility values to the various outcomes even if you do not explicitly do so numerically. Not everyone will make the same decisions based on the same sets of possibilities, as you have (hopefully) seen from the poll results above.

A slightly fuller description of the show is that there are 22 potential contestants, each of which carry over from show to show. Twenty-two boxes are randomly assigned, one per contestant, and a random contestant is selected to be the main player that day. The 21 remaining contestants are joined by a new 22nd for the following show. Each of twenty-two different sums of money appears in one of the boxes; the twenty-two values are known in advance and do not change from show to show. In Britain, they vary from £0.01 to £250,000; you can find a list at the bottom of this page.

The main player then selects five of the remaining boxes; after each one is selected, the content of the selected box is revealed so the contestant knows another sum of money that isn't inside that box. Accordingly, should many of the small values go quickly then the main player has good reason to believe the sum of money inside their box is likely to be high; alternatively, if many of the big values are eliminated quickly, the main player has good reason to believe the sum of money inside their box is likely to be low.

After five boxes have been revealed, "The Banker" (a role played by the producer of the show) will then make an offer to the player of a fixed sum of money to leave the show and conclude gameplay. Should this offer be accepted, the gameplay concludes for the day. The Banker makes this offer through the medium of an on-air telephone call to the host of the show, Noel Edmonds. The Banker's voice is never broadcast and only very occasionally will the main player get to speak to the Banker. The host and the player relay the Banker's comments and the host communicates the Banker's offer. The player then chooses between the titular "Deal" (accepting the fixed sum) or the equally titular "No Deal" (rejecting the fixed sum).

Should the offer be rejected, then three further boxes are rejected in the same way, and an offer is made. At this point, 8 boxes have been revealed, meaning the contestant's box will be worth one of the 14 remaining values. Once again the Banker makes an offer and the contestant chooses whether to "Deal" or "No Deal".

A second rejection reveals a third round with three further boxes being rejected before another offer, with the main player now better informed that there are only eleven possible values for the contents of the box. Another rejection brings a fourth round, three more boxes being rejected, another offer being made and the true value being known to be one of the eight remaining. The possible fifth round eliminates another three boxes before the fifth offer; by now, the main player knows there are only five possible values for their box.

A sixth round is played if required, with three more boxes remaining, and a final offer is made; the main player will either accept the final offer or know that the actual sum they win is one of the two remaining known values. Should the sixth and final offer be rejected, the player then (irrelevantly) may be offered the chance to exchange their box for the last unknown box; the main player's box is opened and the sum listed therein is their final prize.

It really is a guessing game. The main player has no control over whether the next Banker's offer will be higher or lower than the last one, which principally depends upon the extra information revealed by the boxes that have been rejected. However, quickly-established protocol dictates that the Banker will tend to make relatively unattractive offers at the start of the game and relatively attractive ones at the end of the game, largely because the show becomes much less interesting once a deal has been accepted. (The rest of the show is played at a fast rate, demonstrating what the offers that might have been made would have been and the sum of money in their box that the main player turned down.)

It is also to be noted that the Banker's offer need have no relationship to the possible values of the box; however, the Banker is said to want to make the player leave with as little money as possible. Accordingly, the Banker is trying to make the player leave with an offer rather than risking taking the player take sum of money in their box if it is very large, but trying to make the lowest possible offer which is successful in this regard. Suppose two successive shows end up with contestants left with an even chance of £1,000 or £15,000; if both contestants take their box, the most likely conclusion is that one will win the lower sum and one the higher sum (in either order) for a total of £16,000 conceded. Accordingly, making two offers of (say) £7,000 which are accepted costs only £14,000 and represents a saving of £2,000.

With this in mind, the Banker must attempt to assess the player's psychology and work out how little he can afford to offer and still have the player take the guaranteed sum rather than the unknown. It is generally believed, based on the evidence of the shows to date, that the Banker's final offer will be between around 74% and around 106% of the average (mean) of the two remaining values - so should £1,000 and £15,000 remain, one would expect to see offers between about £6,000 and about £8,500. Offers very close to the mean are rare but not unknown; offers in excess of the mean are very rare but, again, not unknown. Earlier offers tend to be lower percentages of the mean of the remaining boxes; see the comments about protocol above and the Banker's preference for a longer show because it's more interesting.

A player therefore stands to receive better offers by convincing the Banker that better offers will be required - that they are prepared to reject a relatively good offer in return for an unknown sum. Intuition suggests that richer players can better afford to be risk-loving than poorer ones; accordingly, the show will tend to reward richer players (or more convincing actors?) more than poorer ones. I would tend to believe this is unfortunate. The show would be more to my taste were the Banker to make more generous offers to poorer players who need the money more badly, or to players who stand to win less money through having been unlucky in their choices of boxes to reject, but this is not my protocol decision to make.

(It is believed that the Banker has no personal stake in attempting to keep contestants' winnings low, other than except possibly for the pride of a job well done and job security. Nevertheless, the moral imperative to give away as little prize money as possible takes... well, if not a hangman's sensibilities then at least a miser's, arguably a thief's. The host paints a black picture of the Banker, which is all too easy to believe. The person who plays the role of the Banker may be far from a miser in real life, but anyone who can be so miserly when such large stakes are on the line has demonstrated themselves to be miserly through their actions.)

The issue of hosting protocol, and the effect that hosting protocol has on contestants' chances, sometimes appears in other contexts. People sometimes talk of the Monty Hall problem, whereby a putative player on Let's Make A Deal (though the same problem has appeared in many other guises) is offered a choice of three doors, one of which hides a car and the other two hide a goat. The player chooses a door; the host, who knows the contents of all three doors, opens one of the other two doors to reveal a goat. The player is then offered the choice to swap the contents of their door for the contents of the third unopened door. Is it in the contestant's interest to swap? If the choice is always offered, the contestant should always swap; if they swap, then they only win a goat if they had originally picked the door with a car, which will happen 1/3 of the time. The remaining 2/3 of the time, they will win the car. Wikipedia has lots of proofs of this and alternate ways of explaining this result if it does not feel convincing.

That said, consider two alternative hosting protocols in the Monty Hall problem. Changing the parameters slightly, it is not necessarily known in advance by the player that the host will offer the player the chance to switch from their original door to the unopened door. If the player believes that it's in their interest to swap doors if the chance to swap is offered, then the host can play on this fact and artificially help or hinder the player's chance of winning the car.

A host who wanted the player to win the car would not offer the player the choice to swap should their initial choice be correct. Accordingly, if the initial choice is correct, the player wins at once; if the initial choice is incorrect, the host offers the player the choice to swap, which they will, resulting in the player swapping to the winning door and so winning. Conversely, a host who wanted the player not to win the car would not offer the player the choice to swap should their initial choice be incorrect. Accordingly, if the initial choice is incorrect, the player loses at once; if the initial choice is correct, the host offers the player the choice to swap, which they will, resulting in the player swapping to the losing unopened door and so losing.

Returning to Deal or No Deal, I tend to believe that people will (internally, if not explicitly) base their decision whether to deal or not upon whether they feel that they have, in some sense, won the game. The most interesting aspect to the show, in my view, is seeing how different people define victory. I tend to believe that people define victory by avoiding losing. If you turn down an offer and later receive a lower offer, you are identifiably losing compared to where you were before, though you may or may not receive a higher offer later. If you decline the final offer and your box turns out to be worth less than the value you were offered for it, you have identifiably lost compared to what you would have received had you taken the offer.

That said, there's another way to look at it which ignores offers that might or might not be made after any offer you receive. Suppose you're in a situation where you're going to win one of £0.01, £0.10, £0.50, £1 or £250,000 and you are offered - say - £30,000 to leave the game. If you take the money, there's a 4/5 chance that you will take £30,000 instead of a tiny sum of money and only a 1/5 chance that you will take £30,000 instead of £250,000. Considering the massive joy to be gained from the highly probable former and the relatively limited pain to be suffered from the improbable latter - because, hey, £30,000 goes a long way! - then thinking about your potential joy and pain, you have won a lot of joy at the risk of relatively little pain.

Another way of looking at it supposes that a player is £10,000 in debt from which they are having terrible difficulty recovering. Any win of £10,000 or more can arguably be seen as a win for that player in a way that a win of less than £10,000 is not. Alternatively, a player might have their eye on some other sum with a particular purchase in mind. On some game shows, a player might try to win a car; on this show, a player might try to win enough money to buy that car. Making less than that sum is a loss, albeit potentially a loss with a consolation prize. Another player might just choose to avoid being embarrassed by ending up with a poor sum of money, however they choose to define that - though when two contestants out of the first 41 won £0.10 and £10 with the next lowest prize being £1,000, then £0.10 and £10 are identifiably fairly poor.

The conclusion is that there is no definitively correct way to play the game. The tactic with the most apparent logic to it is "try to win as much money as you can"; if you are willing to accept this, then we simply get into the realm of a branch of probability called expectation theory. The expected value to you of taking the deal is known; the expected value to you of taking the unknown sum is simply the average (mean) of the possible values. By way of example, you would prefer an unknown choice between £250,000 and 1p to a guaranteed £120,000, simply because half the time you'll win the £250,000, and half of £250,000 is more than £120,000. However, given a choice like that, would you really take the unknown sum? Perhaps the results of the poll above will show there are lots of people who would take the guaranteed sum in real-game situations.

If you're still thinking that you simply want to win as much money as you can, consider the rather pathological case I outlined above: a guaranteed £30,000 or an unknown prize from £0.01, £0.10, £0.50, £1 or £250,000. If you wouldn't take the guaranteed £30,000 then you're gambling a lot of your money for potentially very little reward.

However - however! - it's not quite that simple. You're not just gambling a flat £30,000 against those five possible prizes, you're gambling it against what you might be offered at the two-box stage. If the £250,000 box is one of the three eliminated, then you're going to be choosing between two boxes both worth pennies, so your next offer will be pennies itself. However, if you eliminate three low-valued boxes, then your choice will be between £250,000 and pennies, so you can reasonably expect to receive an offer of at least £85,000 or so and probably one in six figures. The chance of eliminating three boxes and not getting rid of the £250,000 is the same as the chance of picking two boxes at random to keep and having one of those be the £250,000, so 2/5. So the question is not just "do you prefer £30,000 or a 1/5 chance of £250,000?", it's "do you prefer £30,000, a 2/5 chance of an unknown sum likely to be at least £85,000 or a 1/5 chance of £250,000?"

And that's a heck of a decision. That's a decision involving life-changing sums of money, a decision that it's interesting to see someone have to take. That's the reason why this show can be interesting to watch. If you want to think of it like this, you're not just gambling a guaranteed offer against the unknown value of your box, you're gambling it against the unknown value of future offers you might receive. If you manage to keep the high-value boxes until later offer stages, your offer values can be expected to rise; if you eliminate the high-valued box, then your offer value will collapse. You haven't lost everything, because you can still take the sum offered, but the sum offered will be so much less than last time that you can measure the extent of your loss.

Another way of looking at it is that if you want to win as much money as you can, you want your offers to go up until the point you accept one (if you do) and then you want your offers to go down afterwards. Every sum of money that gets eliminated will have an effect on the offer; very roughly, but reliably, eliminated prizes that are less than the rejected offer will raise the average value of the remaining prizes and so raise the value of the next offer, but this will usually be by a relatively small amount. Eliminated prizes that are more than the rejected offer will lower the average value of the remaining prizes and so lower the value of the next offer. If you eliminate a value that is much more than the value of the rejected offer - for instance, if you eliminate the £250,000 box - then the next offer will drop a long way.

Different people will choose to take their decisions in different ways; some people will care about the levels of offers that might later materialise, some people will not. Not every case will be so clear-cut, and it's the shades of grey that are the attraction to the game, but taking a guaranteed £30,000 rather than an unknown prize from £0.01, £0.10, £0.50, £1 or £250,000 will be seen as an excellent result by almost everyone. There's even an argument that if there's a bigger chance that you'll make more money by taking the fixed value than the unknown choice than you'll make less money, and you don't care how much more that more is or how much less that less is, then you should take any offer above the "half-way up" value - the median value, technically.

It's interesting how the range of possible values is shaped and much effect the presence of the £250,000 prize has. Regardless of whether it's in your box or not, if the £250,000 prize makes it to the final two boxes, you can expect to be offered £85,000 or more, which is a huge sum of money whatever you have in your box. If you can get the £250,000 to the last five, then you're looking at an offer of £30,000 or more, which is still a very handsome sum of money - and getting the £250,000 to the final five is not all that unlikely. (It'll happen 5 times in 22, or about 1 time in 4½ - about as often as you get a winning scratchcard from most scratchcard games.) Even if the £250,000 goes, getting the £100,000 to the final five should still see a bountiful reward, particularly if it's backed up by £50,000 or £75,000. So you're looking at "getting at least one of the big two values to the final five", which is not that unlikely an outcome. In fact, it'll happen almost half the time.

In this way, because the offers tend be roughly based on the averages (arithmetic means) of the possible values, the good values are more powerfully good than the bad values are powerfully bad. The effect of the presence of one of the big two in the final five is to guarantee a big offer (probably £20,000 plus), whereas the presence of one of the small two in the final five makes very little difference. I can't imagine there being much difference between the offers made for (one of £0.01, £0.10, £0.50, £1 or £250,000) and (one of £100, £250, £500, £750 or £250,000). When the small possibilities are compared to £30,000, £0.01 is moderately indistinguishable from £750. Or perhaps it's not, but there's the utility theory calculations again.

With this in mind, if you're really getting into the whole "utility theory" concept as it applies to this show, I recommend you to download mr_babbage's utility theory calculator (from Bother's Bar), plug in your own utility values for the various prize levels (if you can quantify them, which is a heck of an "if"!), knock out the values as they're eliminated on the show and work out what level of offer gives you a higher expected utility than leaving things to chance. This is actually an extremely rational way of doing things if you're seeking to maximise your utility from the prize you win; I think most people are tending to do that, but few people are sufficiently good at working out their own utility function that they can do so effectively. Or can they?

Lest we forget, the contestants are doing this all under the pressure of a game show taping, with a host, an audience and cameras aplenty all pointed at them. Anyone who can be rational under such circumstances is doing extremely well; the show and the decisions made on it are testament to the difficulty of the atmosphere. I played a version of the game organised by brigbother at his bar a few months ago (by slightly different rules, when we had just heard about this show overseas) and managed to turn down the top prize of £50 for about a quarter of that. My utility function started off as one thing, but as the game progressed over the course of several days, I had a bad day, wanted out and it turned into a completely different function altogether. That was just for a possible £50 in front of an audience of dozens (hundreds?) when the TV show is written on a scale five or so orders of magnitude higher.

Nevertheless, there are some statistics worth bearing in mind. While I firmly reject the hypothesis that the more money a show gives away, the more interesting the show is, I'm tempted to compare episodes of the same show to each other based on the stakes involved. Frankly, in Deal or No Deal's case, there's not an awful lot else to compare episodes by. (Not nothing, but not much.) Let's say that a show gets interesting when there's an offer of £10,000 or more. Sure, you can get blasé about what constitutes life-changing money in this day and age, particularly if you've watched a lot of Who Wants To Be A Millionaire?, but it's a good chunk of cash by any standards.

It's probably not unrealistic to assume that a five-digit offer will come about should £250k make it to the final eight (which happens about 1 time in 3), should £75k or more make it to the final five (which happens about 55% of the time) or should £35k or more make it to the final two (which happens about 41% of the time). There's a lot of double counting there, but 31 of the first 41 shows (from the stats at the Bar) had an offer of £10,000 or more somewhere along the line. £10,000 not enough to catch your attention? Almost half the shows have an offer of £20,000 or more somewhere along the line, and something like 20% of shows talk about a possible prize of over £50,000. This is a big-money show, especially considering that it's broadcast in a sleepy teatime slot.

Should you find yourself on the show - and good luck about that! - or should you want to enjoy playing along at home, here are some relatively simple statistics.

At the "five boxes remaining" stage, if there is one box whose value is above the offer, there is a 40% chance it will get to the "two boxes remaining" stage (and you can hope for an offer close to half this big box's value). If there are two boxes remaining whose values are above the offer, there is a 70% chance that at least one of them will get to the "two boxes remaining" stage (and you can hope for an offer close to half the second best box's value). Accordingly, if the second best box of the five is at least twice as good as the offer - see the two cases in the poll - then there's a 70% chance that one of them will get to the final two and give you a final offer which should be at least about as good as the one you turned down, quite possibly considerably better.

At the "eight boxes remaining" stage, if there is one box whose value is above the offer - say you have £250k and seven cheapoes - then there's a 62½% chance it will get to the "five boxes remaining" stage. If there are two boxes remaining whose values are above the offer, there is a 90% chance that at least one of them will get to the "five boxes remaining" stage.

(I should point out that I worked out those using a pencil jaydlewis sent me for Christmas. It is yellow and purple and has "is that your final answer?" written on it; accordingly it rocks out.)

There's a more complete Deal or No Deal stats guide here, which tells you the chance of eliminating 2 or 3 of the highest-valued remaining boxes in your next three pulls at any time, but it's possibly too detailed to be of use. The way that the offers play out, and the way that the earlier offers are so relatively poor, I think the distribution of endgames will be something like 10%-20% taking the box, 30%-40% taking the "two boxes left" offer, 40% taking the "five boxes left" offer, 9% taking the "eight boxes left" offer and maybe a very occasional episode with an offer taken earlier still.

Enough mathematics, but it's a rare show that has that much interesting mathematics to it. (And hence it's so rare that I go on about a show in quite so much detail.) What's the show like as a game show?

It doesn't work too well for me. In short, it's a 45-minute show with six real decisions, of which three are interesting. That's a lot of sloth, waiting and guff for very few high spots. Admittedly, those high spots can frequently be very good, but all told I can't find it a great way to spend my viewing time. It's a 45-minute show with two internal ad breaks and I contend that you can easily miss the first two-thirds of the show, because all the interesting gameplay will come in the last segment. When the last segment features decisions about five-figure sums, which is frequent (see above), then the decisions are enjoyable to think about and the last third of the show is fun to watch, to a greater or lesser extent. Whether you buy into the whole of the show as a show or not is another matter. I don't.

The atmosphere is a little strained. The set is ugly and warehouse-themed, for no clear reason other than to save money to divert into the voluminous prize budget. (I do like the theory that the emphasis and tone of the show changed from something overtly upbeat to something more sinister relatively late in the production.) The theme tune and opening sequence suggest a knockabout comedy show, but that doesn't hold true with the way the show goes. The estimable brigbother reckons the show is actually reasonably negative in entertainment, being about setting people's expectations high and knocking them down when the high-value boxes go, or people discovering that they dealt at the wrong time in an extended "look at what you could have won" sequence.

Fewer than half the contestants win the largest sum of money they could have won (i.e. take the last deal before the remaining values go down and their box is worth less than their deal) though about half either win as much as they can or come within 20% or so. It's also true that the show not only generates big winners but also big potential-losers and bittersweet conclusions. For instance, is taking £27,000 when you could have taken £63,000 a good result or a bad result? The shows where people end up with less than, say, the price of a good holiday are pretty clearly unhappy results when the show gives away so much so often. Then you get shows like this one.

There are lots of reasons why people watch game shows. Sometimes you get people who say they watch to see the contestants get easy questions wrong, or lose lots of money, or get humiliated, or worse. Certainly it's true that there are lots of shows where a great deal of the entertainment is fairly negative; arguably most of the biggest hits have had quite a lot of (at least) potential for negativity in among the positive entertainment of seeing winners, achievers, happiness and joy. A couple of off-LJ people I know have said they quite enjoy this show when I regard them as not being particular game show fans, which is unusual, and I suspect the show's negativity is quite a large attraction to them.

Host Noel Edmonds is generally agreed to be coming out of this show pretty well. Although he has clearly seen better days and his youthful pinchable-cheek charm is four long-running formats gone, he can still (slightly over-)act up the show and its prizes in a way that no other host on TV quite does through the media of pacing around the set and overemphasis on the importance of the money. Think of Chris Tarrant after six coffees and you're about right. For U.S. readers, I think the Monty Hall comparison probably rings reasonably faithful.

Noel does remain clearly supportive of the contestants, on the right side for the show and wanting to see big wins, and it only sometimes seems a little forced, there's unusually little to look at in this show and its tendency for close-ups and big reactions, particularly during the Banker-call set-pieces, are far more a factor of the format than Noel. While Noel is always going to be seen as a face rather than a heel (if you "weel") due to his children's TV pedigree, his history is sinister enough that you can imagine him saying to a contestant "I could have you slimed with just a click of my fingers. And your family too." just before the show starts, and everyone would know that he meant it merely by virtue of him being Noel Edmonds.

One thing that a lot of fans of the show get in a way that I don't is the sense of the journey and progression of the show; they get into the contestants and their rapport with Noel. Certainly it's unusual and I'm prepared to believe it's effective that we see the same contestants day in, day out, and a contestant might have been watching the game played any number of times (quite easily twenty or more) before they are selected to be the main player for their episode. All the contestants are on the same side; they all want to see everyone winning and the Banker losing. When there's been a clear win and the contestants celebrate with the main player at the end of the show, that's a big genuinely happy moment. Perhaps it's not just a case of watching a contestant's journey through one episode, you have to watch their whole journey through their involvement on every show.

I've plugged Brig's comparison of Deal Or No Deal around the world before, but it's fascinating reading. The hardcore fans compare how the different versions change the number of boxes that must be opened between offers; I must say that I think that a more triangular distribution of rejections (perhaps 5,5,4,3,2,1 rather than 5,3,3,3,3,3?) might work better as the start of the game really is quite predictable and the action is most exciting at the pointy end. People may be tempted to push their luck further knowing they only have to get rid of one or two boxes rather than three; one might expect to see more high prices and more low prices, whereas the British show has many deals clumped around "the middle" (say, £8,000-£20,000) compared to versions with other box distribution schemes. It wouldn't be expected to change the average level of the payouts, merely their variance.

The whole of Bother's Bar has built up a very strong Deal or No Deal fan community and the real-time bulletin board commentaries people provide while the show is in progress seem to work quite well. The community has built up a bewildering selection of shorthand, algebra and statistics, though I tend to believe they follow their own conventions rather too closely - for instance, the selection of "root mean square" value as somehow being fair is entirely arbitrary. I don't think there's a justification for it, though admittedly it's not completely unreasonable. I guess we all have our own different utility functions, but that one needs rather more justification to me than is given. Oh, the show also has a spurious premium rate phone-in viewers' competition, which nobody cares about.

The show has mostly been broadcast (outside Christmas) in a teatime slot, partnered with Countdown, akin to the old "Quiz Hour" arrangement which tied together half an hour of Countdown with half an hour of Fifteen-to-One. BBC 2 struck back over the years with 45-minute Ready Steady Cook and The Weakest Link and all of a sudden 45 minutes is the accepted length for a teatime game show, which (again) is something that has never felt right to me. The Weakest Link may have advertised £10,000 at teatime, though never paid anything like it, but Deal Or No Deal has blown away prize budget expectations in the slot to the same degree as Who Wants To Be A Millonaire? did in prime time. (Which show gives away the higher average prizes per contestant these days? Deal can't be too far behind.)

The show is doing tremendously well for Channel 4 - indeed, the show is regularly winning its time slot. Considering Channel 4 normally averages around a third of either BBC 1 or ITV, this is an amazing result. In fact, Deal has been the channel's top-rated show in at least one week to date, beating both Lost (yes, that Lost) and The Simpsons. Admittedly it's not the most competitive of timeslots to win and the huge-money game show is something unusual and highly distinctive to try in the slot. I'd be curious to know what sort of demographics the show is getting; while game shows usually skew old and female, I suspect this show skews far more male and left-brained than most. I wouldn't be surprised if it were also Channel 4's biggest "adult game show that draws lots of kids" since The Crystal Maze.

For me, though, it's a show I can respect somewhat bewilderedly rather than enjoy; while it's fun to think about, it's not essential viewing for me. In fact, I don't think I've watched it since the first week, though I have a nasty suspicion I would be sucked in to watching the rest of the episode if I were to get blindsided by it at about a quarter to five. All told, I think that adds up to four out of ten.
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[User Picture]
From:meggitymeg
Date:December 24th, 2005 03:49 am (UTC)
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I wouldn't be a very interesting contestant. *sheepish grin*

Also, you're a giant dork! *loves*
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From:zorac
Date:December 24th, 2005 05:02 am (UTC)
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Quick commentary on my answers:
  1. 5k and 8k seem fairly close, but 15k is much better than 8k, so a gamble seems good.
  2. With the fixed price so near the mean and the pain and gain so close, the gamble is exactly that and not a good deal.
  3. The fixed price is definitely a worse deal here, but again the pain and gain are close, so I'll take the fixed sum.
  4. Again, the expected winnings are rather better than the fixed price, and the 40% chance of a good or really good gain seem to make this gamble worthwhile (whether I'd actually take it in reality...)
  5. This time the offer is really paltry, but the 60% chance pain of losing 32-35k make the gamble just too unpalatable.
[User Picture]
From:bopeepsheep
Date:December 24th, 2005 08:35 am (UTC)
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That was a great (if long <g>) read - thank you. My dad's a huge fan of the show, and got me into watching it recently - the first show I saw was the 31K show you link to! How to get your audience hooked...

I will forward this write-up to him, if you don't mind, I think he'd enjoy it.
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From:jiggery_pokery
Date:December 24th, 2005 11:41 pm (UTC)
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Certainly!

Not really sure what the audience for that entry is, other than "my Friends list at large, because I wanted to write it". I can't imagine people who weren't positively disposed towards game shows being interested, and people who have seen and liked the show have a good appreciation of it already... :-)
From:quidditchmaster
Date:December 24th, 2005 08:58 am (UTC)
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I tend to believe that people define victory by avoiding losing.

Stalemating in chess is "not losing" yet is not a victory... unless you're playing for it while black... Hmm.

Anyway, In the US version, there exists 26 cases. The player chooses one, leaving 25 remaining, and then removes 6-5-4-3-2-1-1-1-1 until there exists one left on the board and one for the player. This makes it significantly better for the US players, probability-wise as they reach the endgame, particularly if the board still contains high level boxes.

Our highest five boxes are $1M, $750K, $500K, $400K, $300K; however, the median values are $750 and $1000. The show I watched, there were three boxes remaining on the board, and possible values were $x, $500, $50,000 and $500,000. I forgot what X was, but it was some low value. The banker offered the player $138,000 to run. The player only has to open -one- box at this point; therefore, a 75% chance to get an offer significantly higher than $138,000. The player refused, and promptly removed: *drumroll* $500K. Ooooooops. The banker offered something stupid like $1700, player refused, and opened $x. Banker offered $25,000 to run with possible $500 and $50K... player dealed, and made the right play. They seemed happy despite losing $113K in the process.

I think they only scheduled the show to run for a week. I hope they bring it back; it's a great show, and given the play I've seen, their prize budget will not be depleted any time soon.
[User Picture]
From:jiggery_pokery
Date:December 24th, 2005 11:45 pm (UTC)
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Good point about victory definition; "I tend to believe that people define victory by avoiding losing" definitely was intended to apply to this show only, though there are other situations where it applies, e.g. games where there are only losers.

The US box removal scheme makes lots of sense to me.

The US scheduling is certainly interesting; people were speculating that NBCU thought it was going to be a disaster and so tried to burn it off when ratings would be low and so it would, at least be a low-profile disaster. However, its strong performance indicates that there's enough to it that it might well be worth a longer run. Perhaps the "burn it off" hypothesis is flawed; in the UK, the debut (and first few series of) Who Wants To Be A Millionaire? was a similar sort of TV event, stripped across the schedule for ten days, leaving just as abruptly as it had started.
[User Picture]
From:sir_quirky_k
Date:December 24th, 2005 09:00 am (UTC)
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I've noted the failings of the root-mean-square 'fair deal' formula (which I myself suggested, in fact) at times. In particular, it seems to overplay the really low values, and can be very harsh in the two-box situation if the values are polar opposites (£25k or so for a £1/£100k decision). That said - I suspect most people, myself included, would consider that deal fair... (the Banker would apparently have offered £48k in that situation, but it occurred in the proveout so you've got to consider the Bullseye Factor there). I don't think we pretend to consider it much more than a benchmark figure, and one commentator will often say 'I think the board is worth £x', where x is a value that is independent of, though usually not dissimilar to, the root-mean-square value.

It's worth noting that the offers appear to have become rather more generous as the show has gone on, and contestants are becoming somewhat less risk-averse. Certainly things have changed since episode four when the player was offered £25k with a final five of £1, £100, £20k, £75k and £250k - and took it. On current form, anyone keeping £250k into the final five can expect to be recipients of a £45k offer. Does this have a knock-on effect on the players' willingness to gamble?

Worth noting: the offers on the US version of the show towards the end, with an offer after each case, have generally been right on the mean; utility theory only seems to come into play when one of the top three or four values is on the board. (See this spoilerific link.)
[User Picture]
From:jiggery_pokery
Date:December 24th, 2005 11:47 pm (UTC)
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Glad to see that there is some sort of feedback loop; I would be disappointed if the show hadn't picked up momentum as it had developed over the weeks, and suspect there may be further natural evolutions to take place. Perhaps you could convince the Brig to sanction a page where you explain all the shorthand used by the DoND posse?

BTW, Friended. Merry Christmas!
From:daweaver
Date:December 24th, 2005 10:19 am (UTC)

Crikey!

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Lots to get through here, well done.

1. Crazy frog of game shows? Or just the game show hosted by the crazy frog?

2. The Banker is acting, surely. Would viewers wish to pick a fight with Ross Kemp based purely on the actions of his on-screen character?

3. One useful thing that's come out of this show (and, in particular, the mr_babbage Utilit-o-meter) is that it's given me an opportunity to think about what various values of money mean. I've picked out four key figures, ranging from "This stops being a holiday and starts becoming serious" to "This will allow me to pay off my mortgage at once and only work to fill my days". In turn, that produces an ungainly quartic curve...

4. Which, I'm pleased to say, gives the answers I instinctively chose on the poll. Except for the last one, the quartic says "Deal at £34,205."

5. Children's game show Raven may have been a bit rubbish this year, but it was still better than Deal.

6. Someone may wish to try using root-mean-square, but taking the 0.5271th root, then raising the mean to the appropriate reciprocal. That gives an initial "fair" deal of £10,000.

7. In fairness to Lost, it was up agains the ratings juggernaut of I'm A Celeb, and regained its top position the following week.
[User Picture]
From:sir_quirky_k
Date:December 24th, 2005 01:03 pm (UTC)

Re: Crikey!

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1. A-ha - so that is where they get the prize money from.

2. Indeed. But the difference is that the Banker is affecting real people. That said, they know what they've put themselves in for, and if you think about it they have effectively been given a free holiday (albeit at a presumably-average hotel in Bristol) and thousands of pounds of free money on the condition that they gamble it.

6. I was indeed considering a twist to the root-mean-square formula. A logarithmic element perhaps? Though your idea of changing the root might be better and certainly easier.
From:daweaver
Date:December 24th, 2005 02:06 pm (UTC)

Re: Crikey!

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2. Good point, which implicitly raises another. Suppose one had to pay to go on Deal..., how much would one pay. I reckon the absolute maximum anyone would cough up would be their expected utility at the start of the game - for my Utility Quartic, that's a few pence shy of £12,800. But if I had that sort of money, would I gamble it on a 50-50 chance of winning more? Certainly not. £1000? Certainly. The line beyond which a person won't pay to play is an individual decision, and might usefully factor into their consideration of risk, and strategy.

6. Root-mean-square and derived functions will take us so far, but I really do think that the future lies in some sort of Generalised Utility Curve, along the lines of Trevor's work last month.

The element I think is missing from the discussion so far: some measure of how usefully distributed the holes are. Removing mostly from the bottom is trivially fabulous; removing mostly from the top is briefly painful but not so bad in the long run; taking things out from the middle leads to awkward decisions later.

Let me mull over some possibilities.
[User Picture]
From:jiggery_pokery
Date:December 25th, 2005 12:00 am (UTC)

Re: Crikey!

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2. Agreed. Given that Channel 4 have aired both Late Night Poker and Grand Slam, perhaps it's a route they might consider. (Or, more realistically, perhaps they might be interested in buying the GSN cash game poker show where players buy in with $100,000 of their own money at a time?)

Why didn't I use this icon for this post?
[User Picture]
From:jiggery_pokery
Date:December 24th, 2005 11:57 pm (UTC)

Re: Crikey!

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2. Excellent point.

I broadly tend to give the "they know what they've put themselves in for" argument fairly short shrift (not just in this case but generally when it comes to game show participation) on the grounds that no amount of watching the show and thinking about it can prepare the contestants for the experience and the emotions they'll feel. Additionally, Banker protocol can change; anything can change without notice, though contestants would rightfully feel aggrieved if the rules were significantly varied.

One thing I meant to say is that people are mooting a prime time version, presumably with a bigger budget. Personally, I can't see this as terribly likely; Channel 4's experience with Celebrity Countdown, where all the ingredients were in place for what looked like a nailed-on racing certainty, showed that teatime shows are thought of as teatime shows and people like to watch for the familiar routine as much as anything else.

Possibly the highest-utility outcome would be for the show to continue on Channel 4 at teatime and for there to be a prime time version on ITV as well, presumably heavily referencing and promoting the Channel 4 original, but it seems most unlikely that ITV and 4 could co-operate so much these days. A more likely outcome is that ITV commission a similar knock-off for prime time. I wouldn't be surprised if well-connected people weren't pitching such formats already...
[User Picture]
From:jiggery_pokery
Date:December 24th, 2005 11:50 pm (UTC)

Re: Crikey!

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3. Would it be rude of me to ask you to get your utility curve out?

5. Alas, UKNova is letting me down here. We can but hope that the BBC roll out their own licenced sharing and short-term retrieval solution wider than the 5,000-user beta test before long.

7. A good point well made.
[User Picture]
From:teppanyaki
Date:December 24th, 2005 01:12 pm (UTC)
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One thing that a lot of fans of the show get in a way that I don't is the sense of the journey and progression of the show; they get into the contestants and their rapport with Noel. Certainly it's unusual and I'm prepared to believe it's effective that we see the same contestants day in, day out, and a contestant might have been watching the game played any number of times (quite easily twenty or more) before they are selected to be the main player for their episode. All the contestants are on the same side; they all want to see everyone winning and the Banker losing. When there's been a clear win and the contestants celebrate with the main player at the end of the show, that's a big genuinely happy moment. Perhaps it's not just a case of watching a contestant's journey through one episode, you have to watch their whole journey through their involvement on every show.

This is what I thought, IMO, is the best part of the show. Trevor, IIRC, was *so* helpful to the other contestants that you wanted him to get the 250K -- and that is what would keep me glued if NBCUni decides to go syndicated. It's a reversal from all the other reality shows where it's a competition.
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From:jiggery_pokery
Date:December 25th, 2005 12:03 am (UTC)
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Sounds very insightful to me.

I'm not sure that syndication would necessarily work, as different markets might be at different points during the run and syndicated shows are famously frequently shown out of order. That said, it could well be extremely well-suited for a network-wide daily show, should NBCU be prepared to consider having a network-wide daily game show.
[User Picture]
From:brigbother
Date:December 24th, 2005 02:29 pm (UTC)
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Excellent read Chris, thanks!

I don't think The Bar has done too badly from Deal or No Deal, which is quite good considering everything came about pretty much by accident, although everyone who works on the commentaries does a tremendous job.

I'm still of the opinion that the banker has to underpay the board generally not out of anti-socialism spirit but otherwise you wouldn't have a show because it would be a no brainer to take the deal all the time. Or rather you would ahve a show, but it would be Cilla Black's Surprise Surprise. Deal or No Deal without agonizing decisions isn't an awful lot really.
[User Picture]
From:jiggery_pokery
Date:December 25th, 2005 12:05 am (UTC)
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Glad you enjoyed it! It's been a while since I've written something like that, hasn't it?

Cilla Black's Surprise Surprise was a successful and popular show in its time, as was Jim'll Fix It. You wouldn't want to do either of those in the DoND style, though, would you?
[User Picture]
From:oinomel71
Date:December 25th, 2005 01:10 am (UTC)
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Ururur now then, as it 'appens we 'ave 'ere a letter from Mandy from Dagenham.

"Dear Jim,

Could you fix it for me to meet Robbie Williams!

Ta ever so,

Love

Mandy (11)"

Well, young Mandy, one of these two boxes contains a ticket to a Robbie Williams concert - the other one contains Chas and Dave. Remember, I've no idea which one you have in front of you.

*ringring*

Hello. Right... OK, thank you.

Well, the Producer's offer is Ainslie from Fame Academy 1. Mandy... Fix or No Fix?


Good post and some good threads - you do indeed cover topics like this remarkably well. Merry Christmas!
[User Picture]
From:jiggery_pokery
Date:December 25th, 2005 01:38 am (UTC)
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That's the ITV knock-off I suggested will probably end up existing above. No, seriously plausibly!

Merry Christmas, yourself!
[User Picture]
From:amuzulo
Date:December 24th, 2005 07:56 pm (UTC)
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Wow, I can't believe I read all that. :)
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From:jiggery_pokery
Date:December 25th, 2005 12:06 am (UTC)
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You've read longer things (Friends-only) in this journal in the past, though... ;-)

I've been browsing old entries of mine (and dezzikitty's!) tonight - I thought this wasn't the longest thing I'd written (and I know angiej has written longer in her LJ in the past, too) but couldn't remember the details.
[User Picture]
From:bateleur
Date:December 26th, 2005 10:56 am (UTC)
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Personally I don't think it's possible to come up with a formula for evaluating the objective worth of one-or-the-other options. The reason is that when considering my own preferences it takes a long time before the higher of the two figures matters at all. This is because the utility of these sums of money varies with what category of expenditure they would be useful for. Something like £35000 may seem like a lot of money, but the things I could get with it are still in the same category at £5000 - it wouldn't really make any difference to my life. However, any 4 figure sum would be enough to treat myself and my friends to a few things, so I'd significantly prefer that to £10. Expected values aren't relevant since I don't get presented with this kind of deal frequently enough for it to matter.
[User Picture]
From:sir_quirky_k
Date:December 26th, 2005 11:15 am (UTC)
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That's an interesting point as well. (And fits in with Noel's 'on the other hand, how many times in your life will you be offered £x,000 tax-free?' catchphrase.)

Ultimately it is individual perspectives of utility that matter in this game, though how much these can be generalised is a very interesting moot point. This show is particularly interesting to analyse in this context...
From:banningy217
Date:January 25th, 2007 07:10 am (UTC)
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In the Philippines, which is from where I am, Deal or No Deal is a very popular game show because it has a unique format. This has a different format from the U.K. version, but follows the same format to that of the American version whereby briefcases are carried and will be opened by gorgeous models and not boxes opened by the other contestants.

The contestant chooses a briefcase which will be set aside and remain unopened up until the game’s end, when the contestant says “DEAL”. Then the contestant will begin to open the rest of the cases one by one to reveal different amounts from 1 peso up to 2 million pesos.

Based on the opened briefcases, the contestant can estimate the odds. After every round of opening the briefcases, the banker offers the contestant a specific amount of money and lets the contestant choose: DEAL means he gets the cash and the game ends; NO DEAL means he will continue with the game and will not accept the cash.

I find the show very entertaining and although there is not much thinking involved, because the player much relies on his luck, it is an exciting show to watch – You’ll never know what number will be revealed, and of course the fun part is watching the game show with your family or friends and discussing what you would do when you are the contestant – take the offer (DEAL) or go on and take your chance on the other unopened cases (NO DEAL)?

Banning Yao
http://www.myfindmorgateloan.info


[User Picture]
From:lenno_cornish
Date:October 29th, 2007 07:53 am (UTC)
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In fact I don't think that I may have any sligh wish to spend my time watching this:)

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