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Deal or No Deal

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File:DOND US.jpg
Early logo of the American version of Deal or No Deal

Deal or No Deal is a television game show format owned by Netherlands-based production company Endemol, known for creating such shows as Big Brother. The format first aired in Australia in 2003 but has since spread internationally, now airing in 38 countries, including the United Kingdom, the United States, Mexico, Canada, Italy, Israel, Tunisia, Romania, Poland, India and Germany.

Format

The basic format of Deal or No Deal consists of a number of cases (usually 26) each containing a different amount of money. Not knowing the sum of money in each case, the contestant picks one case at random which potentially contains the contestant's prize. They then open the remaining cases, one by one, revealing the money they contained. At pre-determined intervals the contestant receives an offer from the bank to purchase the originally chosen case from the contestant, the offer being based on the potential value of the contestant's case. The contestant must then decide whether to take the deal from the bank, or to continue opening briefcases. If they decide not to take the deal and reveal low values then the next bank offer is likely to be higher (as the contestant's case is proven not to contain these low values). Alternatively they risk revealing higher values, lowering future offers from the bank.

The format of Deal or No Deal varies in each country. In the UK, for example, contestants choose from 22 boxes rather than 26 briefcases.

International versions

Australian version

File:Deal or No Deal Cases.jpg
The Australian version of Deal or No Deal
Main article: Deal or No Deal (Australia)

Deal or No Deal airs in Australia on the Seven Network, and is hosted by Andrew O'Keefe, with special appearances by Gareth Nixon as the man at bar. The program debuted in late 2003 as an hour-long program, airing in prime-time on Sunday nights and offering a top prize of $2,000,000. In 2004 the show was reduced to a 30-minute format, airing weekdays at 5:30pm and offering a top prize of $200,000. As of January 2006 the top prize has only been won once – by sales consultant Dean Cartechini.

As of February 2006, a slightly new format, "Double or Nothing", was introduced. After revealing the amount in the case initially chosen by the contestant, he/she must then decide how much of their winnings they would like to gamble. A giant case bearing a "?" is wheeled out by two models and is opened to reveal the word "Double" or the word "Nothing". If "Double" is revealed the amount the contested chose to gamble is doubled and added to his/her initial winnings. If "Nothing", then that amount is removed from the winnings. For example, say a contestant wins $50,000 and bets $10,000 of it. If "Double", he wins $60,000, and if "Nothing", he wins $40,000. A week after the new format came about, history was made, when a contestant bet all his money he had won ($2) and the case revealed "NOTHING", he then received a novelty check with the word "NOTHING" written on it.

A special celebrity version of the show "Dancing with the Deals" ran from the 13th to the 24th of February 2006. The show introduced the TV audience to the stars of the upcoming series of Dancing with the Stars, which is shown on the same network. Each celebrity took turns on each weekday playing for a charity as well as for home viewers. One notable difference was the absence of the Double or Nothing feature, introduced earlier on in the year.

As of 23 February 2006, The CAR has been won 3 times. The latest person who has won it is Russian Born Boxer Kosta Tszyu, he had 3 cases left $5'000 & CAR ($33'000), he was offered a deal of $12'000, he said no deal and when he opened up his suitcase he had won the CAR.

Dutch version

The Netherlands version, called Miljoenenjacht, is hosted by ex-Endemol boss John De Mol's sister, actress Linda. The top prize is 5 million euros.

French Version

The French version, called "À prendre ou à laisser" (which roughly translates to "take it or leave it"), premiered in January 2004 on TF1, and currently is shown every weeknight at 19:05. It is hosted by "Arthur", the vice-president of Endemol-France and popular radio presenter. The 22 boxes represent the 22 regions of France, with each contestant coming from that region. The prizes range from 0,01€ to 500 000€; and include (normally) three "joke" prizes (eg. a cup or a coathanger) and a "joker", containing an amount determined by the number of contestants who answer the "selection-question" correctly. Other factors particular to the French version is that the prize is shared with a viewer who has phoned in to enter the competition; and offers of an "échange" (change of box) are fairly frequent.

Indian version

The Indian version was created by Thakkar and Masrani Associates as Deal Ya No Deal. It premiered on November 23, 2005 on Sony Entertainment Television and broadcasts three nights a week. The first host, Madhavan, quit after fulfilling his 35-episode contract and was replaced by Mandira Bedi [1]. There is a top prize of 1 crore rupees and a gag prize of a stick of Chlormint gum.

Israeli version

In Israel the show is hosted by Moran Atias, a model, actress and Miss Israel winner. The top prize is 1.5 million new shekels. It airs on the Israel 10 channel. In the Israeli version there is a round of trivia questions among ten audience members before a contestant is chosen. The ten audience members are divided into two groups of five. Each group is asked a question, and the first member of the group who answers the question correctly moves to the second round. In the second round, both players are asked trivia questions. The first to get two questions right, becomes a contestant. If a player gets two questions wrong, he or she is automatically disqualified, and the other player becomes the contestant.

Italian version

In Italy, the show is a big hit on leading TV station RAI. The show is called Affari Tuoi, which literally translates to "Your Affairs", and is now hosted by former singer Pupo after two hugely successful seasons with Paolo Bonolis. In the Italian version, there are only 20 briefcases, and each person that has a briefcase is a representative of one of the 20 regions of Italy. The highest prize is 500,000 euros. Along with some petty money prizes like 50 cents, there are also 3 prizes that you want to avoid. They are usually silly things like Salami, or a year's supply of soap, and even a live performance of the host's most famous song, "Su Di Noi".

Mexican version

In Mexico, the show is called "Vas o no vas". The original version is transmitted on Saturday nights and a spinoff version is shown during weekday evenings. While the weekend version shows a more solemn environment (with a top prize of 5 million pesos), the weekday version seems more informal (with a top prize of 1 million pesos). The show is broadcasted by Televisa and hosted by Héctor Sandarti.

Polish version

In Poland, the show is called "Grasz czy nie grasz" and is shown at 7.00pm on Saturdays on Polsat. The top prize is 1,000,000 zlotys. The show is hosted by Zygmunt Chajzer.

UK version

Main article: Deal or No Deal (UK)

Channel 4 in the UK began broadcasting Deal or No Deal from October 31, 2005. It is presented by Noel Edmonds and normally airs from 4.15pm to 5.00pm on weekdays and in the early evenings on Saturdays. On March 4 2006 the Saturday edition was moved to 7.10pm due to the show's ratings success. The weekday show is repeated on Channel 4's digital sister station, More 4 at 6.10pm. The show saw the return of Noel Edmonds after some years away from television. The grand prize is £250,000, and as of early March 2006 has yet to be won. However, two players have been unfortunate enough to win the lowest available prize, 1p. The first of these was Nick Bain, who was joined on March 1, 2006, by Trevor Bruce, who turned down the same offer of £9,900 twice and retained the £250,000 in play until the last box of the last round. Four players in the first 100 shows have had the £250,000 box (one in the 100th itself), although unfortunately, they all dealt earlier on in the game.

US version

Main article: Deal or No Deal (USA)

The United States version, hosted by actor-comedian Howie Mandel, premiered on December 19, 2005 on NBC and ran during the following four consecutive nights. The grand prize is $1 million.

NBC's sister business network CNBC aired episodes of the premiere week of Deal or No Deal starting on December 26, 2005, scoring above-average ratings for the network. The show as aired on CNBC was blacked out in Canada due to programming rights issues and Canadians watched CNBC World programming instead.

Canadian broadcast network Global landed the Canadian rights to the show, and aired the original December 2005 run concurrently with NBC.

NBC brought the show back after the network's coverage of the 2006 Winter Olympics. The show broadcasted each night from February 27 through March 3. It will settle into a regular weekly slot every Monday and Friday at 8:00 PM Eastern/Pacific, starting March 6.

For this second week-long special, the grand prize started at $1,000,000 on Monday, with the top prize increasing by $500,000 each night (i.e., $1,500,000 on Tuesday; $2,000,000 on Wednesday; $2,500,000 on Thursday; and $3,000,000 on Friday); the increased prize was indicated by a red border and a silver background. Furthermore, the $750,000 prize was increased to $1,000,000.

The new maximums only apply to new games, not games in progress.

Mathematical Basis

Optimal Strategy -- When to Deal

Deal or No Deal is a repeated exercise in decision theory -- the player must decide whether he values the banker's offer more than he does the possible prizes. The expected winning at any point in the game is the average of the unopened boxes. This expected value of the player's position is what she should expect to win if she refuses all offers, or what to expect if she were to open her own briefcase. As the game progresses, the possible content -- or expected value -- of the chosen briefcase changes. For example, if the top prize is revealed, then the expected value goes down.

To maximize their expected return without regard to risk, a player's optimal strategy is to reject the banker's offer if it is below the average value of the unrevealed cases. Typically, the banker's offer undervalues the remaining cases significantly at the start of a game. The gap closes later on, giving the player a strong motivation to continue to play early in the game, but less so as the game progresses. Because the banker's deals are almost always lower than the expectation value, "no deal" seems to be statistically a better choice.

However, maximizing expected return is only optimal if the player can replay the game many times. Since each player only plays once, he must balance his tolerance for risk with the expected winnings. This is analogous to modern portfolio theory, where a tradeoff exists between higher average return and investment risk (variously quantified as spread, variance, standard deviation or beta coefficient). A safe investment usually offers a lower average return than a speculative investment. The optimal strategy depends then on the investor's risk aversion.

In addition, optimal strategy depends on the utility or satisfaction of the various amounts of money to the player. For example, most people would agree that the difference between winning $750,000 and $1,000,000 is much less than that between $250,001 and $1, even though the dollar difference is the same. The utility curve for money for a player is non-linear and is subject to the law of diminishing returns. A player who badly needs money may be willing to "cash out" for a bank offer of $50,000 and forgo a chance at $1 million, an opportunity that might fetch him little at the end.

Actual play, as opposed to optimal or "best" play, is also influenced by psychological factors, including perceived luck, fear of embarrassment, and pressure from the audience or friends.

Will We Get To See Someone Win a Million Dollars?

It is interesting to consider: will someone win a million dollars (or whatever is the maximum amount in a case)? In order to win a million dollars, certainly one must be lucky enough to choose the right case, but he must also never accept a 'deal'. But, even if they choose the million-dollar case, most people will not ever win the million unless the second to last case is also a very large amount, since this is the only condition in which a player will typically say 'no deal' all the way through to the last case. Imagine if there are two cases left, one with $1,000 and one with $1,000,000, and the banker offering a deal of $500,000. While certainly wild risk takers might go for it, most people could not tolerate the risk of only going home with $1,000. Alternately, if the contestant doesn't choose the $1 million dollar case initially, but instead picks a high value case ($250,000-$750,000) and then in the process of eliminating the unchosen cases, leaves the $1,000,000 case as the final case.

If the last two cases are $200,000 and $1,000,000, with a deal of $600,000 offered, perhaps many people would say "no deal" and go for it. However, if the contestant were to imagine that deal outside the context of the game they would not go for it: Imagine you have taken the $600,000 banker's offer and gone home. It is sitting safely in your bank account. Then you receive an offer to take a gamble with your money. You are asked to bet $400,000 of your own money on the toss of a coin at even odds. This is the same fair bet as offered in the game, but it's difficult to imagine anyone accepting it. To make the correct decisions a contestant must always think outside the flow of the game.

Certainly if the two cases are $750,000 and $1,000,000, most would go for it. Thus, except for the case of extreme risk takers, we will expect to see someone win a million dollars if (1) they initially pick the million-dollar case (since after repeated 'no deals' they ultimately take home the case they first selected), and (2) the second to last case is in the range of $200,000 to $750,000. The probability of this occurring is (1/26)(5/25), or one in 130. At two contestants per episode, this would mean that the million dollar prize would be awarded about once every 65 episodes.

In the endgame, when the two largest amounts remaining on the board are close to each other in value, this usually leads to a player continuing to open cases one by one since the risk is low. Conversely, having one isolated large value separated from much smaller values gives very high risk and leads to very difficult decisions and more dramatic games.

With some psychological differences, the same analysis also applies if the player does not initially select the million dollar case. The player would simply have to be lucky enough not to eliminate the $1,000,000 case until the very end, reject all bank offers, and choose to switch cases.

In the versions with a higher than $1,000,000 top prize and a $1,000,000 second prize, the odds of winning a million are, of course, much higher. In one episode, an offered deal (with the $75 and $2,000,000 cases remaining) was $1,000,000 (the deal was only hypothetical, as the contestant had already accepted an actual deal). It also brings up the possible, yet extremely unlikely, scenario of the last two cases being $1,000,000 and some higher number.

Comparison with the Monty Hall Problem

When only two cases remain, Deal or No Deal might appear to be a version of the Monty Hall problem. This might seem to be true because there were (say) 22 cases, and suppose 20 of them have shown to not contain the major prize. At the start of the game, the probability that the player did not have the top prize was 21/22 -- if that probability still held true, then a player would almost surely win the top prize by swapping the originally chosen case with the other one remaining.

The reason why Deal or No Deal isn't Monty Hall is that the cases to be opened in Deal or No Deal are chosen at random (by an uninformed player). In the Monty Hall gameshow, the host must know the location of the top prize and must actively reveal a door that does not contain the top prize. Thus in the Monty Hall problem, the host is forced to select non-randomly and thereby give information to the player. Contrast this with Deal or No Deal where the player chooses the cases, and does so essentially at random. This subtle difference changes the conditional probability completely. In fact, for Deal or No Deal with two cases remaining, there is no advantage to swapping the cases - it is equally likely that the top prize is in either of the two remaining cases. It is for this very reason that the Banker can offer the player the chance to swap cases because he knows there is no statistical advantage in doing so. By contrast, a contestant repeatedly playing a 22-briefcase Monty Hall game will win the top prize more than 95% of the time by switching to the last unopened briefcase!

In Deal or No Deal, it is unlikely that 20 cases will be opened without revealing the top prize. In the rare cases where this happens, it is equally likely to be the player's initially chosen case or the last unopened case -- the probability of each outcome is 1 in 22.

Indeed, the Monty Hall problem may be usefully explained in terms of how it differs from Deal or No Deal. The reason that the Monty Hall problem produces an apparently counterintuitive solution is that it appears to work in the same way as Deal or No Deal, but is altered by the introduction of a non-random element.

Analyzing decision making under risk

A team of economists - Post, Van den Assem, Baltussen & Thaler - have analyzed the decisions of people appearing in Deal or No Deal and found, among other things, that contestants are less risk averse when they have seen their expected winnings tumble. "Losers" tend to continue playing the game even if this means rejecting bank offers in excess of the average of the remaining prizes. The findings provide support for behavioral economists, who claim that the classical expected utility theory falls short in explaining human behavior by not accounting for the context of decisions. The study of the Dutch is unique, for the underlying "experiment" Deal or No Deal is characterized by high stakes, a transparent probability distribution and only simple stop-go decisions that require minimal skill or strategy. A recent research report of their study can be downloaded here.

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