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Deal or No Deal

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File:Miljoenenjacht1.jpg
Linda de Mol hosts the original version of Deal or No Deal entitled "Miljoenenjacht"

Deal or No Deal is a television game show format owned by Dutch-based production company Endemol, known for creating such shows as Big Brother and Fear Factor.

Format

The basic format of Deal or No Deal consists of a number of cases (usually 26, but varies in some countries) each containing a different amount of money. Not knowing the sum of money in each case, the contestant picks one case which potentially contains the contestant's prize. They then open the remaining cases, one by one, revealing the money they contained. At predetermined intervals the contestant receives an offer from the bank (run by "The Banker") 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 cases. If the contestant decides not to take the deal and reveals low value cases, then the next bank offer is likely to be higher (as the contestant's case is proven not to contain these low values). Alternatively, there is risk in revealing higher values, lowering future offers from the bank. The aim of this system is to try to make an exciting and suspenseful game.

The format of Deal or No Deal varies in each country. In the UK version, for example, contestants choose from 22 boxes rather than 26 briefcases. The concept of pitting a contestant against an in-house adversary (in this show, the Banker) is unusual, though not unique, among game shows.

National versions

  Countries which have their own version of the show
  Countries which, as a group, have their own version of the show

The show has several versions that air around the world:

Optimal strategy: when to deal

Deal or No Deal is a repeated exercise in decision theory -- the player must decide whether he/she values the banker's offer more than he/she does the possible prizes. The expected winning at any point in the game is the mean of the values contained in the unopened boxes. This expected value of the player's position is what he/she should expect to win if he/she refuses all offers, or what to expect if he/she were to open his/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 mean 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 expected value, "no deal" seems to be statistically a better choice. In fact the only time the banker's deal is above the expected value is when he offers retrospective deals to players who have already accepted one.

However, maximizing expected return is only optimal if the player can replay the game many times. Since each player only plays once, he/she must balance his/her tolerance for risk with the expected winnings. This is analogous to modern portfolio theory, where a trade-off exists between higher expected 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 the difference between winning $1 and $250,001, 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, family or friends.

Will someone win the top prize?

It is interesting to consider the likelihood that a contestant will actually win the top prize in a case. In the following discussion, "$1,000,000" represents the top prize value (as is the case in the United States), and the number of boxes in the probability model is equal to 26 (as in the US and also the original Dutch version; this number varies depending on the location of the show).

In order to win $1,000,000, one must

  • choose the right case, and
  • must never accept a deal.

A contestant chooses the right case with probability 1/26. However, deals near the end of the game usually

  • are closer than usual, but still less than the expected value of the remaining cases, but
  • accurately account for the contestant's risk tolerance in determining the difference between the deal and expected value of the remaining cases.

The following example illustrates the need for the banker to assess the contestant's risk tolerance. Imagine if there are two cases left, one with $1,000 and one with $1,000,000 (i.e. high variance). The banker is contemplating how much to deal. Assume the contestant is rational. If the contestant is then, for example, a billionaire, a $400,000 deal would probably not be accepted as it is far below the mean value of $500,500. That same $400,000 deal would be very attractive, however, to a contestant living in poverty.

Modelling the outcome

To calculate the probability that the top prize will actually be reached, the following simplifying assumptions can be made:

  • Deals are most often accepted when there are two cases left. This is when the deal value is closest to the expected value of the remaining cases.
  • Since the variance of a $1,000,000-or-nothing deal is so high, virtually all contestants would accept a carefully constructed banker's deal in that situation. That means the top prize will be hit only when the two remaining boxes are of high value (say, $200,000 or more)

The likelihood, then, that we have a top prize winner is:

(the number of different two-case combinations with one $1,000,000 and the other $200,000 or more) /

(the total number of different two-case combinations) *

(the probability of picking the correct case of the two left)

Based on the assumption that the contestant will choose to take the banker's offer when there is one case with $1,000,000 and one case with an amount less than $200,000: At 1.5 contestants per episode, this would mean that $1,000,000 prize would be awarded about once every 87 episodes. However, since several contestants have taken deals earlier in the game with the $1,000,000 prize and another amount of $200,000 or more still on the board, these odds are likely too high.

  • The only time the $1,000,000 prize was ever chosen in the U.S. version was by LaKissa Bright, who ended up accepting a bank offer of $215,000, after removing the third highest case, $500,000. Thorpe Shoenle holds the record for most money won on the American version, with $464,000.

This assumption amount set at $200,000 is only applicable to the U.S. version and is quite arbitrary. Similar arbitrary assumptions could be made based on the game boards of other versions. In addition, the assumption that humans exhibit fully rational thinking is rarely true, especially under stressful circumstances like in a game show. The contestant can also be a fame seeker wanting to be a $1 million dollar winner, possibly leading to his taking the additional risk.

Actual simulation results and conclusions

Statistical analysis through a simplified model yielded a pattern that revealed something some might hold counter-intuitive: there is no net deviation of the average winning from the initial mean (expected winning) in the game. In our models, a significant assumption is that the player would deal with any offer that was above some fairly low preset threshold above the expected winnings (the reason for this strategy being that the program was designed to show that it is possible to have an average win above the modest expected win so the virtual player would not be too greedy).

This result may be a bit strange since on first analysis, one would reason that given multiple opportunities, a player would be given extra information in some sense and thus be given an advantage in some sense. To see this, we only need to show that the percentage of time where an offer is made at some time is quite high (>90%); this is quite easy to see since the chances are that after so many offers, it is quite unlikely that somehow all the offers would lie below the average (expected average of all the suitcase values). Given the fact that the cases are distributed with only a few high valued ones actually seem to support the advantage since now there is less chance that a player may open a high valued case during the course of the game (the fact that this would also lower the chance that the player's own case is high valued is mostly irrelevant since the player doesn't know his case value and in most cases, both in actuality and wisely enough, doesn't keep his case) and thereby jeopardize the offers given afterwards. In fact, simulations show this statistic to be true but the overall net winning still only equals the mean, and doesn't deviate above by any statistical significance. How does this happen? In the few times that during the entire game, no offer is made above the expected winning (average of all the suit case values), the player kept his case and found it to be on average far below the mean: so far below the mean that although this happens rarely, the overall average win is pulled back down again to the expected winning. We can explain this by realizing that because there are so few high values, when somehow the player opens one of the higher valued ones (> 100,000), he seriously lowers all of his future offers so that he basically cannot recover again no matter how wisely he deals afterwards (just consider what happens if he should open the million dollar case on the first try).

Of course, this result is precisely a conclusion that the producers want to hide from the viewers, as the purpose of skewing the distribution and giving multiple offers is to provide the illusion of an advantage thereby inducing a false sense of control and strategy as well as encouraging players to aim for abnormally high offers that only end up creating bad deals, longer game play, and lower-than-ideal actual winnings. Thus the distribution of the cases has both advantages and disadvantages that miraculously balance each other out perfectly. With further simulation, it can even be shown that this conclusion is independent of the value distribution.

A possible conclusion from this could be that no decision/deal algorithm no matter how clever will be able to successfully generate wins above the expected winning in the long run, independent of how the case values are distributed. Even if the case values are evenly distributed or skewed toward higher instead of lower values, the distribution's advantages and disadvantages actually "negate" themselves and result in no net advantage or disadvantage, i.e. no average win above the expected winning. In fact, it has indeed been proven mathematically that the optimal decision strategy cannot yield a win above the expected mean as long as the banker chooses not to give offers higher than the expected mean (which is only logical since the role of the banker is to minimize the earnings of the contestants; only on extremely rare occasions during the game, when the contestant has already opened a vast majority of the cases, has the banker been observed giving offers that are higher than the mean of the remaining cases). The proof for this assertion as well as the actual results of the various programs used in these simulations can be found here.

Comparison with the Monty Hall problem

When only two cases remain, Deal or No Deal might seem like a version of the Monty Hall problem. Consider a game with three cases (similar to the three doors in the Monty Hall problem). The player chooses one case. Then, the host chooses a case to open. Finally, the player is given the option to trade his or her case for the one unopened case remaining. The Monty Hall problem gives the player a 2/3 chance of winning with a switch and a 1/3 chance of winning by keeping his or her case. However, statistical testing has shown that there is no advantage in switching in the Deal or No Deal situation. The player has a 50/50 chance of increasing his winnings by either switching or keeping his case.

The Monty Hall problem gives the player an advantage only because the host always eliminates a losing door from the game. In other words, there is no randomness to the door exposed. It is purely based on the player's first choice. In the Deal or No Deal situation, the player chooses one case and then exposes another. The one exposed may very well be the winning case, leaving two lesser cases still in play. Therefore, there is no relation between the player's first choice and the case that is exposed. That breaks the Monty Hall advantage in switching.

Another way of seeing this is to consider that each opened case gives the player, in a sense, some information about his case value and allows him to make a more accurate judgement (and thereby optimize the value of the offers he accepts). In the Monty Hall problem, the opened cases can only be the wrong cases since there is only 1 right case. But in Deal or No Deal the opened cases can be disastrous if the player opens a "right" case (that is, a high value case that, once opened, irrevocably significantly lowers all future offers). Thus, the difference between the two is that the Monty Hall event gives only advantage to the player through information, while in Deal or No Deal, the advantageous information is accompanied by the possibility of eliminating favorable cases. In fact, from the section on experimental modeling, we can see that in Deal or No Deal there is no net advantage in the long run at all because the disadvantages and advantages cancel each other out exactly.

To further clarify the difference between the two problems, we need only increase the scale of the problem. With the Monty Hall Problem: Imagine there were a million doors and the contestant chose one. Then the host opened all but one of the rest of the doors, showing that either the player's door or the remaining door had the prize in, the contestant would inevitably chose the other door, knowing it almost certainly has the prize in. This can be applied to the switching at the end of Deal or no Deal. Imagine there were a million cases, and the player managed to play the game right through to the end, with a winning amount left, and a losing amount left. The difference here is that the player has achieved this place in the game through pure luck (as the previous banker's offers in themselves convey no further information since the offer values are loosely calculated based on the mean of the unopened case values, which the player can already figure out from knowing which cases have been opened already), and so the player's case and the remaining case have the same probability of holding the winning amount of money. Compared to the Monty Hall situation, where the host has eliminated all of the losing doors, here the contestant has by extreme chance managed to get through all the cases with the winning amount and one other amount left over.

Analyzing decision making under risk

A team of economists - Post, Van den Assem, Baltussen & Thaler (report) - 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. A separate experimental study (report) with student-subjects playing the game with scaled down prizes reveals a similar pattern. 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 four economists 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.

This particular study recently attracted some media attention in the US, including coverage on the front page of Wall Street Journal, January 12 2006, and National Public Radio, March 3 2006.

Antecedents

  • The New Treasure Hunt -- 1970s Chuck Barris game show similar in concept to Deal or No Deal
  • The Bong Game, invented by the London-based radio station Capital FM in the 1980s, also tested contestants by offering them increasing returns in tandem with increasing risk.

Official websites

Other websites

  • Deal or No Deal: How to play smart, scenarios with odds from the UK version
  • Deal or No Deal Official UK website with online games
  • Deal or No Deal UK game summaries/Stats/Games/Forum
  • "Deal or No Deal", an essay about the show by Said Shirazi.
  • NEW Deal Or No Deal Fanclub Forum/stats and Live Chatroom
  • Blog for fans of the UK show
  • For all the fans who love Deal or No Deal
  • Deal or No Deal Forum, Gallery, News, Blog, Video and Chat
  • Post, Van den Assem, Baltussen, and Thaler (December 2004). "Deal or No Deal? Decision Making Under Risk in a Large-payoff Game Show". {{cite journal}}: Cite journal requires |journal= (help)CS1 maint: multiple names: authors list (link)
  • Deal or No Deal Calculation
  • Deal or No Deal show tapings Free tickets to Deal or No Deal tapings
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