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Game Theory of Deal or No Deal
By
Luke Berndt
November 24, 2014
Math 89S Game Theory and Democracy
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Introduction
NBC’s Deal or No Deal was a hit game show from 2005 to 2010 known for the
excitement of the contestants wins and loses, the host Howie Mandel’s antics, and the models
holding the cases. Each night the show aired, a contestant chosen from the crowd was confronted
with 26 sealed briefcases full of varying amounts of cash. The briefcases’ values ranged from
$0.01 to $1,000,000. Without knowing the amount in each briefcase, the contestant picks one
briefcase to keep, if they choose, until the unsealing of it at the end of the game (1).
The contestant must then instinctively eliminate the remaining 25 cases and try to win as
much money as they can. The cases are opened and the amount of cash inside is revealed and
eliminated for the contestant to win (1). In the first round six cases are selected, in the second
five are selected, in the third four are selected, in the fourth three are selected, in the fifth two are
selected, and for the six remaining rounds only one case is selected by the contestant. After a
pre-determined number of cases are opened, the participant is tempted by an unknown and
unseen "Banker" to accept an offer of cash in exchange for what might be contained in the
contestant's chosen briefcase. It is at this time that Mandel asks the both famous and important
question: Deal or No Deal? (1).
As each case is opened, the likelihood of the contestant having a valuable cash amount in
his or her own case decreases or increases. The Banker’s offer is also influenced by what cases
are remaining and which ones are gone. As long as the larger cash prizes haven't been opened,
the Banker's deals will only get higher. However, if the conflicted contestant accidentally opens
a case with a bigger cash value, the Banker's offer could be retracted and be replaced with a
lesser one (1). The game either ends with the contestant accepting the Banker’s offer or
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continuously choosing “No Deal” and waiting until the very end to receive the value that is in
their case that they initially chose.
In this essay, I intend to explore the dilemmas of playing this game, as well as the
different strategies contestants could use to maximize their winnings. Different approaches
should be taken at different points in the game and contestants need to be well aware of the
briefcases that are already eliminated. Overall, the best thing for a contestant to do is to not play
with their gut or heart, but with their brain.
Beating the Initial Expected Value
The 26 cases on the show hold values ranging from $0.01 to $1,000,000. These 26 values
are:
To determine the expected value that a contestant would win at the start of the game an expected
value, or an arithmetic mean would need to be taken (2). The calculations for this are:
(.01 + 1 + 5 + 10 + 25 + 50 + 75 + 100 + 200 + 300 + 400 + 500 + 750 + 1,000 + 5,000 + 10,000
+ 25,000 + 50,000 + 75,000 + 100,000 + 200,000 + 300,000 + 400,000 + 500,000 + 750,000 +
1,000,000) / 26 = 131,496.577
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This value means that to beat the initially expected value of the briefcases, or in other words, to
beat the game, a contestant would have to win more than $131,496.58. Because there are only six
briefcases that are greater than the initial expected value, there is only a six in 26, or a 23.07
percent chance of winning the game on the first briefcase pick of the game. There is no way to
approach choosing the first case because the values inside of the cases are completely random.
This percentage serves as a reminder to the contestant that there is about a 25 percent chance that
they have picked a winning case and the following cases that they choose will either lower or
raise that percentage.
Unlucky First Choices
If the contestant is really unlucky and happens to eliminate the highest 13 briefcases right
away, then it is evident that they will not walk away with a lot of money (2). However, they
should still attempt to maximize their money by trying to beat the expected value which is shown
below:
(.01 + 1 + 5 + 10 + 25 + 50 + 75 + 100 + 200 + 300 + 400 + 500 + 750) / 13 = 185.846
The contestant’s expected value is now only $185.85. If the banker offers him or her anything
above this value they should take it. Even though they have not beaten the initially expected
value, and technically did not beat the game, they still did not make a gutless, weak decision by
taking a value less than the expected mean (2).
“Failed” Example
A contestant was in a situation with their own chosen briefcase and one other briefcase.
The two values left were $400 and $1,000,000 and the banker offered him $700,000. After a lot
of debate between him and his family members and shouts of “Deal” and “No Deal” from the
crowd he finally had to make a decision. According to his expected value of $500,200
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((1,000,000 + 400) / 2 = 500,200) the banker’s offer was an extraordinary offer that he should
have taken. This $700,000 was $199,800 more than his expected value and is $568,503.42 more
than his initial expected value of 131,496.58 at the beginning of the game. However, a family
member who reminded him that he came with nothing and $400 was still a profit for the night
egged him on. The contestant risked it and relied on luck treating the game like a lottery (2).
After saying “No Deal” his briefcase was open and it revealed a comparatively measly $400. The
contestant was not as upset as viewers might have expected from someone who just “lost”
$699,600. He was still mildly happy to walk away with some cash in his pocket even though he
could have had much more. This shows the different personalities of people who play this game
and the affect that their personalities have on the decisions they make over the course of the
game.
Expected Value
Beating the mean should always be the contestant’s main goal if they are playing the
game in a smart, statistical way (2). Almost every situation in the game can be solved through
finding the mean and then basing your response to making a deal or not based on if the offer
from the banker is above the mean or not. Obviously, the mean changes as suitcases are
removed, but regardless of the mean at any given time, your goal should remain the same: beat
the mean (2).
An episode in April of 2011 left a contestant with, what to her, looked like a tough
situation, but if she had used statistics it should have been a no brainer. She had 5 suitcases left
that contained the following amounts: $100, $400, $1000, $50,000, and $300,000; and the
banker offered her $80,000 to quit (2). A calculation of the expected value gives her:
(100 + 400 + 1,000 + 50,000 + 300,000) / 5 = 70,300
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Because what she was offered was greater than her expected value she should have said “Deal”
and walked away with 13.8 percent more than her mean was. However, the contestant said “No
Deal” and the next briefcase she picked was worth $300,000 (2). Because of this, her next call
from the banker was a lot lower of an offer. Although it was very unlucky that she picked that
value next, realistically she still should have taken the $80,000. 80 percent of the remaining
briefcases had at least $30,000 less than the banker’s proposed amount in them. The
only guaranteed way she could do better than the proposed amount is to actually be holding the
$300,000 case. This is because if she were to remove more cases and reveal amounts less than
$300,000, the banker’s offer would likely go up to compensate for the increasing mean.
However, there was only a 20 percent probability of this happening. Also the contestant should
have kept in mind that the $80,000 offer is guaranteed. The show would give her cash money
with a 100 percent chance of success (2). The lady eventually accepted the banker’s next offer of
$21,000, which represented a $110,496.58 loss against the starting mean of the game instead of
just a $51,496.58 loss against the starting expected value.
The only problem with approaching the game in this way is that it does not provide for
great entertainment or viewing pleasure (2). NBC wants their contestants to go with their gut
feeling, or to have the crowd influence their decision. Many people would say that contestants
should always be as risky as they can because if they end up only getting a penny, it is still a gain
on their part. These same people say that the thrill of maybe winning hundreds of thousands of
dollars is better than winning very little (2). This brings up the point of the different types of
contestants in this game.
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Contestant’s View of Risk
The contestant’s answer to the simple question of “Deal or No Deal?” strictly depends on
what his or her view of risk is. More importantly, it depends on how risk-averse the contestant is
(3). This concept can be understood through a simple situation. If a player is given the choice
between a sure $50 or a 50-50 chance of getting $100 or nothing, which would he or she take?
They are risk-neutral if they do not care between the options, because they have the same
expected value of $50 (50 x 1 = (100 x .5) + (0 x .5)). They are risk-averse if they would take the
sure $50, and risk-loving if they would prefer to take the gamble on winning $100 (3).
Because preferences are tied to specific individuals, preferences are subjective. When
people engage in purposeful actions, they are motivated by desires that are not necessarily
identical from person to person. In order to explain exchanges, economists must recognize that
preferences are subjective (4). The idea of subjective preferences is tied to the modern utility
theory. Using the utility theory a different question can be asked (3). Given an amount $x, at
what probability level p would you be indifferent between a sure $x or a lottery in which you had
probability p of getting $100 and probability (1-p) of getting nothing? A contestant’s utility for
$x is based on the answer to this question. If $x=$0, then their choice for p will be 0. If $x=$100,
then their choice for p will be 1 (3). If the contestant is risk-neutral and $x=$50, then their choice
for p will be .5 but if they are risk-averse, then they will probably want p to be higher than .5
before the lottery becomes just as desirable as the sure $50 (3).
The modern utility theory gives us three different curves:
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Risk-averse
Risk-neutral
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Risk-loving
Conclusion
Deal or No Deal is a great game show that shows a lot of different personalities and many
different ways to approach the challenge. Although it is definitely still a game of luck, the
actions of the contestants can determine just how much money they will go home with. There is
only a one in 26 chance to win $1,000,000 but contestants can still walk away with large chunks
of money. How a contestant goes into the game thinking they will play it plays a major role in
the outcome of the event.
There are two main types of contestants on this show. There are the thrill seekers who
treat the game like a lottery or like a gambling game in Vegas, and there are the
“mathematicians” who know exactly how they are going to play the game and what moves they
are going to make during the show. If a contestant does not care about missing out on the chance
for $1,000,000 they have a much larger chance of going home with a bigger chunk of money
than someone who is dead set on winning $1,000,000. Using expected value and the mean of the
prizes left on the table allows contestants to know if the offer the banker gives them is adequate
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or not. Overall, players must remember that it is still a game of luck and that there is only so
much they can do to win a substantial amount of money.
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One Page Summary
NBC’s Deal or No Deal was a hit game show from 2005 to 2010 known for the
excitement of the contestants wins and loses, the host Howie Mandel’s antics, and the models
holding the cases. Each night the show aired, a contestant chosen from the crowd was confronted
with 26 sealed briefcases full of varying amounts of cash. The briefcases’ values ranged from
$0.01 to $1,000,000. Without knowing the amount in each briefcase, the contestant picks one
briefcase to keep, if they choose, until the unsealing of it at the end of the game (1).
The contestant must then instinctively eliminate the remaining 25 cases and try to win as
much money as they can. The cases are opened and the amount of cash inside is revealed and
eliminated for the contestant to win (1). In the first round six cases are selected, in the second
five are selected, in the third four are selected, in the fourth three are selected, in the fifth two are
selected, and for the six remaining rounds only one case is selected by the contestant. After a
pre-determined number of cases are opened, the participant is tempted by an unknown and
unseen "Banker" to accept an offer of cash in exchange for what might be contained in the
contestant's chosen briefcase. It is at this time that Mandel asks the both famous and important
question: Deal or No Deal? (1).
To determine the expected value that a contestant would win at the start of the game an
expected value, or an arithmetic mean would need to be taken (2). The calculations for this are:
(.01 + 1 + 5 + 10 + 25 + 50 + 75 + 100 + 200 + 300 + 400 + 500 + 750 + 1,000 + 5,000 + 10,000
+ 25,000 + 50,000 + 75,000 + 100,000 + 200,000 + 300,000 + 400,000 + 500,000 + 750,000 +
1,000,000) / 26 = 131,496.577
This value means that to beat the initially expected value of the briefcases, or in other
words, to beat the game, a contestant would have to win more than $131,496.58. Because there
are only six briefcases that are greater than the initial expected value, there is only a six in 26, or
a 23.07 percent chance of winning the game on the first briefcase pick of the game. This
percentage serves as a reminder to the contestant that there is about a 25 percent chance that they
have picked a winning case and the following cases that they choose will either lower or raise
that percentage.
Beating the mean should always be the contestant’s main goal if they are playing the
game in a smart, statistical way (2). Almost every situation in the game can be solved through
finding the mean and then basing your response to making a deal or not based on if the offer
from the banker is above the mean or not. The only problem with approaching the game in this
way is that it does not provide for great entertainment or viewing pleasure (2). NBC wants their
contestants to go with their gut feeling, or to have the crowd influence their decision.
The contestant’s answer to the simple question of “Deal or No Deal?” strictly depends on
what his or her view of risk is. More importantly, it depends on how risk-averse the contestant is.
This concept can be understood through a simple situation. If a player is given the choice
between a sure $50 or a 50-50 chance of getting $100 or nothing, which would he or she take?
They are risk-neutral if they do not care between the options, because they have the same
expected value of $50. They are risk-averse if they would take the sure $50, and risk-loving if
they would prefer to take the gamble on winning $100.
There are two main types of contestants on this show. There are the thrill seekers who
treat the game like a lottery or like a gambling game in Vegas, and there are the
“mathematicians” who know exactly how they are going to play the game and what moves they
are going to make during the show.
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Bibliography
1. "Deal or No Deal." NBC. NBC Universal, n.d. Web. 24 Nov. 2014.
2. Pearson, Chris. "Deal or No Deal: A Statistical Deal." Pearsonified RSS. N.p., n.d. Web.
24 Nov. 2014.
3. Su, Francis E., et al. "Deal or No Deal." Math Fun Facts.
4. Murphey, Robert. "Modern Utility Theory and Interpersonal Utility
Comparisons."Consulting By RPM. N.p., n.d. Web. 24 Nov. 2014.