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Deal or No Deal
Probabilities
Oracle Tips by Burleson Consulting |
Many people have
been fans of the latest game show "Deal or No Deal".
During the show, the
Deal or No Deal contestants choose to open a series of amounts and
then the "Bank" makes an offer to make the person
choose to continue.
When faced with a
choice, the "Deal or No Deal" contestants are essentially dealing
with a probability problem. On the Deal or No
Deal board, the contestants choose "cases" that contain values from
one cent to a million dollars:
|
Column |
On/off |
avg. $ value |
|
Column |
On/off |
avg. $ value |
|
$0.01 |
1 |
$0.00 |
|
$1,000 |
1 |
$38 |
|
$1 |
1 |
$0.04 |
|
$5,000 |
1 |
$192 |
|
$5 |
1 |
$0.19 |
|
$10,000 |
1 |
$385 |
|
$10 |
1 |
$0.38 |
|
$50,000 |
1 |
$1,923 |
|
$25 |
1 |
$1 |
|
$25,000 |
1 |
$962 |
|
$50 |
1 |
$2 |
|
$75,000 |
1 |
$2,885 |
|
$75 |
1 |
$3 |
|
$100,000 |
1 |
$3,846 |
|
$100 |
1 |
$4 |
|
$200,000 |
1 |
$7,692 |
|
$200 |
1 |
$8 |
|
$300,000 |
1 |
$11,538 |
|
$300 |
1 |
$12 |
|
$400,000 |
1 |
$15,385 |
|
$400 |
1 |
$15 |
|
$500,000 |
1 |
$19,231 |
|
$500 |
1 |
$19 |
|
$750,000 |
1 |
$28,846 |
|
$750 |
1 |
$29 |
|
$1,000,000 |
1 |
$38,462 |
The goal is to
pick small values in order to have a higher expected payoff, and
hence, a higher bank offer. The expected payoff in Deal or No
Deal is essentially probability problem.
Click for the
deal_or_no_deal.xls
spreadsheet that computed the expected payout for any round on
the Deal or No Deal game. To use my "Deal or No Deal"
spreadsheet, change the "on/off" values from 1 to zero as the values
are chosen. The ideal behind
"Deal or No Deal" is simple. For example, consider the
following scenario with four numbers on the board:
$75
$300
$500,000
$1,000,000
The Deal or No deal contestant
has a 25% change of getting any number, and over a given number of
trials, the expected payout of each value is 25% time that value:
Value
Expected Payout
$75
$75*.25
= $19
$300
$300*.25
= $75
$500,000
$500,000*.25 = $125,000
$1,000,000
$1,000,000*.25 = $250,000
$375,000
In this simple example, the expected payout probability is $375,000.
Now, you would expect the Deal or No Deal banker to offer a value
that relates to the expected probable payout, right? In
reality, the bank offers are calculated to manipulate the contestant
(it's what makes the game exciting).
In my observation using the
probability spreadsheet I noted that in the early rounds of "Deal or
No Deal", the bank makes offers that are less than the contestant's
expected probable payout (an incentive to continue). As the
rounds near completion, some suggest that the contestants behavior
can be altered by the "bank", and that the offers (as compared to the expected
probable gains) should increase, proportional to the risk/reward for
each round:
Why offer
above the expected payout? It's because people's attitude will
change when faced with a large-stake high-risk gamble.
For example,
consider a deal where we have two items left, one for $100k and
another for $300k. The 50-50 chance expected payout over
thousands of simulated trials would be (.5*100)+(.5*300)= $200k.
What would you rather have, a $200k sure-thing, or a 50% chance of
getting $300k, with the risk of losing $100k?
Since our downside risk is limited to $100k
(the worst outcome), this decision point can be reduced into a
simpler problem:
You have just won
$200,000 and the right to gamble half of your winnings. You have
these two choices:
a)
You may wager $100,000 on a double-or-nothing, winner takes all,
based on a random 50-50 choice (like the toss of a fair coin), or:
b)
Take $200,000 and quit.
Obviously, this curve is never
linear, but with a hit show like this, the Deal of No Deal bank
offers are carefully crafted to maximize the stress on the
contestant! I'm going to refine my model, in the hopes of
deriving the equation used by the Deal or No Deal banker to
calculate their offers.
But in the end, it’s not the probabilities that
determine the decision, but the human emotion, watching some average
Joe make the largest financial decisions of their life, right on
prime time TV. The raw emotions are almost voyeuristic and viewers
are encouraged to identify with the highly-likeable contestants (and
their adorable families, conveniently, right on the set).
Reader Responses:
I love your deal or no deal spreadsheet.
It added a more realistic feel to our game over the random card bank
offer of the at-home game we had. I added the ability to assign a
percentage of mathematical payout by round at the bottom left. Hope
it helps you figure out the bank offers! Greg
deal_or_no_deal4.xls
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