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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, itís 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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