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Analysis of Deal or No Deal Game

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					            Analysis of Deal or No Deal Game
                                    Sven Laur
                                March 15, 2009


1    Introduction
Deal or No Deal is a popular TV game, which can be analysed purely in terms of
descision theory. As there are many versions of this game, we restrict ourselves
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to the Estonian version (V˜ta v˜i j¨ta). At the beginning of the game, 26
monetary prizes depicted in Table 1 are put into the briefcases. At his point
nobody knows the contents of any briefcase. Then a randomly chosen briefcase
is given to a player. The aim of the player is to sell the briefcase with the highest
price to the banker or take the prize which is in his or her briefcase.

                    0.01              0.50          1.00          5.00
                   10.00             25.00         50.00         75.00
                  100.00            200.00        300.00        400.00
                  500.00         1, 000.00     2, 500.00     5, 000.00
              10, 000.00        20, 000.00    30, 000.00    40, 000.00
              50, 000.00       100, 000.00   150, 000.00   300, 000.00
             500, 000.00    1, 000, 000.00


              Table 1: Monetary prizes in Deal or No Deal game.

    The trades between the player and the banker are organised in rounds. In
the first round, the banker makes a blind offer and the player can either accept
or reject the offer. Before the second round, 6 randomly chosen briefcases are
opened from 25 unassigned briefcases. Then the banker makes an offer and the
player can either accept or reject. The game continues analogously, except 5
briefcases are opened before the third, 4 before the fourth, 3 before the fifth and
2 before sixth and 1 the following rounds. When there are only 2 briefcases left,
the player can swap the cases before the final trade round.


2    Analysis of the Game
Note that not all people value money the same way some are more risk averse
than the others. Hence, one should first convert the prizes x into a utilities U (x)
so that we can use decision theory to find an optimal strategy.

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    First, find an optimal strategy for the banker and the player under assump-
tion that both are risk neutral: U (x) = x. What is the corresponding average
win in this case? How are the wins distributed?
    Second, construct an algorithm that computes optimal strategies for both
players for any valid utility functions Ubanker (·) and Uplayer (·). Consider stan-
dard choices for utility function and estimate the distribution of wins? How
does the optimal strategy of the player change if he or she knows the utility
function Ubanker (·). What is the corresponding distribution of wins?
    Obviously, the complete description of the optimal strategy is too complex
to memorise. Hence, a player can use only compact strategies. As a third
task, formalise the notion of compact strategy and compare them with optimal
strategies. In particular, study how the distribution of wins changes if player
is willing to memorise more facts. This is an open task, the result depends
completely on your formalisation. Nevertheless, I expect the formalisation be
reasonable—a description of a strategy should be “human readable”.
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    Fourth, search for the documented runs of the “V˜ta v˜i j¨ta” game [Elu24]
and determine empirically the approximate utility function for the banker or
prove that banker is acting non-consistently over the games. If so is there a
detectable trend in his or her behaviour?


References
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[Elu24] Short summaries of “V˜ta v˜i j¨ta” in the Internet portal Elu24 media
       section. http://www.elu24.ee/, 2009.




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