deal or no deal

[Duncan Smith]

Chip Turner wrote:
> On 2005-12-26 15:05:21 -0500, james.moughan at sunderland.ac.uk said:
> 
>> I believe not; the Monty Hall problem is biased by the fact that the
>> presenter knows where the prize is, and eliminates one box accordingly.
>>  Where boxes are eliminated at random, it's impossible for any given
>> box to have a higher probability of containing any given amount of
>> money than another.  And for the contestants box to be worth more or
>> less than the mean, it must have a higher probability of containing a
>> certain amount.
> 
> 
> Agreed -- unless the presenter takes away a case based on knowledge he
> has about the contents, then Monty Hall doesn't enter into it.  Deal or
> No Deal seems to be a purely chance based game.  However, that doesn't
> mean there aren't strategies beyond strictly expecting the average payout.
> 
>> Like another member of the group, I've seen them offer more than the
>> average on the UK version, which puzzled me quite a lot.
> 
> 
> I imagine it is about risks.  Many gameshows take out insurance policies
> against the larger payoffs to protect the show and network from big
> winners.  I believe Who Wants to be a Millionaire actually had some
> difficulty with their insurance when they were paying out too often, or
> something.  Perhaps the UK Deal or No Deal didn't want to risk
> increasing their premium :)
> 
> But even the contestant has a reason to not just play the average,
> thereby bringing psychology into the game.  It comes down to the odd
> phenomenon that the value of money isn't linear to the amount of money
> in question.  If you're playing the game, and only two briefcases are
> left -- 1,000,000 and 0.01, and the house offers you 400,000... take
> it!  On average you'll win around 500,000, but half the time, you'll get
> a penny.  Averages break down when you try to apply them to a single
> instance.  On the flip side, if you think about how much difference
> 500,000 will make in your life vs, say, 750,000, then taking a risk to
> get 750,000 is probably worth it; sure, you might lose 250,000 but on
> top of 500,000, the impact of the loss you would suffer is significantly
> lessened.  In the end, it comes down to what the money on the table
> means to *you* and how willing you are to lose the guaranteed amount to
> take risks.
> 
> It's similar to the old game of coin flipping to double your money.  Put
> a dollar on the table.  Flip a coin.  Heads, you double your bet, tails
> you lose it all.  You can stop any time you want.  The expected outcome
> is infinite money (1 * 1/2 + 2 * 1/4 + 4 * 1/8 ...), but a human playing
> it would do well to stop before the inevitable tails comes along, even
> though mathematically the house pays out an expected infinite number of
> dollars over time.  Exponential growth in winnings doesn't offset
> exponential risk in taking a loss because, once you hit a certain point,
> the money on the table is worth more than the 50% chance of having twice
> as much.
> 
> Chip
> 

As you say, it depends on the player's utility function.  But it's not a
straightforward question of comparing the offer to the expected values
of the remaining boxes, even for a risk-neutral player.  At most stages
of the game refusing an offer means that there will be a future offer,
and, later in the game, these tend to be closer to (or even greater
than) the expected value.

Duncan