Chip Turner wrote: > On 2005-12-26 15:05:21 -0500, [EMAIL PROTECTED] 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 -- http://mail.python.org/mailman/listinfo/python-list
