Deal or No Deal

#41
The median dollar amount on the board on DOND is less than $1000. You have a 50% chance of choosing less than that at the outset.

The mean dollar amount of all suitcases is nearly 263,000.

If this were an 'infinite tries' game (like poker), wouldn't this game have a positive expected value of nearly $263,000?
The fact that it is a 'one shot Deal' and NOT an 'infinite tries' game doesn't quantitatively change the +EV... but, getting one shot, most contestants are uncomfortable with the variance.... (risk of "winning" $1 or $.01 are too high).
 

Slinky Bender

The All Powerful Moderator
#42
Easy question for someone to find out: what has been the first offer in each game? Follow up question: does the bank offer an increasing, decreasing, or variable (from game to game) percentage of the mean of the remaining cases as the game progresses?
 
#43
Just empirically, the show's interest is in suspense and entertainment. Minimizing the payout is not a priority. So

# early offers will be lower than the odds would dictate, to keep players in the game.

# Later offers will be calculated to make the decision as difficult as possible. Too high and the contestant will take it quickly. Too low and the contestant rejects the offer out of hand. I would guess the idea there would be to make the offer slightly too low.
 

Slinky Bender

The All Powerful Moderator
#44
So, I decided to try the online game. So far, I've opened 17 suitcases. What's left are:

1) 1
2) 300
3) 5,000
4) 10,000
5) 50,000
6) 400,000
7) 500,000
8) 750,000

The mean of the remaining cases is $215,413 and the offer is $84,031. Sounds pretty bad to me? Let's see what happenes next?

1) 1
2) 300
3) 5,000
4) 10,000
5) 500,000
6) 750,000

The mean of the remaining cases is $210,884 and the offer is $80,560. Still not great? Let's see what happenes next?

1) 1
2) 300
3) 5,000
4) 10,000
5) 750,000

The mean of the remaining cases is $153,060 and the offer is $80,700. Doesn't sound rational, does it? The mena went down substantially, there's only one case left over the offer, but the offer actually went up (slightly)? It sounds lie soeone's risk averseness is going up.... and it's the bank's (or maybe he knows I have the $750,000 case?)

Now it gets more interesting:

1) 300
2) 5,000
3) 10,000
4) 750,000

Got rid of the $1 case: The mean of the remaining cases is $191,325 and the offer is $131,274.

Back when there were 8 cases and the mean was $215,413 the offer was $84,031. With 6 cases, the mena was $210,884 and the offer was $80,560. So now, the mean is lower, but the offer is substantially higher. If my theory that the bank knows what's going on and is making higher offers to knock me off of a winning case is correct, it would play exactly this way.

Guess that theory is wrong (at least on the online game):

1) 300
2) 5,000
3) 10,000

The mean of the remaining cases is $5,100 and the offer is $7,650 (of course anyone should take that deal). But let's keep playing

1) 300
2) 10,000

The mean of the remaining cases is $5,150 and the offer is $5,150

1) 300

So, the best "deal" seemd to be at the point were 4 cases were left and the mean of the remaining cases is $191,325 and the offer was $131,274.

perhaps the algorithm is "positional" in that the best offer is always made at a certain point in the game? (like when 4 cases are left?).
 

Slinky Bender

The All Powerful Moderator
#45
PS I definitely would not have taken the $80,560 offer with the $210,884 mean and 6 cases, I would have had to think a lot about the $80,700 offer with the $153,060 mean and 5 cases, and I definitely would have taken the $131,274 offer with the $$191,325 mean and 6 cases.
 

Slinky Bender

The All Powerful Moderator
#46
Played a second game and something very interesting hapened:

got down to 3 cases: .01, 100, and $50,000. Offer was a little over $25,000. I would have taken this. Picked again and hit the $.01. The offer pretty much did not change. Hard to explain that one, huh?
 

Slinky Bender

The All Powerful Moderator
#47
After playing a number of games, it seems like the best deals get offerede at 3 or 4 cases remaining (when i say "best", I mean highest percentage of mean of remaiing $). I think I've already come up with my "gut instinct" method of play, with only playing about 5 games.
 
#48
Ok, here's my trial run....

I eliminated 1, 50, 75, 750, 50000 and 750000.
My offer was 27478.
No deal.
I eliminated 10, 25, 1000, 100000 and 300000.
My offer was 40725.
No deal.
I eliminated 0.01, 100, 400 and 200000.
My offer was 66951.
No deal.
I eliminated 5000, 500000 and 750000.
My offer was 68001. (baffling!)
No deal.
I eliminated 5 and 300.
My offer was 93940.
DEAL. Winner winner chicken dinner.
-------
So, the computer allows you to "see what would have happened..."

I eliminated two more suitcases and had 500, 10000, 400000 and 1 million remaining. My would-be offer was 239085.

I eliminated the 500 suitcase. My would-be offer was 432000.

Then after eliminating another suitcase, I had 10000 and 1 million left. My would-be offer was 360,000.

(my suitcase had the 10000, btw).


Based on this ONE trial run, I think Bandaid's low-offers-early, more-difficult-offers-later theory, besides making sense, has some validity to it....
 

justme

homo economicus
#49
JackT said:
So, to put this in a language that hopefully I can understand...
Taking your simplified example above (where the bank knows nothing), if the three cases held $1 million and $1, then what would the bank's only optimal offer be?


To me, this stark example of $1 million and $1 really brings to light how PLAYER DEPENDENT optimal strategy really is. (or looked at another way, bank dependent). The example could well be $50 and $1.

The difference between winning $750,000 and $1,000,000 is much less than the difference between winning $1 and $250,001, even though the dollar difference is the same.
The optimal play for the bank will be the smaller of one cent above the player's equivalence for that lottery (the minimum the player will take) or the bank's euiqvalent for the lottery. The problem (I think) is that the bank has no idea what the player's min is, so it has to play it's own min (plus a penny).
 

justme

homo economicus
#50
JackT said:
Based on this ONE trial run, I think Bandaid's low-offers-early, more-difficult-offers-later theory, besides making sense, has some validity to it....
The interesting thing is that this is somewhat consistant with my game theoretical approach to the problem. If the bank can callibrate the player's utility opne, it might be able to make a more self-serving offer late in the play.

The best way to do that callibration is with offers that the player will not expect. Once the player accepts an offer, the game is over...
 

Slinky Bender

The All Powerful Moderator
#51
justme said:
The best way to do that callibration is with offers that the player will not expect.

Do you mean accept?

Also, since it's a game show, the audience wants excitement. If the contestant takes an early offer, there is ZERO excitement in the game for the audience. The further the game goes, the better the ratings. I would imagine the best possible outcome for the show would be to get down to 2 cases, one with the $1 million and one with the 1 cent. So "optimal" for the show is to drive as many contestants in that direction as possible.
 
#53
There was a real dummy on last night...


Only 4 boxes left... all were under $300 cept for one that was $300,000. The offer was $71,000. So with a one in four chance of having the 300,000 she gambled, turned down the sure thing (25% offer) and lost. That's a degenerate I'd like to have playing with me.
 
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#54
I would love to see one of those dickheads take the 1st offer, and walk...LOL. How far in advance do the players know they have been picked? I have seen the " Banker " offer a Hummer and cash, knowing that the player wanted a hummer...
 
#57
They might use a weighted average, where the higher value cases count differently. a 'weighted average' is "a method of computing a kind of arithmetic mean of a set of numbers in which some elements of the set carry more importance (weight) than others." The weighting might change as the game gets later.
 
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