Deal or No Deal easy math puzzle/question

[durron597]

You are invited to play on Deal or No Deal. However, there is a catch: you must pay some amount of money out of pocket to be allowed to enter the game.

What is the most amount of money would you be willing to put up? You only get to play the game once.

For reference, here are the possible values of each suitcase:

$0.01
$1
$5
$10
$25
$50
$75
$100
$200
$300
$400
$500
$750
$1,000
$5,000
$10,000
$25,000
$50,000
$75,000
$100,000
$200,000
$300,000
$400,000
$500,000
$750,000
$1,000,000

And the rules (from wikipedia)

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Before the real game begins, a third party randomly places the possible winnings in the cases, which are distributed to 26 models (identically dressed, which is taken from the original Dutch version but without wigs) who reveal the contents during the game. No one, including the host, knows what amount in the cases. Each contestant receives a new, randomly assigned set of cases. The winnings range in amounts from $0.01 to the top prize (see listing below).


The show's mysterious banker makes an offer to buy the contestant's chosen case.After picking his/her case, the contestant then selects 6 of the remaining 25 cases, revealed one at a time. This is followed by a "phone call" by "The Banker", a mysterious figure whose face is not shown (at times a silhouette can be seen). He purportedly sits in a skybox (situated between the two audience sections) and makes an offer, via telephone to Mandel (his voice is never heard) to buy the contestant's case based on the cash amounts still in play and the player's psychology (supposedly insulting comments have been relayed to the contestant; in one episode a rabbit-owning contestant was informed that The Banker, after making his offer, "was going rabbit hunting"; in another with a boxer, he made a comment seeing if she could "take a punch". ). The player is then asked by Mandel the title question: "Deal or No Deal?".

If the contestant accepts the buyout (by stating "Deal!"), they must lift a cover and press a button to confirm the decision. The game then ends, and the contents of the case that s/he chose at the beginning of the game are then revealed along with the whereabouts of the top remaining prizes. Sometimes, Howie lifts the cover for the contestant, especially during more dramatic decisions. As of lately, he seems to be doing it almost every time. The contestant does not necessarily have to say "Deal!"; only hitting the button is necessary.

Should the contestant refuse the offer (by stating "No deal!"), they must choose five of the remaining cases to eliminate from consideration. The Banker makes another offer, and play continues as before. The Banker's offer may be higher or lower than the previous offer (if a top prize is eliminated, generally the offer decreases; conversely, if lower amounts are eliminated the offer increases significantly). (The contestant can simply close the cover to imply "No Deal" without actually saying the phrase.)

Subsequent rounds have the contestant withdrawing four, three, then two cases from play; should the contestant continue to decline The Banker's offer after this point, they then eliminate one case each time (with an intervening offer from The Banker) until two cases are left. If the player rejects the final offer, they receive whatever cash amount is contained inside the case that they originally chose (earlier shows gave the contestant a chance to switch their case with the one remaining in the gallery, but this offer has not been made in the more recent shows).

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[Nicholasp27]

if i added correctly, then the ev of a random case is $131,477.54, but u get higher ev because u get to make some choices and not just pick one random case and go with it

[DMBFan23]

[ QUOTE ]
if i added correctly, then the ev of a random case is $131,477.54, but u get higher ev because u get to make some choices and not just pick one random case and go with it

[/ QUOTE ]

Nick,

can we use the pure EV? at some point we'll probably accept the banker's offer rather than take a very high variance gamble, right? I mean if there is 1 mil left in one case and 1 dollar in the other, and the banker offers me 400K, call me dumb but I'm probably taking it

[Nicholasp27]

i was just throwing that out there as a starting point to move from

yes, u have to weigh the fact that u get a choice in when u take the offer along with the fact that the banker doesn't ever offer u neutralEV

to figure it out, u'd prolly just do some math; how many cases do u open the first round? do odds of high vs low cases being chosen to determine what your true ev should be after that round; then maybe use about 80% of that ev to be banker's offer...then rinse and repeat for subsequent rounds, etc

[durron597]

Also realize about life EV here... the diff between 400k and 1 mill vs. $1 and 400k is huge

[RoundTower]

[ QUOTE ]
if i added correctly, then the ev of a random case is $131,477.54, but u get higher ev because u get to make some choices and not just pick one random case and go with it

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I don't think you ever get to make any choices that can increase your EV.

I think I would put down all the money I have in the world for this game, but that's significantly less than $131k.

[DMBFan23]

[ QUOTE ]
i was just throwing that out there as a starting point to move from

yes, u have to weigh the fact that u get a choice in when u take the offer along with the fact that the banker doesn't ever offer u neutralEV

to figure it out, u'd prolly just do some math; how many cases do u open the first round? do odds of high vs low cases being chosen to determine what your true ev should be after that round; then maybe use about 80% of that ev to be banker's offer...then rinse and repeat for subsequent rounds, etc

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yeah I was just guiding the discussion in that direction...you are def. right about that as a starting point.

so, what we'd need to know is the percentage of the banker's offer compared to the EV of all remaining cases (this would probably be a function of the variance of the values of the remaining cases, as well). also our own variance tolerance.

but then we need to know, how much are we willing to pay once we know that? for instance, let's use the 131K EV as an assumption, how much of that would you be willing to give up as an entry fee?

[SWB]

[ QUOTE ]
yes, u have to weigh the fact that u get a choice in when u take the offer along with the fact that the banker doesn't ever offer u neutralEV

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On a recent episode, he offered a lady neutral choices after she'd eliminated every amount over something like $500. So there's probably some threshold where the amounts are low enough that it's not in the banker's interest to extend the game by making lowball offers. I doubt that would impact on any overall EV calculations very much.

[T.J. Combo]

I'd pay $1000. Since there are 9 cases below $1000 and 10 cases above $1000.

[Nicholasp27]

but all of the cases above 1k are at LEAST 1k above (only the lowest is 1k more, the highest is 999k more)

however, all of the cases below 1000 are within 1k below 1k

so 1000 is a very lowball offer

also, your first deal from banker will be more than 1k no matter what, even if u remove the 5 highest values