Deal or No Deal

[OrangeKing]

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EV matters when you are facing a decision that you can make enough times so that your gained value approaches the EV. This does not apply in a once in a lifetime gameshow choice.

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Well, EV still matters - you just have to adjust how you calculate it (a lot) for the fact that you really, really don't want to blow your chance at an amount of money that means something important to you.

[Buzz-cp]

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EV matters when you are facing a decision that you can make enough times so that your gained value approaches the EV. This does not apply in a once in a lifetime gameshow choice.

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I simply adressed the issue of whether or not the deal was always -EV.

[rsk111]

The answer to your question is that it is almost always -EV, but that doesn't mean that you say no deal. Here is a copy of a post that I made on the same topic in another thread that explains why.

link
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I have read several threads on 2+2 about this topic and I have yet to see anyone give what I believe is the correct answer (if I just missed it, I apologize). So I decided I'd post what I think and get some feedback.

Figuring out the optimal strategy is really just a bankroll management/kelly criterion problem.

For those unfamiliar with the kelly criterion, here is a good link.

At each decision point (i.e., when asked deal or no deal), no deal is (almost always) the positive EV decision. However, for optimal, long-term bankroll growth, one cannot simply take all positive EV propositions.

For example, if my net worth is $100,000 and someone says that they will give me 11:10 odds on a coin flip for a $100,000 wager (and won't allow me to wager anything less), even though it is +EV, I must decline the offer, to ensure optimal long term bankroll growth. The reasons for this are intuitively obvious, but for a more precise mathematical answer we can simply use the kelly equation:

f* = (bp - q) / b
where
f* = percentage of current bankroll to wager;
b = odds received on the wager;
p = probability of winning;
q = probability of losing = 1 - p.

In the above example f*=1/22. Thus, it would be incorrect (by the kelly criterion) to risk more than 1/22 of your bank roll (i.e., about $4500) on such a wager. Since the offer is for you to wager $100,000, you should decline.

Now, as I am sure many of you are aware, the kelly equation can be modified to provide a more or less aggressive bankroll management strategy, but the conclusions will be similar.

So, how does this relate to deal or no deal? When the bank makes an offer one simply must consider that offer as part of his bankroll. For example, if the bank offers me $50,000, then I should consider my bankroll to be X + 50000, where X is my bankroll prior to coming on the show. if 50000/(X+50000) is greater than the f* derived from the kelly equation (as above or modified to be more or less agressive), then I must pass up the +EV proposition and say "deal". If 50000/(X+50000) is less than or equal to f*, then I should take the +EV proposition and say no deal.

Since everyone's X is different, the optimal strategy will be different for each contestant.

This also explains why, when Donald Trump was on the show and advising the contestant, what Trump said was almost certainly the exact opposite of the correct advice. (For those who didn't watch, Trump said that if he were playing he would take a deal for about $200000, but if he were in the contestant's presumed financial situation he would say no deal)

Also note that the above analysis assumes that "no deal" is the +EV play, in the rare cases when the offer is higher than the average of the remaining cases, the player should always pick the +EV play and say "deal".


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[ouyangl]

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OKay yes your right. But I'm just trying to illustrate the point that the expected value of the cases really is meaningless till when it comes down to two cases

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Unless you don't care about utility at all. If a multi-billionaire is on the show and just wants to have the highest expectation, he'll just take no -EV deal, and be absolutely correct in doing so.

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EV doesnt matter till the last two cases! I dont care whats the EV of the cases because if I say no deal, I'm not randomly selecting a case from the lot and taking it home. All I'm doing is eliminating more cases and hoping for a higher deal!

People on here place way too much emphasis on EV.
If I was offered $900,000 and there were 10 cases left
$1,000,000
$0.01
$1
$5
$10
$25
$50
$75
$100
$750,000

and i only had to choose one case before i get offered another deal...

I wouldn't take the deal just because 4/5 of the time the offer is gonna go up, 1/10 of the time the offer is gonna go down to like $400,000-$500,000, and 1/10 of the time my offer will spike down to like $200,000-$300,000

Who cares what the average of the cases is? What really matters is if my predictions about how the bankoffer is accurate or not. $900,000 I'm getting +EV to take the deal... IF I were to select a case and go home with it after I say okay no deal. But I don't, all I do is select another case and get another offer.

[Skjonne]

Wow. You don't have the slightest clue about what EV is do you?

[OrangeKing]

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OKay yes your right. But I'm just trying to illustrate the point that the expected value of the cases really is meaningless till when it comes down to two cases

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Unless you don't care about utility at all. If a multi-billionaire is on the show and just wants to have the highest expectation, he'll just take no -EV deal, and be absolutely correct in doing so.

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EV doesnt matter till the last two cases! I dont care whats the EV of the cases because if I say no deal, I'm not randomly selecting a case from the lot and taking it home. All I'm doing is eliminating more cases and hoping for a higher deal!


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You're making the same kinds of errors people make all the time when gambling, and clearly don't understand EV. In your example, you'd have to be the village idiot to reject the deal, by the way; 20% of the time you're losing half of your deal or more (by your numbers), and 80% of the time you might gain...what, a few thousand dollars?

Of course, you're never going to get that sort of deal, because it's way too good. But since going to the end and getting the maximum - $1 million - is only slightly better than the deal you laid out, and the deals will only increase marginally at best, and 20% of the time you will have your deal halved at least...well, your idea doesn't make sense from a risk-adverse, utility, or EV perspective.

If you have to bet $20, and have a 90% chance of winning, but you only win $1 when you win, that's a bad bet, for the same reasons your idea of not taking the deal there is awful.

[mostsmooth]

i just watched the guy have 4 cases left, $500k, 200K, $500 (or something small, cant remember), and .01. they offer him $109K? and he takes it? am i missing something?

earlier in the show, another person has 4 cases left $1, 200k, 300k, 400k. he takes the offer~220K, then they ask him to pick another for fun, he picks $1, then they tell him the next offer would have been $330K. yeah right, all of a sudden they were going to stop screwing people and give them awesome deals? friggin liars

[shermanash]

how do taxes factor in? i figure the deals are about neutral to a little -EV, but woudln't the tax % for a million dollar win be higher than the % for, say 100,000?

I don't have any data so this may not make much of a difference, but it's something to consider.

[drbst]

If someone accepts 220k if the possible outcomes are 1, 200k, 300k, or 400k, this cannot be critized too much because the EV of rejecting is only a few percent more. Of course the EV should be the main driver for decisions, but the variance can be important too. Taking 220k has a variance of $0, while the rejecting the deal has a huge variance.

I believe there are situations where the variance dictates not to go for the play with the highest expectation.

For example, final table in a Hold'em tournament. You are the Big Blind and get pocket aces. Everybody up to you goes all-in. If you call you may be the favourite to win the whole tournament right here but if not you are out. If you fold you make at least the second place. I would fold here without thinking.

[monroe]

Hi mostsmooth,

I posted this in one of the other 43 Deal or No Deal threads, but that one is dead now. So...

They've made really good offers before. Here are the games for the six most recent contestants. The first number is the round, the second is the banker's offer, the third is the average value of the remaining cases, and the fourth is the banker's offer divided by the actual value (I label this OFAC in my spreadsheet).

You can see that after round six, there is the possibility of a "+EV" offer. The best two offers I've seen are 2800 vs. 2500 (1.12 OFAC) in game 3 round 9 and 202000 vs. 183500 (1.10 OFAC) in game 2 round 8.

1, 32000, 283374, 0.11
2, 21000, 161138, 0.13
3, 44000, 128734, 0.34
4, 92000, 167634, 0.55
5, 73000, 140012, 0.52
6, 9000, 18015, 0.50
7, 23000, 22506, 1.02
8, 28000, 28341, 0.99
9, 5000, 5012, 1.00

1, 15000, 142145, 0.11
2, 37000, 167839, 0.22
3, 66000, 159712, 0.41
4, 110000, 194478, 0.57
5, 198000, 258463, 0.77
6, 85000, 110155, 0.77
7, 139000, 137675, 1.01
8, 202000, 183500, 1.10

1, 22000, 169644, 0.13
2, 71000, 212183, 0.33
3, 187000, 252886, 0.74
4, 107000, 178969, 0.60
5, 95000, 180292, 0.53
6, 123000, 216200, 0.57
7, 19000, 20250, 0.94
8, 2000, 2000, 1.00
9, 2800, 2500, 1.12

1, 13000, 114592, 0.11
2, 37000, 151706, 0.24
3, 52000, 136381, 0.38
4, 88000, 125011, 0.70
5, 109000, 150013, 0.73
6, 164000, 180010, 0.91
7, 221000, 225000, 0.98

1, 26000, 169581, 0.15
2, 48000, 187107, 0.26
3, 39000, 186934, 0.21
4, 90000, 250713, 0.36
5, 179000, 333400, 0.54
6, 199000, 340080, 0.59
7, 109000, 175100, 0.62

1, 9000, 144171, 0.06
2, 7000, 65227, 0.11
3, 16000, 70671, 0.23
4, 250, 291, 0.86
5, 300, 354, 0.85
6, 350, 365, 0.96
7, 400, 431, 0.93
8, 450, 442, 1.02
9, 650, 625, 1.04