Optimal Strategy for "Deal or No Deal?"

[JohnnyFX]

I'm assuming that the banker doesn't know what's in the players case, but I've always felt that the "actual" value of the case was irrelevant anyway. That's because, unless it came down to the last two cases and the only amounts left were the two highest, it would almost always be correct to take the deal when there are only two cases left.

One point that hasn't been mentioned is that it's actually in the networks best interest if the player doesn't take the deal. That's because it makes better television. The amount that they pay out is going to average out over the season anyway, so making the show entertaining is much more important to them then controlling the payouts.

I also think that the banker "reads" the contestants and factors that into the offer. I haven't watched enough to verify this mathematically, but it seems that the contestants who are more risk adverse or less inclined to gamble get lower offers then the ones who "feel lucky".

Bearing that in mind, it seems to me that the best strategy is to keep refusing the deal as long as there are at least two high numbers that are fairly close to each other. As soon as the low values outnumber the high ones and the highest value is much greater then the second-highest, it's usually going to be correct to deal.

Of course if the bankers offers were actually much closer to the average value of the case then they usually are, then it might be correct to take the deal sooner.

[Double Down]

From a mathematical standpoint, it is never "correct" to take the deal. As long as the deal is less than what the average case is, it's better EV to say no deal. When to take the deal is all about what money means to you individually. If there were 2 cases left, a 1 mil and a penny, and the deal was for $500,000 it's pretty much about whether or not you want a guaranteed 500K or you're willing to flip a coin for a million.

[twoblacknines]

Try for yourself at
http://www.nbc.com/Deal_or_No_Deal/game/
I've gotten up to around 200k so far

[Runner Runner]

Just played the game, got down to 2 suitcases, one with $1m and one with $25. The offer from those cheapskates was only just over $300,000, I declined and won the coinflip for the $1m.

[Jordan]

it is rational to be risk averse, and there is mathematics that can tell you what the suit-case "lottery" is worth aka the "certainty equivalent". It is something like CE = EV - Var/2R. Where R is a function of your bankroll and risk tolerance. basically it is rational to give up some amount of EV to not have to endure risk. search for Izverg posts at bonuswhores.com as he has made many posts bringing this concept to people trying to maximize the value of sticky bonuses.

[rsk111]

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".

Cross posted to other thread in probability.

[bodnotbod]

We have this show here in the UK but the max prize is £250,000.

I'm not too hot on maths, but now that I've become interested in poker I'm thrown into the situation of having to embrace it.

I was really pleased to see this thread cos I knew that there must be something essentially predictable about the offers.

I don't like the show, although it was improved vastly by the comment of one of our TV critics. I don't know about the US version, but in our version we only hear the host's end of a telephone conversation with the mysterious banker. The critic said the programme is much more fun if you go on the assumption that the host is actually mentally disturbed and the voices are all in his head; it adds a note of high comedy to his overly emotive reactions to what he is (supposedly) hearing on the phone.

That may only work here where the host is Noel Edmond's whose TV career fell flat when a member of the public was killed during a stunt trial on one of his shows and, some years later, a child also died in a helicopter accident, the helicopter a part of a charity he was running.

[StregaChess]

[ QUOTE ]
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 criterion problem.

[/ QUOTE ]

Very nice but here's my optimal strategy.

If the amount that I'm offered (taking into account the IRS could rake 50%) will change my life in some sort of meaningful way I take the money. If not I'm playing on.....

Of course that's up to you what that means...

I have a second mortgage of $30K, do I really care..
Not much... so if I was offered $65K forget it..

Let's say I owe $330K on the first mortgage and could pay 40% of it off, 275K (assuming 50% to the IRS and some amount of pissing in the wind)...

OK, I'm taking it

[Runner Runner]

I was watching this show earlier this week and noticed that they are now offering more then the fair value of the suit cases for the last 2 offers. For instance, there were 3 suitcases with $500,000, $50,000 and $500, and the offer was over $200,000. Then the offer when there was just a $500,000 and $500 suitcase would of been $300,000!

I would die laughing if a contestant denied one of these ridiculously good offers which not only mitigate risk, but actually have better expected value then the worth of the prize. Now that would be a true gambler.

[StregaChess]

[ QUOTE ]
Try for yourself at
http://www.nbc.com/Deal_or_No_Deal/game/
I've gotten up to around 200k so far

[/ QUOTE ]


This game is CRACK...


Check this out...

I have three suitcases let to open



The offer is 305,288 to take the money



The values left in the cases are a million, $75 and $500 would you take it?


I turned it down...

I hit the case that's worth $500



Now it says $75 and a million are left and the offer only goes up to $355,308



That’s crack…. I’m taking the cash…



I got the $355,308 and my case contained $75…