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Ask the Wizard #150
- $0.50
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- $200,000
The contestant selects one of the briefcases to be THEIR suitcase. Through a process of elimination, by opening the other suitcases, they try and work out how much money is their case, or whether it would be wiser to take a "Bank offer". The Bank Offers are based on, but not equivalent to, the arithmetic mean of the remaining briefcases. So, if there are mainly large valued briefcases remaining, there is a high chance that the contestant’s briefcase is valuable, and so the Bank Offer will be generous. Conversely, if the player has been less fortunate and opened the more valuable briefcases, then the Bank Offer will be low. What would be the best strategy to employ if you were a contestant on this game? A non-mathematical gut instinct strategy would be ignore the Bank offers and carry on opening cases until either the $200,000 was opened and eliminated, or both the $100,000 and $75,000 were opened and eliminated. What’s the math behind this game, Wizard?
Jacqui from Birmingham, England
Deal or No Deal just started here in the U.S.. The rules sound the same except our prizes go up to a million dollars as follows.
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Here is the flow of the game:
- Player picks one case for himself
- Player opens up six of the remaining 25 cases.
- Banker makes an offer.
- If player declines he opens five more of the 19 remaining cases.
- Banker makes an offer.
- If player declines he opens four more of the 14 remaining cases.
- Banker makes an offer.
- If player declines he opens three more of the 10 remaining cases.
- Banker makes an offer.
- If player declines he opens two more of the 7 remaining cases.
- Banker makes an offer.
- If player declines he opens one more of the remaining cases.
- Keep repeating steps 11 and 12 until player accepts an offer or player has the last unopened case.
The following chart plots the player’s expected value and the banker’s offer.
The most obvious thing to be learned from these three charts are that the first four to six bank offers are terrible deals. The average suitcase has $131,477.54 before any are opened. To only offer $9000 to $13000 the first stage is a deal only a fool would make. However gradually the offers get better. Game 2 shows us the expected values were almost the same as the banker offers towards the end of the game when the player’s expected value was fairly low. However in games 1 and 3 when the expected values were higher the banker apparently was trying to take advantage of the risk averse nature of most people when large amounts are involved. I don’t know if it mattered but the contestant in game 2 appeared to be a gambler who wanted to win big. Based on comments by the host, who communicates to the banker by phone, the banker does appear to take the contestants words and actions into consideration. If I were in the banker’s shoes I would act much the same.
If the player is neither risk averse nor risk prone, and also ignoring tax implications, the player should keep refusing banker offers until one exceeds the average of the remaining suitcases. For most people the progressive nature of the income tax code favors taking a deal. As I have said before I would roughly say the value of money is proportional to the log of the amount. So the more wealth you have going into the game the more inclined you should be to gamble and refuse the banker offers. With such large amounts involved, no strategy will fit everybody. However I can fairly confidently say that the player should refuse the first four to six offers and then take the offers on a case by case basis (pun intended).
Links:
You can play Deal or No Deal at NBC.com.
Archive of past shows.
Mick
That is a good question. Personally I eyeball the borderline plays based on what has already been played, especially on the second hand to be played. If forced I would say using the exceptions is better than the basic strategy only. However the way the basic strategy exceptions were created was on the initial hand only, so they are not entirely accurate for splitting, because the deck composition will be a little different.
Raekeisha from Poghkeepsie, NY
Believe your friends. Often love can cloud one’s judgement, while your friends can see more clearly see what is going on and can give good advice. Lying is also a big strike against this guy. Move on with your life.
Mike from Miami
Because it is so much fun!
Brian from Kennewick, WA
Thanks for the kind words. You are right, if the expected value is less than -1 then you should opt to take the double down surrender. As you said if the expected value for standing in my blackjack appendix 9 is less than -0.5 you should surrender, because you are betting two units. I would only add to your strategy that if the dealer hits a soft 17 then you should also surrender 17 against an ace.
Jason from Boston
I was recognized in Reno when counting once, although it may have been by dealer who knew me personally and tattled. It is a long and bizarre story. Aside from that I don’t play often, nor for high stakes, so the threat I pose is fairly small and not worth the fuss of memorizing another face.
Mark from Las Vegas
Under typical Vegas rules (6-deck, dealer hits soft 17) the house edge by always standing is 15.7%. In the short-run one could still overcome that, but in the long run you’ll lose badly.
Jaimee from New York, NY
You can’t fault him for being a nice guy. What I think is unhealthy is your jealousy.Unless don’t stop trying to keep him on a short leash I think a second breakup is inevitable.
Mike from Westfield, MA
Thanks. If your goal is to win just one unit I agree craps is the best place to start. The don’t pass is slightly better than the pass. However if you lose your first bet I would switch to blackjack. Only when exactly where you started would I go back to craps. This is because you won’t need to double or split to win just one unit, and a hit/stand only blackjack game has a house edge of about 2.5%.
