…and arrive at a “best bet” strategy, or are the ever changing odds too complex to do this?
I suspect that calculating the exactly correct odds would be tougher than most people could manage under the circumstances, but approximating them would be a lot easier for one trained in mental math. This is because the effect that the lower-valued prizes have on the total diminishes quickly.
A rough example: suppose the following cases are still on the board:
$1, $2, $5, $20, $100, $500, $5000, $10 000, $200 000, $400 000, $750 000, $1 000 000
There are 12 cases remaining. Your big-ticket cases are the 1-mil, the 75K, the 400K, and the 200K. These add up to 2.35 million.
2.35 million divided by 12 would be very close to 200,000. (That’s a very good position to be in!!)
estimated value comes out at $197,135.67
You don’t even have to be that bright. You use the time waiting for the banker’s offer to ballpark an average estimated return, and compare it to the offer. It’s a simple formula that someone better versed than I will be along to explain shortly.
Actually, I don’t believe that such calculations factor into the best decision very often.
I’ve only seen the show a few times, but the banker’s offer is always less than the calculated expected value (thanks to the many “strategic” commercial breaks, I had plenty of time to calculate and anticipate various outcomes in the comfort of my parlor.) The disparity generally decreases in the later stages – by design: there is more interest, drama, tension at the higher monetary values, so the producers want to linger at those choices as much as possible. It’d be a very boring show if each contestant took the “sure thing” and bowed out in the low five figures, after 5 minutes of play. On raw expected value, the right choice is almost always to continue, except in unusual cases (most people would take $450K over a coin-flip between $1M and $0.01 – it’s a great payday, and the emotional impact of losing it for a penny could be devastating on you and those close to you)
It comes down to choices based on the value (life impact) of various amounts to you. Would an expected “next bid” of $325K really mean that much more to you, emotionally or practically, than the current bid of $250? How would you feel if you declined it and ended up with nearly nothing? Therefore you have to make other calculations based on the remaining values, like “odds of Busting Out” (winning a meaningless small amount) “Odds of breaking even” (winning an amount comparable to the current bid) and a sense of what the range of likely offers will be at the next offer.
That may sound like just more math, but it is both simpler and more subtle than that: you have to assign emotional values, and I think that these can be broken down to maybe 3-4 categories. You might be thrilled to get a letter with $75K sweepstakes check today, or devastated by an equivalent IRS bill, but if you take $250K when you would have won $325K, I think you’ll manage to live with it pretty easily. For me, the emotional categories of a typical contestant’s run fall fairly easily into “fuggedaboudit”, “nice”, “wow” and “jackpot”. For me, the distinction between “wow” (somewhere in the low six figures) and “jackpot” (high six figures) isn’t really great enough to warrant a separate category, but the host always harps on the highest available possibilities, and they seemed to factor disproportionately in the thinking of the few contestants I watched.
One thing that I rarely see mentioned in discussing ‘deal or no deal’ strategy, is that the choice is not between accepting the deal, or taking home your suitcase - until the end. It’s between accepting the deal and moving on to the next stage. That is, you know that you have a series of offers coming to you, in between opening suitcases in a particular sequence. You don’t know exactly what those offers will be, but you know that they’ll be coming as long as you keep saying ‘no deal.’
This is an aspect of the strategy that’s very hard to quantify, because the banker’s offers are slightly unpredictable. When you’re looking at a particular offer, you have to not only balance that offer against the high amounts or low amounts that might be in your case, but also against the chance that the next offer might be higher or might be lower. One strategic element that I think works pretty well in this situation is to have a ‘target offer’, and to take the deal whenever the banker’s offer goes above that level, no matter what the odds are.
Why am I reminded of
“Wargames”
“The best move is not to play” ?
Of course they could, it’s just a few simple additions and a division. Theres no complex math involved.
Dunno. The contestant ALWAYS comes out ahead (and usually by thousands of dollars), so not playing doesn’t seem like a good way to go. The “losing” state of the game is actually “winning a penny” – but that’s not going to happen very often – the win will almost always be more.
Especially since the bottom numbers are noise, and can be neglected for the estimate.
To follow up on what KP said, behavioral economists have found that value is not linear, that money needed to compensate for losing something in one’s possession is not the same as the money you’d pay to get it, and that people do inconsistent net present value calculations. If I were going to be on that show, I’d figure out equivalent values - $100 K is likely more than 10 x the value of a $10 K prize. I’d say that winding up with $10 would have a big negative value, in that I’d be ashamed of screwing up so badly. Using these, I could calculate what the expected value is for me, and compare that to the banker’s value.
Of course for the first few rounds the bank underbids, so the contestants stay around. Same reason the early questions on Millionaire were so easy, and the producers on Jeopardy like when you clear the boards and show all the answers.
Expected value is not the correct criterion for a decision anyway - utility and variance are the key factors.
For example, let’s say you have five numbers left. In one case, you have $1,000,000, $1000, $500, 20, and .01. Expected value: roughly $200,000.
In the second situation, you have $500,000, $300,000, $200,000, 1000, and .01. Expected value is about the same.
In the first case, the banker offers you $200,000 to quit. You have to pick one case. The upside of this choice is that you knock out a low number, and the offer goes up to maybe $250,000. But if you pick the million, you’re wiped out. And bear in mind that you probably aren’t playing for a million dollars - you’re playing for a better deal (i.e. if you get to two cases, you’d have to be crazy to risk $500,000 to get an extra $500,000, unless you’re rich). So really, you’re risking a one in five chance of losing everything for a 4 in five chance of having your money go up by 20% or so. Or if you decide that you’re going to try to pick 2 more cases then quit, your odds of hitting the million and being wiped out go up to about 35%, and in the 65% case you’ll get an offer of maybe $350,000.
In my opinion, unless you’re already wealthy, you take the deal. The risk is too high. The added utility of another $150,000 on top of your $200,000 is nowhere near enough to compensate for the loss of utility in going from $200,000 to zero.
But look at the second situation. Here you’ve got a chance to knock out two low numbers, and the worst-case scenario is that you draw the $500,000, in which case you still have $125,000 or so. If you pick $200,000, the offer won’t change much. If you pick the two lower ones, the offer goes up substantially. And there’s no chance of being wiped out on a single pick.
So a perfectly legitimate strategy in the second case would be to say, “I’ll pick a case - if it’s $500,000, I’ll take my money and walk away. If it’s a low value, I’ll think about picking another”.
Interestingly, in keeping with what others have said, there is an argument to be made for this view after the initial stages.
There is a substantial cost associated with each stage after the first few. Game Show psychology obscures it, but I ask you to imagine that Howie Mandel announced at the end of an episode “The winner of our ‘I woulda…’ da\rawing, taken from the list of viewers identified by our Spymatic This-Means-You Viewer Monitoring System is… J. Random Straightdoper! J. Random has the option of paying us the amount the contestant just accepted/declined, and taking over the game where it left off.”
Would you, J. Random Straightdoper, take the offer? Well, if the last offer were $100 you might do it for the kicks. Even at $1K or $10K, you’d likely to consider it (if you had the disposable funds), unless “the board” was stacked against you. At some point, you probably wouldn’t settle for anything even close to a breakeven or reasonable Expected Value. You’d want an EV 2-3 times (or more) the amount you were paying. Equal pain ain’t worth equal gain, because the loss would no longer be just a sterile number.
I think most of us can imagine continuing to play even after an offer that was very significant to our lifestyle, but very few of us would invest that same amount from our bank account to pick up a game in progress, all other things being equal. Of course, this is a slightly irrational decision in mathematical terms: the offer is a legally binding promise, and by giving it up, you are giving up an asset that is yours for the asking. For equal total assets, the loss/investment is the same whether the source of the funds was an offer, a bank heist, a lifetime nest egg, a sweepstakes, blackmailing Cecil or the long lost will of a dearly departed pet.
Yet how many of us would pay $50K-100K to pick up the game at a late stage, even with a modestly favorable board? In the end, we have to live with ourselves and our decision. What’s the cash equivalent of driving yourself to drink, telling little Janie that Mommy left and she can’t go to college, or eating a bullet?
The correct answer (our choice and Meeko would seem to suggest) would be not to play. Yet, with the same bank balance (including the offer), you might continue to play – and you’d be completely rational to do so, given the typical underbidding of the offer vs. EFV. The real justification is in the psychology.
I do it as I watch the show. It’s not hard at all, actually, since you really don’t jhave to start making decisions until (at the very earliest) the fourth round of picks. I’ve gotten pretty good at quick in-my-head math, being a nutty baseball fan and all.
Sam does a very good job of explaining what the REAL decisions are. Sam doesn’t come out and say it but just bear this in mind; the theoretical ideal strategy is almost always to refuse every deal. The banker will, almost invariably, offer an amount that is below the average expected outcome. His first three offers are all ridiculous lowballs, but after that he comes up closer to the expected average outcome.
The only thing that really matters, therefore, is to figure out at what point your risk aversion forces you to get out of the game. Most of the players on the show tend to go one or two steps too far, really, in part because
A) They’re under enormous emotional pressure, by the situation, the audience, and Howie Mandel, and
B) They’re picked for it.
Mandel is a terrifically mendacious little bastard - I love the job he does on this show - because he keeps the contestants going back to what they might theoretically have in their case when what you SHOULD be thinking about is what the next deal is going to be. Assuming you are an ordinary schmoe, you just never, ever want to get down to the point where your next choice could totally torpedo your next offer.
Okay, to me, in this context, ‘not to play’ does not mean accepting an early offer and not CONTINUING to play. ‘Not to play’ means, do not ever go on the show and play it at all.
From a strict mathematical, expected-life-value-of-money standpoint, ‘not to play’ is worse than any other strategy, because you get 0 dollars. Of course, there are other considerations, (whether you think it’s worth the money to be on national television, time and effort required to send in an application to be on the show, etcetera,) that could make ‘not to play’ more worthwhile.
Well, most of us live close enough to paycheck-to-paycheck that plopping down 50-100K wouldn’t be as simple as going down to the branch bank and withdrawing a fat bankroll. I myself would pay $10K if the expected payout were $20K or more, if my suspicions about illegality were allayed (the only people to require to you pay money in order to get more money back are Nigerian spammers.)
If I were playing I’d use the same strategy. Less than $10K isn’t really worth anything to me, so I’d stay in until the offer were at least that. After that I’d fold unless the expected payout were substantially higher than the offer - around twice as high. If the offer were in the $100K range I’d take it even if the expecteed payout were $300K – $100K is all I need.
It’s interesting to note that the median value in the first case is $500, while in the second case it is $200,000. That’s a much better metric here than expected value, and explains why the second case is so much more appealing than the first.
As others have said, you can reasonably work out how the banker’s offer compares to the odds.
Sam Stone makes a useful point about how the money is distributed (one large amount remaining means you possible top prize and risk are both higher).
For myself, I would first decide how much I wanted to win. Then keep going until there are less than four boxes with this amount left. Then take the offer.
It’s pretty clear that the contestants really don’t make decisions based on an expected value criterion, so whether they can make quick calculations in their head is pretty irrelevant. And that’s good from a tv show standpoint. It’s more entertaining to see people let their emotions and irrationality rule their decisions. Until the game gets close to the end state, the deals from the banker are also set up for entertainment. The deals are ridiculously unattractive at the beginning so that the contestant would play on. As cases get eliminated, a lot of contestants are driven strictly by the remaining possibility (not probability) of a big payoff. It’s interesting to hear some of their strategies. One strategy I heard was: “As long as there is at least one unopened case with a payoff that’s larger than the banker’s offer, decline the deal.” But there is also a tendency to be more risk averse as the number of unopened cases dwindle. I really think that the popularity of the show comes from a desire to watch people acting crazy and making boneheaded decisions.
Just to be clear, when only a few cases are left, the banker may offer an amount higher than the expected value. I believe it was the May 9th (or 8th?) show, a contestant had four values left:
500
75,000
100,000
200,000
And the banker’s offer was $100,000. That’s more than $6,000 above the expected value. The contestant was funny too, he kept saying “I’m a gambler, I’m a gambler” at this point, making you think he wasn’t going to take it. Then, he took it (good move too, subsequent offers were lower and he only had the 75000 case). I guess he meant, "I’m a gambler, so I can calculate odds in my head 
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