This question is for those who have seen the show on CNBC.Is there a formula that the "banker" uses to formulate his offer for the contestant's briefcase?I was just curious if anyone had thoughts on the matter.

I have tried to figure this out as well. At first, I thought it was simply the expected value of the briefcase amounts with equal probability in each case......in other words....just the average. It did not take long to find out that this was not it. Then I suspected that they were taking the average, and then reducing that amount by a certain percentage (5-10%). That also turned out to be wrong. I have noticed that they are sometimes above the simple average, and sometimes below it. There does not SEEM to be any consistency in the offers, but maybe I am missing something.

I've only watched this show about 3*2/3*70% (3 shows for 2/3 of the show with 70% attention) but I think the banker's offer undervalues the pot in the early rounds and approaches fair value in the later rounds. I'm sure that some dedicated fans have carefully tracked the path of revealed amounts and banker's offers to deduce the banker's algorithm. I'd also imagine that time-remaining in the show affects the offer -- if the show needs to fill a few minutes, the banker presents a low offer to keep the player playing.

I thought something like this as well. It seems that there must be some sort of subjective pricing, and not simply a math model. As with all other television, the interest is in keeping the viewers eyes locked in on the decision.

I guess that viewers at home root for a bad decision when stakes are higher ... this was discussed already here:Deal or no DealAnd the conclusions were about the same.

(I'm in the UK, so my comments refer to our version of the show!) There is a human factor in the offers made by the banker. I'm sure a fixed, fairly simple algorithm is used to produce an initial number, but then the production team are able to alter the amount as they see fit. Perhaps they want to coax the contestant with an unusually high offer, or anger them with a lower figure. Those factors are taken into consideration before the final amount is given.

I think one should consider the motivations of the various parties involved.Depending on how the show is financed, the production company may not be able to take the risk of multiple people winning $1 million (especially if there is undetected cheating). So they may insure the contestant's winnings. For US networks, they likely do not need insurance (the networks have a lot of money). In other territories, it's more likely that the producers need to purchase insurance (this is because US TV budgets are much higher than everywhere else and the networks aren't part of large conglomerates).The producers want to milk their airtime as much as possible. The more contestants passing through the show, the more money is given away. So if they keep the contestants playing longer, the producers will save money.From the insurer's point of view, it's possible they can't take that much risk and would like to hedge their bets. This might possibly explain why they would offer higher than the board average. (Maybe?? I'm not sure about this.) But otherwise, the insurer would want to increase their expected return.Contestants are stupid: The producers of the show screen contestants for likable and interesting people, to make the TV show better. This likely weeds out mathematical types. It's extremely likely that (almost all) the contestants will not maximize their expected return. So if the banker makes crappy offers, people will actually accept them and not wait out for the higher than board average offers. e.g. in Wheel of Fortune, it would make sense for contestants to keep getting letters instead of solving right away (especially if they are behind the leader)... but real people rarely do this.--- The format of the TV show is sold to other countries at a profit. Kind of like a TV franchise. It's likely that the producers of the original show share statistical data about contestant's game behaviour to try to maximize their expected return (winnings and airtime).---I've noticed that the banker doesn't try to 'screw' the contestant. Suppose two cases are left. It's possible that the banker knows what the amount is in the contestant's suitcase. If there is $1 and $1,000 on the board, the banker should really offer <$500.50 (always) / lower than the contestant's expected return. They could even offer $1 (by offering higher than this, the banker sets a precedent for the future where they might induce the contestant to take the banker's offer when they shouldn't; so the banker doesn't necessarily want to offer $1). Sometimes they break this rule (they offer higher than expected return). I'm guessing this is because the banker doesn't know what's in the contestant's case (this would safeguard against cheating / banker+contestant collusion).It would also make bad television for the banker to screw contestants by offering then $1 if it's known that $1 is in the case. So that's another possible explanation.---The explanations I've given don't really explain all the times when the banker makes offer greater than the contestant's expected return. Sometimes the board has very low numbers but the banker will still make those sorts of offers. They don't really need to hedge their risk in that case, so I'm stumped.

I think they make offers above the expected return BECAUSE ITS A GAME SHOW.Every other game show offers prizes - so why not this one.And why do you offer a prize? To make it exciting. There is nothing better for the viewer than to see a greedy player gamble away a really good offer.If a player seems and sounds like they are cautious then they will probably give low offers, but an out and out gambler won't make interesting TV unless you give him something to think about.I don't really think the cost of the prizes is a major factor in the banker's decisions at all. The host is likely to be earning more per show than the prize money...

Mike, it could be that making offers above expected return makes the show more exciting. However, I don't think the average viewer actually calculates the expected return. Most contestants it seems don't do it... I wouldn't expect the average viewer to be doing it. They'd either need a calculator with them or they'd need to be good at adding up numbers in their head and then dividing. And honestly... few people develop the skill of mental math (or game theory).---The cost of the prizes do matter. Suppose they allot $200k for the prize money for each contestant, and say there are 7 contestants (expected return is somewhere around $120k for the Canadian version; not an accurate figure). Suppose the contestants as a whole win more than the budgeted $1.4M (which can definitely happen; it would likely happen if somebody won $1M). In such a situation, the show would default on its obligation to pay and somebody would be in big trouble. To default would be really bad; if anything, the broadcaster would not want this as it might endanger their broadcast license (it looks bad on them if they broadcast a show where the contestants don't get paid). To not insure may be way too risky.Insurance might also be highly desirable as it fixes the cost of the show. The broadcaster may be obligated to spend a certain amount of $ on productions due to gov't regulation... it works out better if the production cost per episode is a fixed amount.

A possible explanation for above expected return offers is to protect the insurer against most contestants behaving rationally. Suppose rational behaviour = maximizing expected return. In that case, if the banker only made negative ER offers, then most contestants would never make a deal. This would make volatility very high, which would be bad for the insurer. Whereas if they offer above expected return offers (even only a fraction of all offers), then a rational contestant would take that offer. This would lower volatility for the insurer. The insurer might never make neutral ER offers (neutral ER compared to what's on the board) because (A) the contestant might not have the mental math skills to evaluate that well(B) there are some agency problems since it's possible that the banker might be trying to maximize his ER (e.g. screwing the contestant) and so therefore it's not clear if the offer is actually neutral ER.2- If you look at the Kelly criterion, the insurer would maximize its growth rate by reducing its volatility. So it might be sensible for the insurer to give up a little expected return in exchange for lower volatility.The Kelly criterion would also apply to the contestants... but it doesn't change much if the banker gives offers above the ER of what's on the board.

Assuming the banker doesnt know the value of the contestants case,the answer should be the expected value of the remaining n cases since the values of those n cases are posted on the boardsum values / number cases remaining =expected informationless value of any casehe varies around this number to mess with the contestant or induce a particular response.

I discussed this with Dan Goldstein (occasionally blogs here) a while ago, he pointed out that there are already many papers on this (and that was a year ago):Blavatskyy and Pogrebna [2006c], Bombardini and Trebbi [2005], Botti, Conte, DiCagno and D'Ippoliti [2006], Deck, Lee and Reyes[2006], Mulino, Scheelings, Brooks and Faff [2006], Post, van den Assem, Baltussen and Thaler [2006], De Roos and Sarafidis[2006], Baltussen, Post and van den Assem [2006], Blavatskyy and Pogrebna [2006a][2006b].See:Decision Making Under Risk in Deal or No Dealhttp://papers.ssrn.com/sol3/papers.cfm?abstract_id=881129Deal or No Deal? Decision Making under Risk in a Large-Payoff Game Showhttp://www.tinbergen.nl/discussionpapers/06009.pdfIs a Dollar in the Hand Worth Two in a Lottery? Risk Aversion and Prospect Theory in Deal or No Dealhttp://papers.ssrn.com/sol3/papers.cfm?abstract_id=877012Loss Aversion? Not with Half-a-Million on the Table!http://www.iew.unizh.ch/wp/iewwp274.pdfRisk Attitudes in Large Stake Gambles: Evidence from a Game Show http://comp.uark.edu/~cdeck/vas%20o%20no%20vas.pdfP