Behind the beautiful models, shiny-headed charismatic host, and shaddowy banker figure lies a deceptive mathematical game. Contestants on Deal or No Deal are asked to choose one case out of 26 which are presented to them by 26 lovely models. Each case contains an amount of money between one penny and a million dollars. The case they have chosen is brought down and kept by their side, though they do not know its contents. The contestant then has the chance to open six cases other than the one he chose, thus giving him more information – by process of elimination – about what is in his case. The contestant has a clear view of a board which tells him which values remain unopened in cases held by the models. Enter the banker. There is a figure affiliated with the show, whose purpose is ostensibly to minimize the amount of money won by the contestant, known simply as the “banker.” After the contestant has opened six cases, the banker makes him an offer – offers him a sum of money to walk away from the game with, leaving the money in his case, whatever that may be. The contestant can then either choose to accept the offer and go home with the offered sum of money, or reject the offer and continue opening cases – to make deal or not. This continues until a person accepts an offer, or until all the cases have been opened. There is now a two player game involving the banker and the contestant. The banker chooses what values to offer the contestant, and the contestant chooses to either accept or reject.

The mathematics of Deal or No Deal are ostensibly simple. At each decision point, it is easy for the contestant to determine how he should best respond the offer of the “banker.” This is easily done by determining whether continuing on with the game or accepting the banker’s offer would expect to give the greater value. The expectation value of continuing the game is simply the expected amount of money the contestant will get from the unopened cases. This is simply the average of the remaining values. If this average is greater than the banker’s offer, then the player should continue the game. Unfortunately, this is not what happens. Contestants on the show seem to consistently choose the offer, even when it’s well below the expected value of the remaining cases. Francis Su explains to us in this article that there is more to this picture to be able to effectively explain the behaviour of contestants. She introduces the idea of risk into the game. For example, say you are presented with the choice of either $500 000 or the chance to win a $1 000 000? Even though the expectation value is the same in both cases, most people would likely agree that the former is the better choice. This introduces the idea of risk-adverse, risk-neutral, and risk-loving individuals. Risk-adverse individuals would take the $500 000 without question, risk-neutral people would not have a preference, and risk-loving people wouild rather take the chance for the $1 000 000. It turns out that most contestants on Deal or No Deal are risk-adverse, and this is exactly what “the banker” wants. If many risk-adverse contestants appear on the show, and they consistently take offers from the banker that are lower than the expectation value, this maximizes the value for the banker, who gets to play the game many times. So only one question remains… What would you choose?