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How to win Deal Or No Deal

#### Written by Deirdre Westenbrink

Have you ever wondered whether there is a winning strategy when playing Deal Or No Deal? The game show might seem like a game of chance, as you choose a suitcase at random and are betting against a banker. However, the opposite is true: you can play according to a strategy when playing Deal Or No Deal.

### What is Deal Or No Deal?

Let’s start with how the game works. In this article, we will mostly go into the US version of the game. However, the basics of the discussed strategy can also be applied to other versions of the game. On the US TV-show, a contestant is faced with 26 briefcases. In all of these briefcases, various amounts of money have been placed, which ranges from \$0.01 to \$1,000,000.

Then, the contestant has to choose one of the briefcases and cannot open it. After choosing a briefcase, the contestant plays a number of rounds. In each round, the player is asked which briefcases to open (several in the first rounds, decreasing in number in the later rounds). Then, when the briefcases are open, a banker offers the contestant a sum of money for their briefcase. The contestant is then asked a question: ‘deal or no deal?”. Hence, they can then either take the money or reject the offer and continue to play another round.

### First: some numbers

Before we dive into the winning strategy of the game, let’s start with some numbers. The expected value of the suitcase you picked in the beginning is \$131,477.54, the average of the 26 cases. This might seem a great value for your initial suitcase, but only 6 out of the 26 cases actually have greater values than the expected value. Moreover, the median of all the cases is only \$875. Furthermore, before any deal is offered, the contestant must select six cases to be eliminated. This results in the expected value of the cases at the time of the first offer ranging between \$13,420.80 and \$170,916.25 and a median ranging between \$350 and \$17,500. Hence, we have a great variety of playing conditions when the player has to make their first choice. Thus, a general strategy can be useful.

### Banker’s perspective

As this is a two player game, it is interesting to know how the banker thinks and what motivates them to make an offer before we discuss the contestant’s strategy. The banker wants to minimize the expected value of the prizes that the contestants win. As in ‘Deal Or No Deal’ multiple rounds are played within the airtime of one hour, the banker does not seek to minimize this expected value per game, but per hour. Hence, for the banker it is beneficial to let contestants play as long as possible. For example, the expected value is lower if one wins \$100,000 in an hour than if one wins \$50,000 in a quarter. Hence, the strategy of the banker is to give small offers in the beginning, motivating contestants to keep playing. What can we conclude from this? As the optimal banker only makes offers to make the expected value per hour of play as low as possible, the contestant in some way ‘pays’ a premium if they decide to end the game early.

### Contestant’s perspective

Now we move onto the contestant’s perspective. An easy strategy might be to only accept the offer of the banker if the offer is above the expected value of the offer (thus taking the average of all remaining suitcases). However, this is where game theory comes into play. Why? Because it matters how risk-averse a contestant is. A person can be risk-neutral, risk-averse and risk-loving. This can be explained using a simple example. Suppose someone is given the choice between \$50 and a 50% chance of getting \$100. The person is risk-neutral if they are indifferent between the options, risk-averse if they would take the \$50 and risk-loving if they would take the gamble. Hence, the strategy for a person who is risk-loving is to accept any offer that is higher than the expected value or else keep playing.

However, what if you are risk-averse? Now, utility theory comes into play. The risk-averse person will accept an offer that maximizes their utility and not the expected value directly. This can best be explained using an example. Suppose that a contestant has two suitcases left: \$1 and \$1,000,000. Now, the bank offers \$350,000. The contestant might now prefer the \$350,000, since that person maximizes their utility. The utility diminishes as a person gains more money. Here, the utility of the first \$350,000 was worth much more than the possibility of receiving another \$650,000.

### Monty Hall Problem

In the US version of Deal Or No Deal, an interesting component is added. When only three cases remain, the contestant has to open one of the two cases that is not their own. Then, the contestant is given the option to trade their case for the other closed suitcase that remains. At first sight, this might look like the Monty Hall problem, where the contestant has a 2/3 chance of winning when switching and a 1/3 chance of winning by not switching. However, this is not the case. In the Monty Hall problem, the host uses their secret knowledge of the prices, making sure a bad choice is always revealed. However, in Deal Or No Deal a random choice is revealed, making sure that after the first case is opened, the contestant has only a 1/2 chance of having the highest amount of money among the remaining two cases.

### Can we now go home with the jackpot?

Unfortunately, no, this was not a guide on how to get away with the jacket of 1,000,000 dollars. Rather this gave us a strategy depending on whether we are risk-loving or risk-neutral. Moreover, the optimal strategy in Deal Or No Deal is not determined by just looking at the expected value, but also by the utility function of the contestant.