Deal Or No Deal: A bored persons maths on how to play it properly

After watching some Deal or No Deal with my mum out of sheer boredom, i decided to use some python to do some simple analysis of the game theory.

Now using the much more stingy ITV boxes of 1p to 100k instead of channel 4 and simiulating 10,000 games, we can plot the expectation value (average money value of the boxes), the average money in the lowest box (in ~68% of games) and the average highest money in the boxes for each round (again in ~68% of games). If you're wondering why I chose 68% of the games, it has a statistical purpose but its too boring to explain, you can think of it as cuting off the extreme values where someone gets lucky or unlucky like still having lots of really high value boxes left or really low boxes left

Fig 1. A plot of the expectation value (average amount of money in the boxes) as the rounds go on. Scales on the y-axis amount remaining are in a log scale (i explain this later if you dont understand right now). Light blue region you can think of as risk, the bigger the region, the higher the risk.

Now many of you may not understand this graph, ill explain it in plain english below:

Firstly, ill explain the numbers on the left side. The small numbers above the right of the 10 are the number of zeros in the number eg. 10^5 is £100,000, 10^2 is £100. On the bottom is which round it is. Now you understand that, everything starts to make a lot more sense

Blue dots (the average value of the boxes):

The blue dotted line basically doesn't change through the rounds, basically the average money in the boxes is always the same no matter the round.

Green and red dotted lines:

The easiest way to think of it is basically that the bottom values in red tend to get picked off as the game goes on as we expect and vice versa for the green, the high values get picked off as well but as there are a fewer big money boxes, they are less likely to be picked off.

The light blue region (the most important one):

The risk, somewhat obvious, but as you go on, the risk of going home with less (or more) increases.

TLDR/conclusion;

If you go on deal or no deal, the banker never offers the average (or expected) value as otherwise you should always take the bankers first offer. The banker tends to lowball you for the first few rounds, 1-3. and by rounds 4 and 5 will start to give you more serious offers.

If by round 4 your values are super scattered, eg a few close to 1p and a few close to 100k, you should probably take the bankers offer, otherwise play on until round 5 or even 6.

Now this is by no means a comprehensive guide of what you should do and if someone wants to do some further analysis, please do. Some of you might find this interesting, mods take it down if you think its too mathsy for casual uk