I am not sure of how much math is involved the banker offers half of the average value of the board. All you have to do is take a look and decide when if the value of the offer is that much greater than the value of the top prize over the value of the offer.
Lets say you have a preference function that is linear. So if you go from 1$ to 2$ you are just as happy as if you go from $1,000,000 to $1,000,001. Then you should always stick with the suitcase.
If for instance you have a curve that is more logarithmic. Going from 1 to 10 is like going from 10 to 100 and 100,000 isn't much different than a million, then you should take the bankers offer early. 10,000 gets you 5 happies but 1,000,000 only gets you 7. Take the first offer almost every time.
Most people's curves fall somewhere in between. $1 is okay. $10 is a free lunch. $100 is a special treat. $1000 is christmas. $10,000 is a car. $100,000 is a house. $1,000,000 is a nice house. How much more is a house worth to you than a car? Does a nice house really matter that much more than just a house?