From a forum discussion at the PM website:

The game of Monopoly with private banking

1. Place a sticker over one square (say the  Go to Jail square) saying “You have been robbed and half of your money has been taken”. When a player lands on that square he must hand half of his cash to the banker who we will hereafter refer to as the Public Banker, The Public Banker in reality represents the rest of the economy including government and also thieves.

2. One or more parties may enter the game as private bankers whose involvement in the game is by private arrangement with players with whom they do business. 

3. Private bankers do not have to enter the game as players with a piece. 

4. Private bankers may enter the game as players with a piece whenever they wish.

5. Private bankers must remain players with a piece once they have become one.

6. Whilst not a player with a piece, a private banker cannot be robbed, cannot own property and cannot collect rent.

7. Once a player with a piece, a private banker can own property and collect rent but is also subject to being robbed.

8. Records of deposits with a private bank should be held by the depositor and signed by the bank.

9. Any loan agreements and  records of repayment should be held by the creditor and signed by the debtor.

10. Default on any properly held and signed written agreement will result in bankruptcy and exit from the game.

11. In the case of the bankruptcy of a private bank, no creditors may be compensated until all depositors are fully compensated.

12. Board prices and rentals may be universally inflated by unanimous agreement of all players with pieces.

It will probably work best if there are no more than one private bank per three non-bank players.

The purpose of the game is to demonstrate that once there is insecurity there is a demand for banking and that this inevitably leaves banks with a proportion of depositors' money that they never need to use. It is of course somebody else’s money that they are supposed to be keeping safe but the game play for a private bank is about how to take advantage of that money that is never called on whilst fulfilling their active obligations.

It may take some time for private bankers to learn how to gain the upper hand but it will come from taking advantage of their customers ambitions to get ahead in the game. The above rules represent a Wild West of banking to illustrate the most basic dynamics. The game can be developed by experimenting with the following which more closely represent the status quo:

1. Private banks must always hold assets (money owed to them) to balance liabilities (money they owe). This stops them from just writing themselves up credit and spending it.

2. The Public Banker establishes a Central Bank where private banks can deposit their money and settle between each other. This enables private banks to enter the game as players with pieces without risk of losing their money.

3. The Central Bank offers to create new money to buy the assets of banks (promises to pay them money) whenever they are short of cash. This enables private banks to operate with very low levels of cash that would otherwise be far too risky.

Having demonstrated the dysfunction of the status quo, monetary reforms can be modelled:

1. A private bank is not allowed to do anything with depositors money, even the proportion that is never called on.

2. A private bank is not allowed to write up credit in a deposit without holding the equivalent amount as cash which has no other role other than to be available to the depositor.

It would even be possible to model the Positive Money transition in mid game, following a democratic vote of all players:

Rules 1 & 2 above apply following monetisation of existing deposits with private banks and transferring corresponding assets of those banks to the Public Banker. 

It is important that the game should remain competitive and interesting for all players. There will be quite a lot of play before private banks start to get on top of things but once they do, the frustration of non-bank players will increase, especially with banks supported by a Central Bank. This of course represents how things are. Conversely, with monetary reform in place there isn't much a private banker can do other than serve as a non-combatant in the same way as the Public Banker.

By the time these crises of participatory interest have been reached, everyone will have learned a great deal about the workings of money and banks and the pivotal role of the rules that are applied.

I suspect that there will be no need to tutor players in how banks do their trucks, and that keen players will find out for themselves when given the same opportunities and mandated with the same moral looseness, as players of Monopoly are. I would very much like to see this hypothesis tested.

I know that Monopoly is not the best model of an economy but it is a popular way for a group of people to pass time and with a small modification, one square that takes half of your cash, much of the workings and motivations of banking can be revealed.