Today's introduction of the new one pound coin has nudged me into contemplation.

Isn't it a bit weird that our that society still relies on increasingly complicated metal disks as a way of exchanging value? How deeply rooted is cash in our society? Is it ever likely to be replaced completely by electronic money?

A quick consideration of amusement arcade games has now tipped me into thinking cash is here to stay. Sure – many machines can be updated with a contactless chip reader or some form of micro-payment acceptance app. But there is one arcade machine which is totally resistant to electronic money. It's the coin nudger.

The coin nudger is the machine where you drop one coin at a time into a slot, then the coin descends onto a shelf, nudging into the back of a pile of randomly stacked coins. If you do it right, (requiring expert timing, spin and possibly a degree of saliva) your coin will push coins off the front edge of the shelf to create a coin jackpot. Where would we be as a society if this game didn't exist anymore? Even my post-Millenial children dump their mobile devices when offered the chance of emptying a bucket of coins into this machine, in return for, well, a slightly smaller number of coins. I suspect dopamine may play a part in this equation (as it does in many payment transactions).

I was considering the skill and mathematics behind the 'coin nudger', and came across a geek on the Mathoverflow site (don't ask). He posed the following question:

"...Are there known limit laws for this game? If I specify a distribution of coins on the table, and then start dropping coins in randomly, what can be said about how the expected number of dropped coins fluctuates, per turn. Consequently, are there various phase transitions as a function of coin density? ...What will the distribution of coin falls look like as a function of the table width? Do the boundary conditions (the side walls and the pusher) create interesting "modes" in the coin falling distribution? I would think that this has to do with sand stacking cascades and Kardar-Pariss-Zhang growth but, do not have much experience in this area. Or perhaps this is just a simple Galton box that produces a normal distribution?"

I was impressed by the analytical rigor of the question – I image clever management consultants address tricky payments strategy questions using similar techniques all the time. One of the answers went like this:

"…I'm not a math guy so frankly I have almost zero understanding of what you are asking. But I can tell you as a person who operates these games they pay out 30-50% depending on how they are adjusted. The main mode of adjustment is the side holes where money falls into and goes into the bottom of the machine (this is the operators profit) which can be opened and closed to adjust the payout. If the hole is mostly closed they will pay out close to 50% of what they take in. If it's open they pay out around 30%.” 

You may have to ignore why a guy with no maths knowledge is browsing the maths geek site. Also you don't really need to know the maths (unless non-linear stochastic partial differential equations is your thing) to see that the demands of payment system users (say a Generation Z child) will create unusual service opportunities for Payment Service Providers (like a Coin Pusher operator).

And what about that 50% maximum payout rate? It's worth comparing it to one of those automatic coin changing machines you see in supermarkets these days. I was initially horrified to see people willingly to pay 12% to use these machines to automate the process of converting lots of small metal disks into a smaller number of bigger disks. Now though, I have reframed this as an 88% payout rate – a bargain! I will urge my children to stop visiting the Southwold Pier amusement park and spend their time instead playing with Tesco's Coinstar machine down the road in Lowestoft.

The payments industry is cursed with needing to handle a broad spectrum of behaviours – not all sensible. So predicting the end of coins or cash based on rational economics or maths is a mug's game. The fully cashless society will only happen once some gifted psychologists and sociologists join forces with maths/economist geeks and come up with subtle ways of nudging us all off.