## Making change in Zimbabwe without coins

This is a message in a bottle.  I’ve got an idea for addressing a problem in Zimbabwe and no idea how to reach the people I’d like to share it with, so I’m going to see if perhaps it can propagate to them.

A couple days ago, the New York Times had an article on an unusual problem in Zimbabwe: lack of coinage.  The country has adopted the US dollar to deal with hyperinflation, and dollars are in common circulation.  But coins, which are a lot heavier and harder to ship, are in short supply, which poses a problem any time people make a non-whole-dollar purchase.

One blog post suggested a switch to electronic currency using mobile phones.  I’ve no idea how well such technology has penetrated Zimbabwe.  So I wanted to suggest a lower-tech solution.

There’s a pretty simple trick, related to a lottery, for “rounding off” purchase prices that could eliminate the need for change.  I’ve seen it in print several times, though I can’t find a reference at the moment.  Suppose you want to make a purchase of 33 cents.  You pick a random number between 0 and 99.  If the number is less than 33, you pay a dollar; otherwise you pay nothing.  On average, you pay 33 cents.  But in fact, you pay either 0 or 1 dollars—no coins are needed.  More generally, if you owe k cents, you pay a dollar if your random number is below k; otherwise you pay nothing.

Of course, on a single purchase, you might be concerned about the possibility that you’ll be overpaying by a factor of 3.  Indeed, it could happen that you pay a dollar when you only owe a penny (a 1 in 100 chance).  And conversely, the store owner might worry about being paid nothing.  However, over a large number of purchases, these outcomes will average out—each customer will spend about as much as they were supposed to, and each store will receive about as much as it was supposed to.

To implement this, all you need is an acceptable way to generate a random number.  It needs to be low tech.  And it needs to be acceptable to both customer and merchant—each should be confident that the other cannot cheat them.

So here’s one low-tech approach that could work.  Each participant keeps a small deck of 100 cards (or other random scraps of paper) in their pocket, numbered from 0 to 99.  To work out a payment, each participant reaches into their pocket, grabs a random card, and lays it facedown on the table.  Together, the participants then flip their cards.  Compute the sum of the two numbers shown, subtract 100 if it’s larger than 100, and use the resulting number (which will be between 0 and 99) as the random choice for the change protocol.

This is a good protocol because it works even if one party cheats.  To see this, suppose that the customer is playing fair, and truly choosing a random 0-99 card from their pocket.  Then it doesn’t matter what the merchant does.  For suppose the merchant chooses card k (possibly in some biased, cheating way).  Then, since the customer is choosing randomly, the resulting sum is equally likely to be any on of the numbers between k and k+99.  Once we subtract 100 from the larger outcomes, we find all numbers between 0 and 99 equally likely, as desired.

As a simplification, note that the customer can make their choice at home, putting just a few random cards (or, just as effective, a few scraps of paper with random numbers written on them) in their pocket in the morning and using them for the days purchases.  So long as they don’t purchase from the same merchant twice, the merchant has no information they can use to cheat the customer (conversely, if a customer returns and uses the same card, a cheating merchant can take advantage of that knowledge to ensure they get paid).   As another variant, the merchant could keep two decks, one to be used by the customer and one by the merchant.  The customer will want to inspect their deck, but this would be pretty easy if the cards were kept sorted.

Is this practical?  I’m not sure.  But the “only” thing it needs is some paper.  Plus some basic mathematical skill, but, importantly, the same mathematical skill needed to compute the cost of two items—a skill I think we can safely assume of anyone who is making purchases.

I’d love to float this idea past someone closer to Zimbabwe, who might be able to comment on its feasibility.  If you know someone, please pass it on!