# More on “Five Down”

by on 8 May 2010 · 2 comments

Yesterday’s puzzle “Five Down” stimulated a fair amount of discussion both in the post’s comments section and via email. I also exchanged emails on the topic with the author of Futility Closet (which is where I came across the puzzle) and he told me that the puzzle generated a lot of correspondence for him too.

All the commenters on the blog came up with the correct solution, but there are quite a few different ways of looking at the problem, all of which help provide insight into the nature of money. Since that is a common topic for this blog, I will consider some of these perspectives here.

First, the solution itself. The question asked was “What was lost in the whole transaction, and by whom?”. Taking the “whole transaction” to include the banker finding the counterfeit note in the first place, the answer is that no-one lost anything, subject to a couple of assumptions. These assumptions are that the banker actually owns the bank and so the bank’s gains or losses are his gains or losses (otherwise we would have to conclude that the banker was up \$5 and the bank was down \$5), and that the banker and his wife pool their finances (so we treat her debt with the butcher as his debt).

The first way to think of the problem is a variation of the comment from James. Imagine that the \$5 was not counterfeit at all and all the same transactions took place with a genuine note. But then imagine that when the banker closed the bank at the end of the day, taking notes and coins back to his safe, the \$5 slips from his hands and is blown into the fireplace. There it is quickly consumed by the fire. Earlier in the day, the banker had a windfall of \$5, but then he lost the same amount to the fire. He gained in the morning, lost in the evening and, although perhaps disappointed to have lost the \$5 again, he was even on the whole transaction. No-one else involved lost either as they had simply performed legitimate transactions, clearing various debts, using a valid \$5 note. The question now is, how is anyone any better or worse off in this scenario than if the note had been counterfeit all along? The answer is, they are not.

Now that approach gives the right overall answer, but it may be unsatisfying to some as it doesn’t take account of the fact that a whole series of “invalid” transactions took place with the counterfeit note. This too can be clarified. If the note had been real, then the banker made a gain when he found the note, but finding a counterfeit note involves no gain, because it is worthless. In that case, the gain for the banker comes when he is able to discharge his debt with the butcher using a worthless note. So, he is still ahead early in the day, but the timing is slightly different. With a real note, the gain is in the finding and the transaction with the butcher is a neutral fair trade (legitimate \$5 in exchange for a discharged debt). With a counterfeit note, the gain is delayed to the next step in the sequence. Of course, in receiving the counterfeit note, the butcher makes a loss. But then the butcher makes a gain when he in turn is able to discharge his debt to the farmer with a worthless note. And so on. Each person in the chain loses when they receive the \$5 but has an offsetting gain when they use it to settle a debt, leaving them whole on the transaction. The chain continues all the way back to the bank, which loses \$5 when the laundry woman settles her debt with the dodgy note. Assuming, as we are, that the bank’s loss is the banker’s loss, this simply offsets the gain the banker had when first paying the butcher. Again, everyone comes out even. Of course, if someone other than the banker had been left with the note, they would have been down \$5 and the banker up \$5. Having the transactions complete a full circle is a key part of the puzzle.

The final perspective is a more technical one. At the heart of money is the notion of a debt. Money is essentially a more convenient way of managing debts. If I buy a cow from a farmer and sell a meat pie to a patron at my restaurant, we could simply agree to record various debts: I owe the farmer one cow, the diner owes me one cow. Of course, this is inconvenient (not to mention risky) as we all have to maintain records denominated in a whole range of different commodities and I don’t really want to discharge my debt to the farmer by giving him a cow back. He has plenty already. Nevertheless, this points to the origins of money. In the excellent (if lengthy) treatise “What is Money” is it observed that “for many centuries, how many we do not know, the principal instrument of commerce was neither the coin nor the private token, but the tally”. Indeed in the Five Down puzzle, there are a whole string of tallies. Each of the players in the story has kept track of a debt due to them and one they owe to another. If the merchant did not owe the laundry woman but instead owed \$5 to the farmer, the merchant and the farmer could simply agree to cancel their debts to one another. It is not so easy when the debts extend in a longer chain. Nevertheless, if one were to assemble all the parties in a single room and ask them all to consider their respective debts discharged, they should all readily agree. After all, they all owe \$5 and all are owed \$5 and it is much easier for everyone if that effective net zero position could really be zero without the fuss of worrying about chasing debts. It would be different if someone was owed more (or less) than they owed. We might call this simultaneous discharging of all the debts “multi-lateral debt netting”. In theory it is very attractive, but in practice we cannot get everyone in the same room to get it done. Effectively, the counterfeit note serves the purpose of facilitating multi-lateral debt netting. Because everything nets out evenly in the story, the counterfeit note can achieve the netting just as effectively as real money. The extra feature real money offers is that if the netting does not quite even out, those owed more than they owe can hang on to the money and use it for netting again in the future. Not so with the counterfeit money: once it is discovered, it loses its power to work. The solution to the puzzle lies in the fact that no debts were left over.

I will end this discussion by reprinting a very similar story that one of my email correspondents sent to me (as I understand it, it is not new but has been updated to fit the times).

It’s a slow day in a dusty little Australian town. The sun is beating down and the streets are deserted. Times are tough, everybody is in debt, and everybody lives on credit.

On this particular day, a rich tourist from down south is driving through town , stops at the local motel and lays a \$100 bill on the desk saying he wants to inspect the rooms upstairs in order to pick one to spend the night in.

He gives him keys to a few rooms and as soon as the man walks upstairs, the owner grabs the \$100 bill and runs next door to pay his debt to the butcher.

The butcher takes the \$100 and runs down the street to repay his debt to the pig farmer.

The pig farmer takes the \$100 and heads off to pay his bill at the supplier of feed and fuel.

The guy at the Farmer’s Co-op takes the \$100 and runs to pay his drinks bill at the local pub.

The publican slips the money along to the local prostitute drinking at the bar , who has also been facing hard times and has had to offer him “services” on credit.

The hooker rushes to the motel and pays off her room bill to the motel owner with the \$100.

The motel proprietor then places the \$100 back on the counter so the rich traveller will not suspect anything.

At that moment the traveller comes down the stairs, picks up the \$100 bill, states that the rooms are not satisfactory, pockets the money, and leaves town.

No one produced anything. No one earned anything.

However, the whole town is now out of debt and looking to the futureĀ  with a lot more optimism.

And that, ladies and gentlemen, is how the Australian Government’s stimulus package works!!!

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1 Luis May 8, 2010 at 9:41 pm

Credit risk is missing in this solution. The banker, by missing the fact that the note is not real, puts the butcher at risk of discovering the bill is fake. Just because it turned out that nobody noticed until the bill returned to the banker, it does not mean each person in the chain did not carry the credit risk on the banker during the process (whoever discovers the bill is fake would need to trace the chain back to the banker – not necessarily possible, and then the banker would have to agree/admit the bad bill was his, and then the banker would have to come up with a real \$5 bill and he may not have one). So the bankers owes everybody along the chain, he benefited by having ‘borrowed’ \$5 from whoever was holding the fake note (the fake note was effectively a very hard to enforce IOU from the banker). Another way to think about this is: If the banker had 2 bills, one fake and one real, and we gave him the choice to use either knowing it will go back to him at the end of the day, should he be indifferent about which bill to use? No, he would be better off using the fake bill EVEN knowing it will come back to him, because he then gets to play with the real bill for the day – a net benefit.

2 Stubborn Mule May 9, 2010 at 3:05 pm

Luis: credit risk is an important consideration here. In fact, that is a big part of the reason why all the participants should be happy to have multi-lateral netting. While someone owes them something, they have credit risk to their debtor and getting rid of this without any losses is a good outcome for all of them. I’m not sure I’d agree thought that everyone along the change really has credit risk to the banker if he starts circulating the counterfeit \$5. If the farmer accepted the note and then realised it was fake, he would have to take it up with the butcher not the banker, whether or not the butcher in turns decides to then follow up the banker.

The scenario you propose in which the banker has a choice between paying with a good note and a fake one is a good example of a phenomenon rather like Gresham’s Law. It reminds me of a common experience for Australians: discovering someone has given you a New Zealand 20 cent coin your change (it is worth somewhat less than an Australian 20 cent coin and no-one is obliged to accept it). Most people in this situation would be inclined to pass that Kiwi coin on as soon as possible (if they can) in preference to using an Australian coin.

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