# How much is "Any amount less than \$100" worth?

Just a silly philosophical riddle/paradox, but people seem to like to discuss these sometimes.

Suppose there were a kind of ticket which could be redeemed once for any amount the owner likes up to, but not including, \$100. (So it could be redeemed for \$40, or for \$99, or for \$99.50, or even for \$99.999993, but not \$100). This is a philosophy question, not a real-world question, so let’s go ahead and assume money is infinitely divisible (yes, in the real world, people round at some point, and don’t worry too much about arbitrage exploiting the rounding).

How much is this ticket worth?

On the one hand, it’s clearly worth at least \$40, and at least \$99, and at least \$99.50, and at least \$99.999993, and so on. So it’s worth at least as much as every value less than \$100.

On the other hand, it oughtn’t be worth as much as \$100. Because no matter what, it will eventually be redeemed for less. So it’s worth less than \$100.

But where does that leave its value?

[Note: I chose this phrasing in terms of money just because it was convenient. But it can be phrased in other terms just as easily; all that’s needed are a number of options with no best option]

Certainly, they are a thought this discussion brings to mind. But can they really save us? Let us elaborate the paradox further.

The considerations in the post above lead us to conclude this ticket is worth the maximum value less than \$100. This is not in itself contradictory; you could hope to make sense of that as \$100 - (some infinitesimal minimum positive value).

But we might argue there is no maximum value less than \$100, like so: Given any values A and B, with A < B, we might reasonably say that, for example, a ticket for a 50-50 lottery between A and B has value strictly inbetween them. And so between any two distinct values, there are others.

When the difference between \$99.9999999… and \$100 is smaller than any possible purchase (i.e. “How much for that grain of sand?” “Oh, about a trillionth of a cent.”) then you have \$100 for intents and purposes.

If you’re willing to exchange \$100 for \$99.9999, say, then someone can you trade with you to gain \$0.0001 for free. If they do this ten million times (perhaps with different people, Superman/Office Space style), they’ve suddenly gained a free thousand dollars.

You don’t need an end-good to be purchaseable for \$0.0001 for this difference to matter in the aggregate.

Of course, in the real world, finances ARE rounded. But this is a question about some idealized philosophers’ economy.

Since you’re only allowed to redeem the ticket once, it’s worth no more than \$99.99.

Anyway, that was my first thought. But since you could combine it with another purchase, any fractional amount may be sufficient to cause the total to round up, making it worth \$100.00.

In theory the mill, or 1/10 cent, is still a valid unit of money. I don’t know how you’d get credit for one.

I suppose you could accept coupons that say “cash value 1/100 of one cent”.

Since the limiting value of the coupon is \$100, it’s hard for me to see its value as anything less than that. Sure, the actual redemption is strictly less than than 100, but epsilon can be arbitrarily small, you know? No matter who holds it with the purpose of acquiring a \$99.999999999999 item, there is always potentially another person who could squeeze a little more value of it.

A real-world example would eventually reach a rounding limit, but philosophically, I can see the coupon being passed from person to person indefinitely, each new bearer anticipating that the next bearer would place a slightly higher value on it. The trick is, for me at least, that it doesn’t have to be redeemed in order for it to have value. No one has to take a loss when redeeming it. It could be like a note that is anticipated to have slightly more value to the new person than to the old person, and with everyone collectively anticipating this, it could therefore take the trading value of \$100 with no one actually deciding to redeem the note. It could work like money in that sense.

I don’t see the coupon functioning as money as problematic, since prices at the store could still adjust even on a philosophical level (or the parity between normal dollars and coupon dollars could be lost) if too many coupons were issued.

I hope that’s an answer in the spirit of the question.

Let me propose an analogue, then, to see how you would feel about that.

Suppose you had a blank check from God. You could redeem it for any finite amount of money. (Well, naturally, if you redeemed it for values far greater than the existing world money supply, there would be terrible inflation; it’s hard to say what it would even mean to have that much money. So let’s say instead, if we must, that you could redeem it for any finite amount of hard end-goods you like.)

How much is this blank check work? Is it worth infinitely much? If somehow you came into possession of infinite wealth, would you give up infinitely much of it for this blank check? But you’d be a fool to pay infinitely much for it; you’d take an infinite loss when you redeemed it.

Yet it seems, at the same time, its value surely must be at least as great as any finite amount of wealth. And thus it wouldn’t do for its value to be any particular finite value, either.

Should we treat this the same as or differently from the “Anything less than \$100” example?

I think first we need to agree on a definition of “value” or “worth.” One possible definition: The value is the highest amount an intelligent purchaser would be willing to pay for the coupon/check/item. Under this definition each would have undefined value, as there is no maximum of the set of amounts purchasers would be willing to pay.

That reminds me of the St Petersburg “Paradox”, and I instinctively feel it has a similar (practical) solution.

Maybe this is too real-worldy, but demand is always considered to be effective demand. Not just willingness to purchase, but willingness and ability. So the price of the blank check is, to you personally, whatever resources you can individually throw at it to purchase it. Liquidate all assets, and borrow absolutely as much as the lenders will lend, and that is the price to you personally. Now that’s the price you can pay, not the value, but the potential consumer surplus is going to be unbounded no matter who it is trying to make the purchase. The price is always the limit of what you can pay, whether you’re Bill Gates or a bum on the street, but the value is unbounded, also no matter who you are. I’m not sure it’s meaningful to put a number on something unbounded like that.

Saying that the cost of buying the blank check could be infinity doesn’t seem meaningful for me, even in a conversation like this, because infinity isn’t really a number, right? (I don’t think Cantor stuff applies…)

We have a blank check from God. I suppose we can add Satan’s Check Cashing Company – which is to say, a typical real-world check-cashing service – and Satan can lend to you without limit. How much do you borrow? As much as Satan is willing to lend. But even then, I can’t see infinity as an actual amount for a loan. You’re going to settle for some N for the loan to buy the check, and then you’d better cash the check for at least N + 1. We have a clear number on price, but its potential consumer surplus is unbounded, always unbounded, and therefore I can’t see that any specific number on the value has any clear meaning.

This is sort of like owning a priceless work of art. It can be incredibly valuable, but priceless in the sense that there’s no way to ascribe a particular price to it. The result is that such things are sold at auction. Their value is based on the interest and means of the potential buyers at the time of the sale. It will certainly become more valuable over time, but may sell for somewhat less tomorrow than today.

And I think it is different from the <\$100 ticket. Infinitesimal fractions of a penny would be non satis considerare.

Hm, I like this tack. Or, essentially equivalently but letting us avoid calling it “undefined”, we could define value as something like the set of things an intelligent purchaser would be willing to exchange for the item, whether or not there is a most desirable element in that set. (Although there WILL always be a most desirable element in that set; namely, the item itself…)

There still seems something rather interesting to me about the valuation of an object whose only value comes from its ability to be redeemed, and which necessarily is redeemed for less than the value it holds before being redeemed. But I don’t know what it is I want to say about or do with such a phenomenon; of course, like anything paradoxical, the thing to do is eventually get to a mental state where it’s not interesting at all, just a familiar brute fact, and perhaps the only thing I should be doing is jumping right to that state.

This is pretty much the nature of economic value. Think about whether the value of the coupon exists after the possible redemption has occurred. If the holder has the opportunity to trade and doesn’t, does he have a piece of worthless paper afterwards or something he values? If it’s just worthless paper, then the lost value is just a tax. There may be a little ex ante value lost to the person who holds the coupon, but unless it stops him from making the trade, it causes no distortion.

The paradox resolves itself if you think of what value means before and after the redemption.

This reminds me a bit of surreal numbers. But mainly I wanted to point out that you seem to be allowing the item itself (or, I should think, an equivalent item) to be among the things offered in exchange. This means you’re not limiting yourself to things with already defined values. If you do this, “value” may not necessarily be a real number.

I suggest that the limit will be when it’s not worth anyone’s time, i.e. “my time is worth more than the 0.00000000001 cent profit I can realize on this, even if I sold it to another investor one second after I bought it.”

Right, though even if we did do such imitation, there’s no reason to expect a real number, nor should we expect it to, given the intuition that this ticket is worth infinitesimally less than \$100 (since we’d be willing to trade any real number of dollars less than \$100 for this ticket, but not \$100 itself). Actually, we should probably take value as characterized by three sets: what you would prefer to trade for the object, what you would prefer the trade the object for, and what you would be indifferent between the object and. So value is just a way of referring to the object’s status with respect to some preference relation; there’s no a priori assumption that this can be measured by a real, and no problem with violations of Archimedean-ness.

The connection to surreal numbers seems apt; surreal numbers are the transfinitely inductive attempt to solve X = Dedekind cuts of Xs (without boundedness or “can’t be just above/below an existing X” requirements), and this approach takes values of objects as given by Dedekind cuts of (values of) objects.

One might still imagine some paradox with “What’s worth more? A ticket which can be exchanged for anything to which an intelligent agent should strictly prefer \$100, or a uniform lottery between such a ticket and \$100?” One could argue that an intelligent agent should strictly prefer the latter to the former, but should also find the former just as good as the latter. But at this point, it just seems like some ridiculous word game…

This. The reality of life is that even quantiities that are theoretically divisible ad infinitum have effective quantums where further division serves no serious social purpose. What can I buy for a trillionth of a cent? What would I be willing to do or part with based on a promise of being payed a trillionth of a cent? Would you be willing to open a door for someone for a payment of a trillionth of a cent? Certainly you could make much more working minimum wage at a fast food restaurant.

This would not apply if you could get arbitrarily many “any amount less than \$100” certificates. You could then, say, get a few trillion of them and then gaining or losing a millionth of a cent on each one is significant.

Also, many common substances like water, gold, or gasoline are not infinitely divisible at all, even in theory, because they are molecular or atomic in nature. If you divide a gold brick repeatedly using some method that always resulted in at least one division, you would end up with groupings of individual gold atoms, each of which cannot be further divided without making the result some substance other than gold. The same thing applies to water. Eventually, after applying your arbitrary separator for a sufficient amount of time, you’re going to get individual H2O molecules. Dividing these will result in hydrogen, oxygen, and/or hydroxide rather than water. Still, you might be able to get away with infinite division on paper e.g. by taking two “This certificate is worth a half atom of gold.” that is issued by the same person or organization and attempting to redeem them for one gold atom. Whether or not such could be accomplished depends on law and policy. Still, half a gold atom is not a substance that is made of gold.

Bolding mine. It is worth \$100. Say, for example, you take the \$100 ticket and purchase something that costs \$40 (if it is from Captain Dave, there is NO CHANGE and NO REFUNDS!) In this case you take a \$60 loss. On the other hand, if you purchase something worth \$99.9999 you take a \$0.0001 loss. The two situations are very similar, in that you are not using the full purchasing power of the money. If you wish to properly assign a value to the ticket you will use the supremum of the cost of what it can purchase.

This leads to a couple of ideas. First, its value could be the cost of the most expensive thing worth less than \$100. Second, if money is infintely divisible cheese should be too. Since you can purchase any quantity of cheese less than \$100 the value of the ticket is exactly \$100.

How does this follow?