Is it possible to peg the value of a digital good to anything?

Recently I had a big discussion with a person in /r/Bitcoin modmail about whether or not it is possible to peg the value of a digital good to anything. The discussion came about due to them being against Mastercoin links being posted on Bitcoin subreddit. Since I think the topic of the discussion is interesting, I would like to elaborate on some examples and counterexamples.

The person I was arguing with has a strong belief that it is not possible to peg the value of a digital good to traditional currency or other items. I would argue that it is only the case under some circumstances. Lets begin...

For all intends and purposes, pegging a value of digital good to something is pretty much identical to pegging a value of any good to something, so there isn't much point in discerning those two. Similarly, in these examples currencies can be viewed as goods - we're talking only about the value of something

First of all, what does it mean to peg a value of one thing to another? The obvious answer is that one good is exchangeable for another good and vice versa at a constant rate no matter the circumstances. 100 cents are under most circumstances exchangeable for $1 for example (unless you are running Coinstar or the like). For analysis one can also try breaking down the peg of value to the minimal and maximal exchange rate - this will be useful for our analysis down the line. In order for a good to be pegged to another good, it needs to have pegged minimal and maximal exchange rate.

So, under what circumstances can one type of good be pegged to another, and under which circumstances does this fail?

So our first consideration are two finite goods - this includes gold, silver, Bitcoin or the like. There is a fixed supply of a given good and nobody can make more (barring asteroid mining). If I hold a bar of gold and a bar of silver, I can start pegging some of their value. If I know the total amount of gold out there, for example, 1 billion bars of gold, I can put my silver bar up and say that I will pay anyone wishing to trade their bar of gold 1 billionth of my bar of silver. Despite this exchange rate being ridiculous, we have just pegged the minimal exchange rate of gold to silver - as long as I can be trusted to pay up, the value of the billion bars of gold will never be smaller than 1 bar of silver. The same can happen in reverse - I can put up my bar of gold to peg the minimal value of silver. The same could be true for pegging minimal exchange rate of Bitcoin to gold - if someone is willing to pay X amount of gold for any Bitcoin they are given and they have 21 million times X amount of gold, they essentially have created a floor for Bitcoin's value. However, pegging the value in this way might not be very useful in real life, since the spread would be too big. People would only resort to this exchange during extreme situations.

Now, what happens if a person holds all of given good there is? Say, someone created MyCoin, premined themselves 21 million MyCoins and started selling them for 1 BTC a piece, as well as buying them back for 1BTC a piece. This situation is more stable - nobody would buy MyCoins for more than 1BTC, since the creator would be selling them for cheaper, and nobody would sell MyCoins for less than 1BTC, since the creator would be buying them for more. If the ratio is chosen correctly, there would always be enough MyCoins to be sold for BTC. However if there aren't enough MyCoins, the value stops being pegged the moment the creator runs out of MyCoins - since they can't sell any more of them, their price can go up through the roof according to supply and demand, and we end up with our original situation. Clearly, one can only create floors in this situation, although if one makes the floor high enough (enough of finite good and proper exchange rate), the floor will equal or exceed the ceiling and the peg will hold.

Now lets look at an opposite example - goods that are not finite.

Lets say we have two digital currencies - Likes and Upvotes - each one being created by a different companies. Both Likes and Upvotes can be created in any quantity determined by the company that owns them. Since we have a free market, Likes and Upvotes can be traded for one another freely. If someone that is not one of the two companies tries to peg the value of Likes to Upvotes, they will run out of one or the other if one of the companies decides to flood the market. However, it is possible for either one of the companies to peg the value of their currency to the other one. They need to specify an arbitrary exchange rate, be it 1:2, 2:1, 1:1 or anything, and just create enough currency for anyone turning in the opposite currency. Again, this would only peg the value one way - if you can create Likes in any amount but have only a limited amount of Upvotes, you can't be selling Upvotes forever, you will run out of stock at some point. Well, unless you start the system that only created Likes for Upvotes at a given exchange rate, then you will always have enough currency to back up the demand for cashing it in. However if that is not the case, the system can still work. If both companies would have some fixed exchange rate, then they both can have as much Likes and Upvotes as needed - they would just buy them off one another. If the ratio does not match, say 1 Like is being sold for 2 Upvotes, but 1 Upvote for 1 Like, then some crafty traders can start trading the currencies ad infinitum flooding the market with infinite currencies. However, if the ratio is identical, we have a pegged value. Both companies essentially are creating interchangeable currencies.

Obviously however, currencies that can be created at any arbitrary amount are pretty much worthless to begin with ;).

Now, for the final consideration - pegging the value of a finite good to an infinite one. In other words, gold standard, pegging Bitcoin to USD and so forth.

So the initial state of the system is blank. We have our finite good in hands of some people, but our infinite good is not yet in circulation. Only one company can create the infinite good, be it a mint or a private corporation. They decide to say, issue silver notes - one note is exchangeable for 1 gram of silver, and vice versa - 1 gram of silver is exchangeable for one silver note. They get their first buyers, they start issuing their silver notes and keeping the silver they get in exchange. If someone wants to get their metal back, they can do so as well. If the company operates well and owns all of the silver there is, their notes are still worth the same - they wouldn't go up in value in proportion to silver, since the notes can always be exchanged back to silver. The value of both silver and the notes can rise and fall, but they won't be different. If everyone cashes their notes in, they will get their silver and the value will still not change. So we can see in this example, that it is possible to peg a value of finite good to infinite good, as long as the infinite good is fully backed by the finite good.

So what happens when we have infinite good that is not backed by the finite good? In other words, what happens when someone tries to peg value of Bitcoin to USD? Well, Bitcoin is already distributed in the system, and so is USD. There are two ways to approach this situation - if one pegs the value at the current market rate, say, $100=1BTC, then again, they create a floor. If the price of Bitcoin goes above $100/BTC, whoever is selling Bitcoin at that price will sell all of their inventory and have nothing to peg. If the price goes down to $100/BTC, we hit the floor and the exchange rate is stable. Clearly, it's not possible to peg the value of finite good to infinite one this way.

What happens when one tries to "catch up" with their infinite currency and buy up the finite good to back up their infinite good? Well, it would be one heck of an inflation / deflation boom that's for sure. First, the creator of the infinite good (USD) would need to buy up as much finite good as they could for as little as possible. Then, having some amount of the finite good and knowing how much infinite good there is in circulation, they can peg the value. The amount of infinite good in circulation divided by the amount of finite good in the possession of the infinite good issuer would be the exchange rate. So if there was, say 10^12 USD in circulation and the Federal Reserve had 1 million bitcoins in their store, they would have to peg the value of 1 BTC to 1 million dollars. Only then would it be possible for the currency to stay pegged - they could print any amount of USD needed to cover people cashing in their BTC, and if people decided to turn their USD into BTC, they would still have enough to cover that too.

The situation can of course only work if there is a single entity creating the currency, or all of the entities are cooperating to keep the peg. If someone starts to churn up too much of the infinite currency without having the finite currency to back it up, everything breaks down.

All the examples presented here also only work if the entities creating infinite currencies can be trusted - if one starts printing gold certificates without backing, we turn to fractional reserve banking and have to endure inflation, if not runs on the banks.

So, to sum all up, it is possible to create a floor value for any pair of finite or infinite good or currency, but it's not possible under all circumstances to peg the value of the two. This can only happen if one of the currencies can be created as needed (or there is enough of it created with a proper exchange ratio to essentially be an infinite currency) and is fully backed by the other currency - be it finite or infinite.

So what are the examples of one good being pegged to another? Well, we have our gold standard, where gold certificates (infinite currency) are fully backed by the finite amount of gold (finite good). Soon we will have MintChip, which creates a digital Canadian Dollar currency (infinite currency) that is fully backed by the physical currency (infinite currency). Finally, we have systems like Ripple that allows anyone to issue their own IOUs (infinite currency) that can be backed by anything, be it finite or infinite currency.