We’ve shown so far that even in instances where a single person has control over the endorsement behavior of the entire global economy, users of index wallets — including those endorsement dictators — still have a selfish incentive to fund public goods and to purchase from less wealthy vendors.

We used this assumption of an endorsement dictator to derive this inequality, which states that as long as the inflation (left hand term) is less than the profit on a sale, then they should make the sale.

r in this case is from the perspective of the shopkeeper after they receive the payment. The shopkeeper is thinking, “I paid $750 for this item, if I sell it for $1000 I’ll have made $250.” They have their own list of endorsements which informs what counts to them as $250. Now a new customer comes along with new money and the shopkeeper has a choice. If they legitimize the new currency by endorsing it above 0, they’ll endure inflation, but they’ll lower the price for their customer.

Both of these effects rely on the endorsement dictator assumption. The fact that inflation occurs at all is due to the fact that the shopkeeper’s endorsements are mirrored by everyone else. Meanwhile, the fact they can change their endorsement without concern for whether the next person they pay will accept it at the new face value (whether r_before is the same as r_after) is also thanks to the endorsement dictator assumption.

What happens when we loosen this assumption and instead assume people don’t have global control over endorsement behavior?

### Terminology

vendor: a person who sells something.

customer: the person who buys from a vendor

shopkeeper: the first vendor we’ll consider

supplier: the shopkeeper’s vendor. Meaning one of the supplier’s customers is the shopkeeper

To get a sense of this we can swing to the other end of things and consider a person whose endorsement is ignored, we’ll call them “mute”. They set endorsements, but no one listens to them.

If we start with this equation again:

In the case of a mute, we’re comparing inflation vs the profit less the conciliation they’ll have to pay on their next payment. The money they receive is not worth anything more that what everyone else is willing to endorse it for. For simplicity, let’s assume perfect consensus among everyone this mute might buy from.

We know that inflation is 0, because no one else is matching their endorsements.

In other words. They get to keep whatever profit they don’t pay out in conciliation in their next payment. So now that inflation is always 0, it just becomes a competition between profit and cost from conciliation. If profit is greater than conciliation loss, make the sale.

We can unpack the conciliation function as

Where now I’ll use E_i as the endorsement set by the shopkeeper’s suppliers — the people the shopkeeper will buy from.

Assuming perfect competition, this pushes the shopkeeper to increase their endorsement until their profit margin disappears (exactly what we expect of perfectly competitive markets anyway).

However, what’s more interesting is the generalization to a spectrum between mute and dictator. I’ve wanted to do this for a while but haven’t found a framing I much like. One way to approach it could be:

Start with our equation from above:

But now scale inflation by some parameter f which encodes the degree to which our shopkeeper’s endorsement is followed by everyone else in the market.

Now consilience is some function that’s dependent on (at least) how much profit was received and how much the endorsement of the market follows our shopkeeper’s endorsement.

This then tells us the two things we need to inherit from our above equation.

We need to know the resulting endorsement from the rest of the market, which is some combination of E_i and e_i, not just E_i We need to know from the supplier’s perspective what proportion of the incoming payment is in the disputed currency. Note this is not the same as the proportion in the payment as the payment is only a fraction of the total wallet’s value For replacing E_i we can linearly interpolate between e_i and E_i depending on f using . I don’t know how to justify this it just seems right For replacing p_i we need to know how valuable this wallet was from the perspective of the supplier before it received the payment. We’ll call this V. We’ll also need to know the value of the payment from the perspective of the supplier. Let’s call this Q. Q / (Q + V) is the proportion of the wallet comprised of this payment, and so the fraction of that which is comprised of the disputed currency i must be Qp_i / (Q + V)

This leaves us with this equation:

NOTE: This is incredibly wrong. shouldn’t r also be scaled?

This can be written equivalently as:

The primary reason to treat Q differently than q is that our shopkeeper considers the incoming payment to be worth q, but to the shopkeeper’s supplier it’s a totally different value depending on their endorsements.

The bummer with this approach is that now we have to know Q and V.

V is conceptually easy. It’s just

In this case we’re imagining that we inherit the market value of the shopkeeper’s wallet and then modify that value based on the supplier’s E_i and P_i.

This works while you still have markets for each currency. But those markets will start to break down once all currencies become too bundled with one another. At that point, you’ll have to use the dynamics of following itself to inform the market value of a particular currency. For now we’ll ignore that process and just assume we have an oracle with tells us the market value of a shopkeeper’s wallet.

From there, finding Q is almost identical.

The key major weakness of this approach is that it assumes homogeneity of external endorsements (as if all E_i are the same for everyone) which almost certainly is not the case. But it should be useful, and I shudder at what it will take to consider heterogenous markets. Gotta go brush up on geometric algebra and vector fields.

This new equation seems to give us our desired dynamics. If for simplicity we consider the case of a single disputed currency we get this:

Which looks a bit better after cancelling the rs:

← With some parameter assumptions percent of post mint money supply made up of new tokens m Charlie’s wallet value v wealth Charlie w_charlie wealth Charity w_charity // not used Permit endorsements greater than 100%? endorsement Charlie e_charlie external starting endorsement E perception of quantity of payment Q perception of Charlie’s wallet value V

As we change followed f (0 is mute, 1 is dictator) Post-endorsement Inflation:

Post-conciliation Profit:

Inflation and Profit as Charlie Followed More

0 I’ve picked some parameters that are particularly informative (reload the page to reset if you changed them). Here we see that around 20% followed the inflation outstrips the profit. Then around 85% followed the profit outstrips the inflation again.

It’s not guaranteed that either of these reversions occur. It’s possible to have inflation always greater than profit, or profit always greater than inflation, or situations where they only cross once (inflation catches up to profit or vice versa).

This makes perfect intuitive sense. As you’re more followed you create more inflation (or deflation if you decrease your endorsement). Inflation always grows linearly with a linear increase in being followed. If everyone follows your every move we once again have a dictator. If no one cares what you think it’s the case of a mute.

One way to dramatically complicate our analysis (which I almost don’t mention because I don’t want to be discouraging) is by pointing out that a wise player probably counts their followership as part of their wealth. After all, if you had the ability to publish a memorandum that said “Stock X in my portfolio should be doubled in value relative to all other stocks” then your ability to do that could generate significant wealth. You could lock those earnings in by increasing your endorsement of a currency right before paying someone who follows you. They would increase their endorsement similarly, and then you pay them with your overvalued currency, and then you return it to where it was originally after the payment.

This would imply that your wealth depends on your followership. This problem becomes worse when we consider that as you become wealthier people probably follow you more, because you can afford to make larger purchases, which means a supplier can sustain a lower profit margin while achieving the same or greater total profit.

We can view this feedback loop as a problem or a certain type of opportunity. Leave it up to the player to determine how much to value their wealth, including the wealth conferred by their market dominance and followership, all we really know is that if there are feedback loops between followership and wealth a wise player accounts for those as part of their wealth. Otherwise, I have no idea how to model that, so I’ll just ignore it and assume that if market dominance is part of what’s making you wealthy, then you’re wise to include that in your wealth.

I don’t know how to rule out the possibility that there’s such thing as a followed parameter that’s greater than 1 or less than 0. If we assume that 100% of the population responds to what you do, then we still have to answer the question, “how much?” If they can only mirror your endorsement 1:1 then sure, you can only have a followed value between 0 and 100%. But why can’t someone decide that any move you make in your endorsements (aka your prices) they’re going to make the opposite move? So you double your price they halve theirs. I’m not sure why they would do this, but they certainly can. On the other hand, if you double your price they could quadruple their price. Once again, I’m struggling to know why they would do this, but I can’t see why they couldn’t do it. For this reason,

Anyway, that’s a very long answer to a simple question.