## The Details of FRP

Let's say Bob has created some digital content that he wants to sell on Upstart. Based on the time he put into the project, he decides that he wants to make \$350. Bob posts the item, and users begin placing bids. After 6,000 people have placed bids we can draw something like the first graph.
This shows how many people are willing to pay a given price on a log-log scale. It shows that there are a few people willing to pay a lot, and a lot of people willing to pay a little. For instance, only 3 people are willing to pay \$10, but 250 people are willing at least \$1. Economists call this a demand curve. We can use this information to figure out the best price at which to sell Bob's item.

Now we have added a graph of the revenue Bob would earn if the item were sold at a given price. For instance, if he charged \$1 to each person who bought it, 250 people would buy it and he would make \$250. We see there are ups and downs, and there are several prices Bob could choose that would work pretty well. However, none of the prices he might choose would generate the \$350 he was hoping for (marked with the black line)

If we wait a little longer maybe more people will place bids and he will reach his goal at the \$1 price or the ¢50 price. But, if we allow some degree of inequality in payments, we can facilitate the sale much earlier. On this last plot we have added one more curve: the amount of revenue Bob would generate if everyone price paid their full bid. We see that there is more than enough money available -- almost \$800, which far exceeds the \$350 Bob wants -- but this means people would be paying different amounts. For instance, if 1,250 people pay ¢10 or more, with an average payment of ¢28, we could reach the revenue target right now! This allows for a much larger audience than the 250 people paying \$1, and everyone pays less in the process. But how much inequality in payment is acceptable? If you sold it to all 6,000 proplr paying ¢1 with an average of ¢5.83, the relative burden is greater on those who bid more. What if one person bid a really small amount like ¢0.01, but technically paid. Should they get access to Bob's item?

We assess this by comparing the ratio of uniform revenues at two different prices. “If everyone paid ¢10, we still get more than 50% of the revenue that we would get if everyone paid \$1.” This value of 0.5 we might call the fairness factor (f) and could be decided by the creator when they post the item. A value of f = 1 would imply that everyone must pay strictly the same amount, while a value of f = 0 would imply that people can pay basically zero and still get access when the item reaches its target. As you lower the fairness factor, you are allowing for a more uneven distribution of the burden but allowing the sale to happen sooner. Presently we plan to let the creator choose their value of f, since we don't have an objective basis to choose a value. It is clear that the subjective value of art and the marginal utility of money is different among individuals, and thus it is reasonable to allow for some people to pay more than others if they are willing to do so, especially since it ultimately increases the audience and lowers the price for everyone. This is balanced against the incentive created for people to bid lower if they know that they might get it cheaper. We will default to a value of f = 0.5. and see how it works as the website grows.

After the collective purchase occurs, the creator is paid. They can move on with their life, on to the next project. The algorithm still runs, trying to bring the content to more people. Now there is a specific price at which the item is sold. Recall that there may still be people who bid earlier who do not own the item, since the price is above what they bid. Each new user who comes along has the option to buy at the current price to get access immediately, or to bid a lower amount as before. The revenue from each additional sale does not go to the creator, who already reached their goal. Instead, it is redistributed the other users who already bought the item, thus lowering the effective price for everyone. In the case of f = 1 (everyone paid the same amount) the revenue from each new sale is precisely enough to refund other users and lower the effective price to a new value of R / N, where R is the revenue target and N is the number of buyers. If the sale was nonuniform, then the new revenue is redistributed so that the price falls more slowly than R/N but refunds proportionally to the outstanding burden of those who paid more. Occasionally a new sale might bring the price down enough to “unlock” other existing bids, and thus facilitate multiple sales simultaneously and lower the price still further. In the long term, an item may plateau at some price. Or, it may continue to fall until it is essentially free. The website becomes a collection of content which is maximally available to users.