Pricing fashion: DeFi devs seek the perfect mathematical curve for limited-edition goods

A nascent effort is underway to bridge the world of Ethereum and consumer goods that appeal to the Gen Z crowd. And beneath the surface, developers are exploring how to leverage decentralized technologies and mathematical models to best build those marketplaces.

Last week, an Ethereum-based startup called Foundation launched its first fashion marketplace, sporting a minimalist website and attention-grabbing gradients that speak to the aesthetics of a young and digital-centric generation.

Behind such hype culture-inspired design is a growing field in Ethereum-based applications, where startups like Foundation are building markets that automatically price items with algorithms borrowed from DeFi known as Automatic Market Makers. Yet selling physical goods is different from exchanging tokens, and some features unique to art trading may make a direct copy-and-paste unsuitable. 

"You want to find a curve that rewards early backers of a project… and in an ideal world, you want to have a curve that finds the true market price as soon as possible and maximizes the amount of dollars captured in that project for the creators. What that looks like exactly is something that we are going to learn over time," said Jacob Horne, co-founder of Zora. Last month, Zora hosted the launch of a tokenized, limited-edition cassette by Grammy-winning DJ André Allen Anjos, known popularly as RAC.

To price a sock

The prototype of a DeFi-enabled merchandise market is Unisocks, a project led by the Uniswap team to sell graphic socks represented by SOCKS tokens. 

Unisocks makes use of Uniswap's existing market-making protocol. Like other tokens on Uniswap, SOCKS are traded in a constant product market that follows the formula x*y=k, with x being the amounts of SOCKS,  y the token SOCKS is quoted in, and k as a constant. The price of each pair of socks is expressed as the ratio between x and y while k remains the same.

Although Uniswap's protocol has many desirable properties suitable for exchanging tokens, there are some complications that an x*y=k curve may not be able to address when it comes to the trade of limited-edition goods.

First, the nature of "limited-edition" dictates that there will not be much liquidity on the market. With the x*y=k curve, the total liquidity is reflected as the constant k. The less liquidity in the market, the smaller k would be — thus creating the condition for the market to become potentially more volatile.

A comparison between the price movements of SOCKS and Saint Fame's FAME token, which also uses Uniswap to create a market for selling limited-edition T-shirts, highlights this observation. 

FAME's three-week rolling volatility is consistently higher than SOCKS, surging to its all-time high around 100 days after its launch. Meanwhile, FAME has an initial supply of 100 — five times less than that of SOCKS. As such, FAME's constant k is less than that of SOCKS's.

Mathematically, the smaller the constant k, the fewer x tokens are needed to move the same amount of y tokens in or out of the market, and therefore, even a small trade size can cause a substantial price change.

As such, doing a large trade becomes more expensive when k is low, since the listed market price diverges more from the price at which the trade actually settles due to the large order. 

Additionally, a constant product market also presents designers with tremendous difficulties when selling the last few items of a collection. This is because with a x*y=k curve, the price increases quadratically as demand surges, which may make the last few items too expensive to find any buyers. 

"The main goal that you are looking for in this use case [of limited edition good sales] is the ability for the tokens to actually sell out. For Uniswap it is very difficult, almost impossible for the last token to be sold," said Horne. 

"That can be great because it could turn out that the goods being sold are in fact worth that much, but for a lot of time maybe that's not the case. So we have been looking at curves that can allow for the ability to sell out," he added.  

Picking the right curve

To this end, both Foundation and Zora are exploring bonding curves — rather than a constant product function — to price tokens.

Bonding curves define the relationship between token supply and price, in which the price rises as more tokens are issued. In 2017, Simon de la Rouviere first proposed this concept, and specified at the time that this mechanism was suitable for "curation markets," where people can share things and ideas they create and attach value to them in the form of tokens. 

As de la Rouviere explained, the curve is bonding because it is designed in such a way to reward "participants for being early and buying tokens in that project." As more participants buy tokens, the token supply in the pool increases and the price goes up. Participants are also discouraged from selling tokens back too early, since the price will be unfavorable during the early stage when there are not many tokens in circulation. 

Depending on the functions being used, bonding curves can have different shapes.

A quadratic curve will continue to grow at a relatively fast pace compared to logarithmic and sigmoid curves, while the latter two see their prices increase significantly early on but smooth out as more tokens are in circulation. 

The Zora team is currently exploring the logarithmic and sigmoid curves, according to Horne. He believes that these curves could help incentivize market participants to support an art project and take advantage of the fast price increase — while also preventing dramatic, unbounded price surges later on.

Foundation, on the other hand, has yet to disclose what kind of bonding curves they are using, although CEO Kayvon Tehranian told The Block that their protocol allows designers to set minimum and maximum prices for their goods. 

"We built out our own bonding curves that let you manipulate parameters of the bonding curves to create the market dynamics you want for your goods," said Tehranian. "A lot of the markets that are being created follow the Uniswap curve, we don't believe that it makes sense for redeemable tokens." 

Still, it's early days for such applications, and the world of Ethereum-based merchandising markets is tiny and new. One can expect further experimentation in this area to find a suitable pricing curve that matches some of the elements unique to trading limited-edition goods.