Advancing Decentralised Funding Mechanisms

Different Fundraising Scenarios Designed With Sigmoidal Token Bonding Curves

Titian Steiger
May 29 · 11 min read
  • At Molecule, we are investigating different bonding curve set-ups as a new means for continuous fundraising around pharmaceutical discoveries.
  • Part of our research is focused on Sigmoid Functions that might be well-suited for the pricing of Intellectual Property.
  • Parameterisation of Buy & Sell Curves enables the creation of fundamentally different funding scenarios. Selection needs to be based on the underlying, real-world economics.
  • Dynamic curve adjustments are needed to account for unpredictable research outcomes.
  • Explore different curve set-ups yourself using our Interactive Dashboard.


Various projects are investigating blockchain-based financing options and Token Bonding Curves (TBCs) are emerging as a new mean for fundraising. The team around the Commons Stack and others see more potential in crowdfunding for philanthropic efforts. Aragon is building a fundraising mechanism for DAOs formed on their platform, while Continuous Organisations describe a new business model that could be applied to any for-profit enterprise.

Wilson Lau has published a great article that summarises the different TBC fundraising mechanisms and my post partly builds on his preliminary work.

At Molecule, we are investigating TBCs to fund and govern open-source, decentralised research initiatives around the development of new therapeutics. Since these undertakings are usually very capital intensive, most will require an initial fundraise to get things started. But let us keep this need in mind for later and focus on the aspect of continuous fundraising while the project is ongoing. This is one of the most significant advantages compared to traditional VC financing which depends on periodic interruptions of entrepreneurs that need to focus on finding new capital and company re-evaluations based on specific business figure snapshots.

Pricing Curve Selection

A crucial part in designing any system exploiting TBCs is choosing the right pricing function. Other projects investigate linear or polynomial curves but at the current stage, we think that Sigmoid Functions might serve our purpose best. These S-shaped curves are commonly used to model markets that stabilise after a specific period and can also be found naturally when looking at the growth of populations in new environments. It could be suitable for intellectual property or general discoveries because once a certain idea has been proven to work and its features developed in more detail, its market value should become measurable and find a stable price. Applied to the process of drug development, market participants purchase more shares along the curve as more positive data is revealed from clinical trials, or vice-versa for negative results. Ultimately, when the majority of information has been discovered and the drug is approved to go to market, the market value stabilises, enabling commercialisation and pricing.

Of course, this is an oversimplification of the economic processes involved. Furthermore, the asymptotic behaviour of these functions requires the selection of a maximum token value before actual price discovery has occurred — which also prevents late-stage investors purely benefiting from increasing token supplies — and this is a major difficulty that needs to be overcome (more about this in the last section).

Due to Solidity’s computational limitations around fractional exponents and logarithms we are currently using an algebraic sigmoid curve with parameters to control the maximum price (a), supply at inflection point (b) and slope steepness (c) according to the following functions:

The Price Function p(x) returns the price for a single token at a specific supply while the Collateral Function C(x) returns the total capital needed to mint or burn a specified amount of tokens.

Visualisation of Price and Collateral Curves for a Sigmoid Function.

Fundraising Scenarios

Capital locked up in the curve collateral can generally be regarded as a drawback and an economic inefficiency when using TBCs for automated market making. Long-term investors not interested in making profits around price speculations want their money to be put to good use and not sit around in some smart contract on the blockchain. One potential way to remove funds from the buyback reserve to pay for actual expenses or work contributed by collaborators is to use different bonding curves for buy and sell prices. Putting the Buy Curve b(x) above the Sell Curve s(x) allows each transaction to be taxed and the fee is safely withdrawn from the contract without the risk of decollateralising the curve.

Let us explain this functionality with a simple example. An investor is willing to contribute $1000 to a new research initiative. With a Tax Rate of 90%, $900 will go to a separate funding pool that is used to cover future spending while $100 will be locked up in the collateral reserve to provide liquidity for token holders that want to burn their stakes by sending them back to the bonding curve contract.

By varying buy and sell function alignment, we can come up with different fundraising scenarios and a selection of them will be explored in more detail in the upcoming sections.

The different curve arrangements are compared using the required collateral as well as two tax metrics. First, we will use the Tax Rate that relates the amount going to the funding pool and the actual buy price at a specific supply according to the following conditions:

This places the value of t(x) between 0 and 1. Next, we can directly look at the amount of collateral that goes to the funding pool compared to the total buy price, the Tax Amount: Aragon

Using exactly the same logic, we can also define metrics around the raised funds using the collateral functions B(x) and S(x) which results in the Fund Rate f(x) and the Fund Amount F(x).

Scenario 1 — Constant Transaction Taxation

One of the most simple funding scenarios is created by taxing each transaction with a specific, absolute amount. This leads to constant funding amounts that do not depend on the token supply before and after new orders.

Starting out with a set sell curve, we can derive the buy curve by simple mathematical transformation.

Plotting these functions shows the parallel buy/sell curves and a tax rate that decreases with supply.

Price/Collateral Curves and Tax/Fund Metrics For Constant Transaction Taxation.

Since every buy transaction is taxed with the same, fixed amount, the tax rate decreases sigmoidally with increasing token supply. Investment risk is independent of minted token supply since the same fixed amount could be lost for each token that is bought from the contract. In the beginning, barely any funds are put into the curve collateral pool and selling back tokens to the contract will only recuperate small amounts. Vice versa, the amount in the funding pool increases linearly and can be used to kick-off initiatives with high, initial capital requirements. By tuning the parameter k, this effect and the actual fundraising amount can be steered.

Scenario 2 — Bell Curve Taxation

Using Sigmoid Functions enables the possibility to create a rather exotic fundraising scenario by changing the inflection point of the sell curve:

This produces a bell-shaped tax rate curve meaning that only transactions at intermediate supplies pay significant fees and almost no funding can be raised in very early and late stages, which is clearly visible when looking at the fund amount in the lower right-hand corner of the below graph. The peak of the curve can be positioned using the inflection point of the buy and sell curves. The high tax ratio, in the beginning, can be explained through a small inaccuracy in the curve set-up since they do not perfectly start at the same initial price. Applicability of this arrangement might be rather limited since it might be conflicting for intermediary investors to carry all risk and funding.

Price/Collateral Curves and Tax/Fund Metrics For Bell-Shaped Transaction Taxation.

Scenario 3 — Decreasing Transaction Taxation

The first two scenarios were based on shifting the entire curves either vertically or horizontally. What we also can do is adapt either the start or end price of the token. Let us begin with sketching out the first idea. When raising the initial buy price above the sell value, the maximum price needs to be corrected so that both curves level off at the same value:

The result of the transformation is shown in the figure below.

Price/Collateral Curves and Tax/Fund Metrics For Decreasing Transaction Taxation.

This plays out very similar to Scenario 1. Most funding is raised at low token supplies and since both curves reach the same maximum price, almost no further capital will accumulate for high supplies. The buyback reserve stays low and early investors can only sell at high discounts. The tax ratio starts off high and goes down to zero which matches common risk considerations for more traditional funding models.

Scenario 4 — Increasing Transaction Taxation

The opposite scenario can be obtained by starting from an initial price close to the origin but increasing the maximum price of the buy curve by k units:

Using the definition of the tax rate, the scaling factor k can be expressed as a function of t that remains constant along the fundraising process:

In the beginning, almost no funding is raised through curve taxation. It only starts increasing when the minted supply gets close to the inflection point. This might be suitable for undertakings that start with enough capital for the initial phase but require more for future growth when adoption rates start increasing rapidly. The equations already showed that this is the only scenario where the percentage tax rate is independent of the token supply and in that sense is the opposite of Scenario 1, where every transaction pays the same absolute fee.

Price/Collateral Curves and Tax/Fund Metrics For Increasing Transaction Taxation.


Exploiting curve taxation might allow the creation of entirely new business models from which both founders and investors can profit alike. Companies are encouraged to publish new results as soon as they come since this will lead to increased attention and thus funding to further business growth. Investors on the other side can come up with continuous valuations based on this data and benefit from immediate liquidity guaranteed by the TBCs. Due to smaller ticket sizes, the markets should be opened up to the public. This gives rise to liquid crowd-financing options without ever needing costly Initial Public Offerings (IPOs).

The scenarios outlined in this post show another dimension of the deep design spaces surrounding TBCs. Let’s quickly summarise their major features:

  • Constant Transaction Taxation:
    Constant Transaction Taxation
    Sigmoidally decreasing tax rate
    Supply-independent investment risk
  • Bell Curve Taxation:
    Taxation only at intermediate supplies (depending on curve inflection points)
    Highest investment risk where curves diverge
  • Decreasing Transaction Taxation:
    Most funding raised at low token supplies
    Suitable for projects with capital needs that decrease over time
    Highest investment risk for early supporters
  • Increasing Transaction Taxation:
    Low, initial fundraising amounts
    Constant tax rate
    Capital needs and risk increase with supply

Open Questions & Future Considerations

Choosing the “right” curve placement and consequential funding behaviour will require a lot of research, modelling and testing around the underlying — and industry-dependent — economics. We are planning to apply best practices from complex system designs before any of these algorithms are used to manage actual financial assets. We are collaborating with thought leaders in this space and results will be published in future posts.

For our specific use case, the discovery & development of new therapeutics, we believe that something similar to the increasing taxation of Scenario 4 provides a good starting point. The sigmoidally increasing tax amount matches the quickly growing development costs (clinical trials are much more expensive than wet lab experiments) while the new drug candidate moves along its development pipeline. Due to the huge complexity of the different stages and the unpredictability of trial results, we either need to be able to adjust the pricing curves dynamically or only use them as a bootstrapping mechanism before shares start trading in a free market. It is unlikely that a single function will be able to match real-world IP valuation. The necessary adjustments and decisions could be governed by the token holders themselves.

To simplify the funding scenario design, we assumed a constant supply increase without people selling their tokens back to the smart contract. Every initiative will go through some rough patches where shareholders lose confidence and liquidate their assets or sell them for some profit-taking. This additional parameter of time will further complicate the curve selection and also means that the total amount of funds raised mainly corresponds with trading activity and not necessarily asset value.

Snapshot from the Interactive Fundraising Playground.

I briefly touched on the challenges that sigmoid curves bring with them in the section about the Pricing Curve Selection. First of all, the asymptotic behaviour can make late-stage supplies and valuations rigid and adjustments along the IP development life cycle become even more important. Furthermore, in all the outlined scenarios the final sell price is either equal to or below the buy price. This means that late-stage investors cannot purely benefit from increasing attention and token supplies. Once the whole market has been bought out, higher prices could be achieved in secondary markets if the demand continues growing. Another possibility to benefit from investments is tied to the new therapeutic achieving regulatory approval and generating positive cash flow. Revenue can be distributed directly to the shareholders or mechanisms such as Sponsored Burning can be applied to increase value without minting new tokens.

Having reached a solid starting point for IP pricing models, our research is currently focused on finding innovative solutions for the outlined, open questions and we are very excited to tackle these challenges together with the whole Curation Market Ecosystem. We have created an Interactive Dashboard that allows anyone to play around with these different fundraising scenarios, come up with their own conclusions and hopefully come one step closer to deciding how it might suit their use cases. Please reach out to us if you have any feedback to share, want to collaborate or are looking for help with your own designs. We are looking forward to hearing from you!

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Molecule Blog

An open source ecosystem to fund discovery of molecules and accelerate innovation.

Thanks to Molecule.

    Titian Steiger

    Written by

    Head of Product at Molecule Protocol. Product Strategy, Fintech & Token Economics.

    Molecule Blog

    An open source ecosystem to fund discovery of molecules and accelerate innovation.