Bonding curve simulation using Incentivai

Results of collaboration with Ocean Protocol

Tokenised ecosystems (e.g. curation markets) require effective mechanisms for distribution of their native tokens. Bonding curves are a promising continuous token model where early adoption is incentivised and price is adjusted based on current supply.

It is however extremely difficult to reason about the likely user behaviour resulting from a particular bonding curve setup. We believe that modelling and simulating the system can provide invaluable insights and guide the design process.

1. Introduction

This report discusses the results of testing a Bonding Curve setup using the Incentivai simulation tool. The purpose of the analysis is to observe the behaviour of greedy Machine Learning agents to identify behaviour patterns likely to occur once the system is deployed to real users.

This introductory analysis focuses on the behaviour of greedy speculators (users seeking profit rather than purchasing asset tokens for their functionality) and the overall characteristics of the system. Follow-up analysis should include and not be limited to:

  • model for utility of the asset token (usage and potential intrinsic value)
  • prevalence and characteristics of possible front-running attacks
  • alternative characteristics and parametrisation of the bonding curve
  • embedding the bonding curve simulation as a component of the large system being designed

The plan of this report is as follows. The next section summarises key findings and conclusions. Sections 3 and 4 introduce main concepts behind token bonding curve and the Incentivai simulation setup, respectively. Experimentation results are discussed in section 5 with a focus on agent strategies in section 6. Finally, visualisations of 5 simulation runs are included in the Appendix.

2. Key findings

  • Asset token price consistently reaches and oscillates around the value of 50,000 (given the bonding curve setup and assumed users’ budget)
  • Asset token price is more volatile when its initial distribution is uneven

3. Bonding curve

Quadratic bonding curve

Bonding curve is a token pricing model where the buy price, tp, is determined by the current supply, tc, of the new token (source). The new token is referred to as asset token and its price is relative to the reference token which is assumed to be the means of exchange. In this analysis, the curve follows equation:

Therefore, the cost of purchasing volume v tokens can be determined as the relevant area under the bonding curve from equation:

Speculators may take advantage of the system by buying into the asset token, waiting for others to follow and exiting their position. This attack is sometimes referred to as pump and dump since speculators attempt to artificially increase the token price which drops substantially as soon as they sell back.

Sections 5 and 6 analyse this kind of behaviour in more detail. The main consideration is to find the right time to exit when token price is unlikely to inflate any further.

4. Simulation setup

The bonding curve system is represented as an environment where ML agents can observe its current state and possible actions: no action (skip) or buying/selling tokens subject to their budget constraints. Agents estimate which action is the most beneficial to them based on their objective which in the greedy case is to accumulate as much wealth as possible (be it in asset or reference tokens).

The simulation setup allows for observing the behaviour of the system as a whole and, in particular, the evolution of asset token price with time. Additionally, the Incentivai tool gives insights into strategies considered by agents by looking at their confidence in choice of optimal action and other actions that have been considered (see section 6).

Simulation assumes two types of users taking part:

  • Agents who follow simple heuristics and form the background for the analysis. They are willing to buy into the system but are constraint by their budget and unlikely to take on significant risks.
  • Machine Learning agents who make decisions based on the current state of the market and their estimate of what is likely to happen in the future. They are willing to take on risks and speculate as long as the estimated consequences of their actions are beneficial to them.

5. Results

Quadratic curve is an aggressive pricing model where a lot can be gained by speculators who manage to buy in with significant volume at a low price and then sell at the right time. Scenarios A and B show different simulation realisations of the same initial setup.

What one can observe is that the behaviour of users who are the first ones to participate in the bonding curve scheme has a big influence on subsequent system performance. In particular, uneven token distribution makes the token price more volatile. The asset token price moves around a similar equilibrium (50,000) yet with a larger amplitude.

5.1 Scenario A

Figure 2 presents scenario A where one user (agent 18) manages to buy 100 tokens early on. The blue curve represents asset token price changing over time. Important events (high-volume transactions of volume 30 or above) are marked on the chart.

Scenario A

In the scenario above, agent 18 manages to take control of large proportion of tokens early on. This results in the token price being very volatile since the agent is able to single-handedly cause large price movements.

Agent 18 sells most of their asset tokens early on and then decides to enter another large position (volume 50). Once token price inflates further, they are able to exit and finally sell most of the rest of their tokens later (volume 60). Each of these high-volume transactions significantly moves the price.

  • Sell 70 in timestep 101
  • Buy 50 in timestep 137 and sell 60 in timestep 240
  • Sell 60 in timestep 347

As a result of this uneven asset token distribution, its price moves significantly up and down the 50,000 value reaching maximum around 70,000 and dropping to as low as 30,000.

5.2 Scenario B

In scenario B (figure 3), 5 different users buy in early on which results in a more even distribution of the asset token.

Scenario B

It is still possible to observe similar enter-exit schemes performed by individual agents. They are however of lower magnitude and spread among different users rather than dominated by one:

  • Agent 19: Buy 30 in timestep 6 and sell 30 in timestep 46
  • Agent 16: Buy 30 in timestep 333 and sell 30 in timestep 340

As a result, asset token price is less volatile when oscillating around the value of 50,000.

6. Analysis of agents’ strategies

The Incentivai simulation tool uses custom code thanks to which it is white box. It is possible, to some extent, to unpack and understand decisions taken by agents. They approximate their optimal strategy based on what is likely to follow in the future before they choose what to do now. This gives insights into why an agent decided to take a particular action.

6.1 Scenario A

In scenario A (figure 2) timestep 101, agent 18 decides to sell 70 out of their 100 asset tokens. The table below gives insights into that decision by showing the top 3 short-term strategies considered by the agent. As an example, strategy I would involve selling 70 asset tokens immediately and 30 at the next occasion (subject to agent’s trading frequency).

All three top strategies involved selling almost all tokens very soon. Interestingly, strategies II and III suggested holding off, presumably in the hope that other users would purchase tokens and increase the price in the meantime. Agent estimated that the risk involved would be too high and chose to sell 70 asset tokens immediately.

6.2 Scenario B

In scenario B (figure 3) agent 16 buys 40 asset tokens early on (timestep 4) and does not make a high-volume sell until timestep 199. We can analyse decisions made along the way that have led to holding rather than selling early.

At timestep 7, the price has already reached 36, 100 but agent’s estimates suggest that it is likely to go further up. In fact, the optimal strategy they have found is to buy 5 more asset tokens with the plan to hold and sell 40 later.

At timestep 86 when the price has gone up further still, agent’s decision is still to hold but there is less confidence around it.

Finally, at timestep 199 when price has crossed the 55,000 mark, selling immediately becomes the estimated most beneficial action to take.


Modelling and simulation with ML agents allows for estimating the likely behaviour of users interacting with the bonding curve system. Greedy agents where found to buy tokens and exit when the price was estimated as unlikely to increase further. This resulted in price volatility which was particularly high when token distribution among users was uneven.


Scenario A
Scenario B
Scenario C
Scenario D
Scenario E

PDF version of this report can be downloaded here.

If you found that analysis interesting, please follow Incentivai or visit the website to learn more. If you and your team would like to use the Incentivai simulation tool for your analysis, please reach out.


I would like to take this opportunity to thank Trent McConaghy, Dimitri De Jonghe & Fang Gong from the Ocean Protocol team for their input and support. Also, I found the insights and resources on bonding curve models published by Simon de la Rouviere invaluable.