Stake Weight Score

The formula for stake-weight function $S$ is expressed as follows:

$$S(w) = \frac{ \frac{1}{1 + e^{- \frac{(w - W_t)}{W_t}}} - c}{1 - c}$$

Where:

• $S$ is the stake weight factor between 0 and 1.

• $S$ strictly increases as $w$ increases.

• $w$ is a variable representing the current stake amount on the IP Asset.

• $W_t$ is a constant target stake weight for any IP Asset.

• $c=\frac{1}{1 + e}$ is simply a scaling factor to ensure $S$ starts at 0 and satisfies $\lim_{w \to \infty} S(w) = 1$

For better visualization, we graph the stake weight score instead of just the stake weight factor alone using the formula below.

$$\text{Stake Weight Score} =M \cdot S(w)$$

Assuming a root IP Asset, we set the maximum validity score $M=100$. By setting a target stake weight of $W_{t} = 10000$, we would produce a curve that:

• Score starts at 0 when there is no stake on the IP Asset.

• Before the existing stake reaches the target stake, any additional stake to the IP Asset has an amplifying effect on increasing the score. In other words, as $w$ approaches $W_t$ ​ from below, the rate of change of the score increases $\lim_{w \to W_t^-} \frac{dS(w)}{dw} > 0$.

• After the existing stake exceeds the target stake, any additional stake to the IP Asset has a diminishing effect on increasing the score. In other words, as $w$ approaches $W_t$ ​ from above, the rate of change of the score decreases $\lim_{w \to W_t^+} \frac{dS(w)}{dw} < 0$.

• The score asymptotically approaches 100 as stake weight continues to increase.

Target Stake Weight & Reward Distribution Mechanism

Consider the target staked amount for any asset as the minimum threshold required for the asset to be seen as of holding any legitimacy. Until this threshold is reached, the IP Asset's validity is highly uncertain and risky. This aspect should apply to both creators looking to innovate on the asset and stakers evaluating whether to delegate their stake towards the asset. Stakers should be especially cautious and perform detailed scrutiny before staking on such an asset. Those who do still choose to stake early prior to the target threshold is met, bear significant risk and presumably did their due diligence. Furthermore, they have had a larger influence on the stake since any additional stake during this period has an amplifying effect on increasing the score. Therefore, these early stakers should be heftily rewarded in return.

Example

In practical terms, early stakers should earn a higher yield on their stake compared to later stage stakers. For example, we can consider the early stakers’ stake weight prior to the target stake threshold (i.e. first 10,000 stake in graph) as doubled for reward calculations. Say the total incentives is 1,000 IP tokens and the total stake is 40k. The split for rewards would be:

$$\frac{1000}{(10,000 \times 2 + 30,000)} \times (10,000 \times 2) = 400 \, \text{IP} \, \text{for the 10k stake (staked before the threshold)}$$

And for the stake after the threshold:

$$600 \, \text{IP} \, \text{for the 30k stake (staked after the threshold)}.$$

This split results in a 50% higher yield for the early stakers than later stakers. Such approach achieves the following goals:

• Encourages early staking on new IP assets, helping bootstrap the legitimacy of the IP.

• Rewards early stakers with consistently higher yields than later stage stakers since they take on more risk and likely conduct more due diligence.

• Still incentivizes slightly late stakers, as the risk is less, yield is reasonable, and their participation continues to have significant impact on boosting the IP Asset's score.

• Discourages very late stakers (e.g., when the total stake exceeds 5x the target) as it's most likely not game theory optimal to join the later stage staking pool and dilute yield among everyone. Their additional stake also has minimal impact on the overall score.