Time Decay Factor

Time decay factor $\beta(t)$ is expressed as follows:

$$\beta(t) = \frac{1}{1 + e^{k \cdot (t - t_{\text{mid}})}}$$

Where:

• $k$ controls the rate of decay. A higher $k$ results in faster decay.

• $t_{\text{mid}}$ is the midpoint where the decay transitions from slow to fast (e.g., $t_{\text{mid}} = 5 \text{ years}$).

For example, setting $k_{1}=1$ and $t_{\text{mid}} = 5$ would produce a curve that:

• Starts close to 1 when $t = 0$.

• Slowly decreases until around year 3.

• Decays faster between years 3 and 7.

• Then slows down again, asymptotically approaching 0 by year 10.

Key Points:

• $\beta(t)$ starts decays from 1 to 0 as time progresses.

• $\beta(t)$ is non-linear.

Time-Weighted Approach

At the early stages of an IP asset, disputes are rare because the asset is often underutilized or unnoticed. As a result, the absence of disputes doesn't strongly indicate the asset's legitimacy. For this reason, we rely heavily on the stake amount as a signal of legitimacy during this period. However, as time goes on and the asset gains more attention and utilization, it starts generating economic value. Since innovation and remixing on existing IPs tend to grow exponentially, the time-weighted decay function follows a sigmoid curve rather than a linear one.

As the asset becomes more prominent, economically motivated parties (such as the true IP owner) are more likely to scrutinize it. If there are still no disputes after this point, it's more likely due to the asset's true legitimacy rather than a lack of attention. As such, the legitimacy score should increasingly depend on the absence of disputes over the stake weight. By shifting the emphasis over time from stake weight to being dispute-free, we achieve the following goals:

• Creators can establish a strong legitimacy signal on their IP Assets early on.

• Over time, creators can offer fewer incentives to maintain this legitimacy.

• Early stakers are rewarded for taking on the slashing risk and doing due diligence on verifying the IP Asset's legitimacy.

• Late stakers, who try to farm incentives with little to no risk, can't exploit the system under this time weighted decay system.