XPower: Adaptive Rewards Rates

— Hasan Karahan, MSc ETH Zurich

Proof-of-Stake (PoS) Sybil protection has a serious problem: inequality and hence, centralization. The problem with proportionally rewarding stake is that richer holders have a relatively smaller expense than the poorer ones, giving the latter an unfair advantage regarding rewards. This advantage translates into a creeping centralization over the course of time.

Therefore, we propose the following scheme to ensure long-term decentralization:

• Split a population of stake holders into groups based on their stake size, i.e. discretization. Represent those groups by NFTs and then name them with the corresponding SI-prefixes: unit, kilo, mega, giga, tera, peta etc.

• To each group offer a flat rate of let’s say 1%, which would correspond to proportionality. Alternatively, use increasing reward rates such that the larger holders would have an initial incentive to upgrade their holdings to the largest possible NFT levels.

• Introduce adaptive reward rates with the goal of keeping the total income of each group equal: This simple rule avoids any group dominating the others and curbs the worst excesses of proportionality, which leads to the dystopian Pareto distribution, where 20% of the population control 80% of the resources (or worse).

The rewards can be distributed as a token, which can then be used as stake on a PoS based blockchain. Further, this centralization-resistant mechanism can be built right into the stake and reward scheme of a blockchain to offer long-term censorship-resistance.

Initial Reward Rates ¶

Below you see (on a logarithmic scale), that we have opted – in the case of XPower tokens – for increasing initial reward rates per NFT level. This super-proportionality serves as a major incentive to attract capital to the protocol. No NFTs have been minted or staked yet:

XPower NFTs: Initial Reward Rates

Minting UNIT, KILO & MEGA NFTs ¶

Now, we are going to deposit 1’048’575 XPower ODIN tokens and mint 575 UNIT, 48 KILO and 1 MEGA NFTs:

XPower NFTs: Minting UNITs, KILOs & MEGAs

Adaptive Reward Rates ¶

After minting and staking the NFTs, the reward rates self-adapt: Since, the UNIT NFTs are in a tiny minority – in terms of the deposited 575 ODIN – their rate jumped to 607.87% (from 0%). Similarly, the KILO NFTs are also in a minority – with respect to the deposited 1K ODIN. Hence, their rate increased to 7.28% (from 1%). Finally, the MEGA NFTs absolutely dominate the distribution – with 1M ODIN deposited. Therefore, their rate decreased to only 0.35% (from the initial 2%):

XPower NFTs: Adapted Reward Rates

Note: In the example above, the initial rates adapted immediately to the new stake distribution, since they have been active for only a very short time (just a few blocks). However, if the initial rates would have been prevalent for a longer duration, then the adaptation would have also taken longer!

Actual -vs- Target Rates ¶

Since now, the UNIT NFTs have a very high reward rate of 607.87%, as rational actors, we mint more of them increasing their number from 575 to 1M. Now, the number of deposited ODIN for the UNIT and MEGA groups is the same, namely 1M tokens.

Therefore, their target rates should also be the same, which with 0.68% is indeed the case. For UNIT NFTs the target rate decreased from 607.87% down to 0.68%, while for the MEGA NFTs the target rate increased from 0.35% up to 0.68%. Further, the KILO NFTs are now in a relative minority. Hence, their target rate increased from 7.28% up to 14.22%.

XPower NFTs: Actual vs Target Rates

Note: The actual rates lag behind the target rates, with the lag duration being dependent on how long the actual rates have been active for: The longer an actual rate remains at a certain value, the longer it takes to approach a new target value. For a detailed discussion regarding the mathematical relation between the actual and target rates please see our blog post on variable reward rates.

Upgrading NFTs ¶

Since now, the KILO NFTs have with 14.22% a higher target rate than the target rate of UNIT NFTs, which is 0.68%, it makes sense to upgrade the UNITs to KILOs – once, the former’s actual rate falls below that of the latter:

XPower NFTs: Upgrading to Higher Levels

Above, all UNITs have been unstaked to upgrade them to KILO NFTs, which will result in converting the 1M UNITs into 1K KILO NFTs. Further in the future, if the actual rates would change in favor of the MEGAs then the 1K KILOs could then be upgraded to 1 MEGA NFT.

Conclusion ¶

We have shown, that it is indeed possible to design a reward scheme that results long-term in a distribution that is flat enough, and where the “rich” (owners of the higher level NFTs) collectively cannot dominate the “poor” (owners of the lower level NFTs).

Further, thanks to short-term incentives the “rich” individuals always have a reason to upgrade their NFTs, and hence become differentiable from the “poor” – without the need to resort to drastic privacy-infringing measures like KYC or digital IDs.

Having such a uniform-ish distribution and then maintaining it over time is absolutely critical to ensure censorship resistance on a PoS blockchain. Otherwise, the entire system would become dominated by a few “oligarchs”, who could then arbitrarily impose their will on the rest, or who could be forced by out-of-chain powers to do so.