Simplify when the existing prover wins the auction for the next period

[ggonzalez94]

This shouldn't happen often, so it is not a big priority IMO, but would still be a good mechanism to have.

Right now, if you are the prover for the current period and you are outbid the logic to kickstart the next period is started, even if you go back and beat the best bid.
This works fine from the correctness perspective, but has a few drawbacks:

• You have to deposit a livenessBond again, since your current bond is locked as stake (less capital efficient, especially if the bond is big)
• You have to go and finalize the period, which is less eficient in terms of proving costs

Ideally, if possible we should just extend the current prover's period instead of starting new one.

[xiaodino]

It sounds like a nice optimization! I have a few questions to better understand the motivation and its effects, especially as I'm interested in contributing to implementing it. 😊

To summarize my current understanding:

• When the kickstart logic for the next proving period runs, we check if the newly selected prover is the same as the previous prover. If it is, we "just extend the current prover's period" and reuse the existing liveness bond rather than requiring a new deposit, saving gas and improving capital efficiency.
• To ensure proper economic incentives, should the new bid’s offeredFee always be higher than the prover's previous bid? (I think so) This prevents the prover from intentionally increasing their bid repeatedly to delay proof submission and maximize profits. We want to incentivize prompt proof submission, not encourage delays.

I’m curious about your thoughts on this approach—especially regarding incentives and potential edge cases we might need to consider.

[ggonzalez94]

When the kickstart logic for the next proving period runs, we check if the newly selected prover is the same as the previous prover. If it is, we "just extend the current prover's period" and reuse the existing liveness bond rather than requiring a new deposit, saving gas and improving capital efficiency.

Correct, that's what I had in mind

To ensure proper economic incentives, should the new bid’s offeredFee always be higher than the prover's previous bid? (I think so) This prevents the prover from intentionally increasing their bid repeatedly to delay proof submission and maximize profits. We want to incentivize prompt proof submission, not encourage delays.

I believe this is implicit. This means they have to outbid their previous bid by at least a certain percentage, so they are not incentivized to do it randomly(only when they are outbid by another prover). Also I don't think this delays proof submission, since the livenessWindow for their publications would still be in place, meaning they can be evicted in case they have an old enough publication they haven't proven.
If you want to work on this I can assign the issue to you :)