Cardano reward formula: total ADA supply VS total staked supply


Parameters \sigma and s and z_0 of the pledging reward formula ( are said to be given in proportion to the total amount of ADA supply.

However, when it comes to the pool performance factor, the doc is using \beta / \sigma as a measure of how “ideal” the mining pool performed. However, it seems to me that this factor is false since \sigma is computed with respect to the total ADA supply and not the total amount of staked ADA.

Indeed, I believe that the probablity for a mining pool to mine a block is “total ADA delegated to the pool” / “total ADA delegated in general” and not “total ADA delegated to the pool” / “total ADA supply”.

Could you possibly help me on that?

Thank you!

Hello @tcosmo and welcome

z0- is relative pool saturation (as in how close to max allowed the pool funding is)
s- is stake pledged by the owners (ADA promised by pool owner)
\sigma- is total amount of ADA in that stake pool

So, these 3 variables have nothing to do with total ADA supply. Maybe that’s where the confusion is.

Beta/Sigma is just pool rewards adjusted for pool performance. So, if pool produces 100% of assigned blocks, then it gets all the wards. If pool misses or doesn’t produce blocks then only % of produced blocks are counted as Beta (in value 0 to 1). This formula just adjust for actual server/pool performance.

So, the formula you linked to has nothing to do with probability of stake pool to get a block. It is only a formula on how stake pool rewards are calculated.

Formula for probability of stake pool to get a block to mint is influenced by:
amount of ADA staked (up to maximum saturation)
amount of pool owners pledge (up to maximum of 22.5% I think, may be a bit different)
and random selection process that is influenced by previous 2 ( They call it weighted average)

So, simply put. If you have fully funded stake pool with large owner pledge and good servers that don’t miss any blocks you can get max rewards. Also, you may get lucky and get extra blocks.

Hope this helps :slight_smile:

You missed that bit:

Note that z0, σ and s are all relative, so they are fractions of the total supply, as they all lie between zero and one.

Also, I linked more than the formula, I linked the entire article, which talks about mining probability, cf:

The rewards that are produced by this formula are now adjusted by pool performance: We multiply by β/σ, where β is the fraction of all blocks produced by the pool during the epoch.

Anyway, someone has already opened an issue on github because of this problem: