α-consensus β-reward scheme for hybrid Stake-Space consensus

Creator

Created

Mar 18, 2023 9:31 PM

Goal

A consensus and reward distribution scheme with two tunable parameters trading off between Stake or Space secured consensus and rewards.

The $\alpha \in [0, 1]$ parameter defines whether the consensus is secured by Stake or Space.

The $\beta \in [0, 1]$ parameter defines the share of rewards distributed towards Stake and Space.

$\alpha = \beta = 0$ : results in a pure Proof of Stake, where the Space doesn’t benefit consensus participants.
$\alpha = \beta = 1$ : results in a pure Proof of Space, where the Stake doesn’t benefit consensus participants.
$\{\alpha = 0, \beta = 1\}$ and $\{\alpha = 1, \beta = 0\}$ are degenerate cases which do not reward consensus participant.

Definitions

$ϙ_{\textrm{stake}}$ - share of the stake for given participant
$ϙ_{\textrm{space}}$ - share of the space for given participant
$E[ϙ_{\textrm{reward}}\ |\ \textrm{winner}]$ - share of the target block reward transferred to a participant given that they won a block

\begin{align} P(\textrm{winner}) &= (1-\alpha) \cdot ϙ_{\textrm{stake}} + \alpha \cdot ϙ_{\textrm{space}} \\ E[ϙ_{\textrm{reward}}] &= (1-\beta) \cdot ϙ_{\textrm{stake}} + \beta\cdot ϙ_{\textrm{space}} \\ E[ϙ_{\textrm{reward}}] &= E[ ϙ_{\textrm{reward}}\ |\ \textrm{winner} ] \cdot P(\textrm{winner} ) \\ E[ ϙ_{\textrm{reward}}\ |\ \textrm{winner} ] &= \frac{E[ϙ_{\textrm{reward}}]}{P(\textrm{winner})} \\ E[ ϙ_{\textrm{reward}}\ |\ \textrm{winner} ] &= \frac{(1-\beta) \cdot ϙ_{\textrm{stake}} + \beta\cdot ϙ_{\textrm{space}}}{(1-\alpha) \cdot ϙ_{\textrm{stake}} + \alpha \cdot ϙ_{\textrm{space}}} \end{align}

Equations 1 and 2 define the two primary goals of the scheme, securing the protocol (by selecting winners) depending on the $\alpha$ and giving out rewards depending on $\beta$ .

Equations 3, 4 and 5 derive the share of the block reward that a given participant should receive when they win a block. When $\alpha = \beta$ the $E[ϙ_{\textrm{reward}}\ |\ \textrm{winner}]$ is equal to $1$ . Equations 4 and 5 are sound, as division by zero cannot arise because the participant cannot win a block.

Goal

A consensus and reward distribution scheme with two tunable parameters trading off between Stake or Space secured consensus and rewards.

The

\alpha \in [0, 1]

parameter defines whether the consensus is secured by Stake or Space.

The

\beta \in [0, 1]

parameter defines the share of rewards distributed towards Stake and Space.

\alpha = \beta = 0

: results in a pure Proof of Stake, where the Space doesn’t benefit consensus participants.

\alpha = \beta = 1

: results in a pure Proof of Space, where the Stake doesn’t benefit consensus participants.

\{\alpha = 0, \beta = 1\}

and

\{\alpha = 1, \beta = 0\}

are degenerate cases which do not reward consensus participant.

Definitions

ϙ_{\textrm{stake}}

- share of the stake for given participant

ϙ_{\textrm{space}}

- share of the space for given participant

E[ϙ_{\textrm{reward}}\ |\ \textrm{winner}]

- share of the target block reward transferred to a participant given that they won a block

\begin{align} P(\textrm{winner}) &= (1-\alpha) \cdot ϙ_{\textrm{stake}} + \alpha \cdot ϙ_{\textrm{space}} \\ E[ϙ_{\textrm{reward}}] &= (1-\beta) \cdot ϙ_{\textrm{stake}} + \beta\cdot ϙ_{\textrm{space}} \\ E[ϙ_{\textrm{reward}}] &= E[ ϙ_{\textrm{reward}}\ |\ \textrm{winner} ] \cdot P(\textrm{winner} ) \\ E[ ϙ_{\textrm{reward}}\ |\ \textrm{winner} ] &= \frac{E[ϙ_{\textrm{reward}}]}{P(\textrm{winner})} \\ E[ ϙ_{\textrm{reward}}\ |\ \textrm{winner} ] &= \frac{(1-\beta) \cdot ϙ_{\textrm{stake}} + \beta\cdot ϙ_{\textrm{space}}}{(1-\alpha) \cdot ϙ_{\textrm{stake}} + \alpha \cdot ϙ_{\textrm{space}}} \end{align}

Equations 1 and 2 define the two primary goals of the scheme, securing the protocol (by selecting winners) depending on the

\alpha

and giving out rewards depending on

\beta

Equations 3, 4 and 5 derive the share of the block reward that a given participant should receive when they win a block. When

\alpha = \beta

the

E[ϙ_{\textrm{reward}}\ |\ \textrm{winner}]

is equal to

1

. Equations 4 and 5 are sound, as division by zero cannot arise because the participant cannot win a block.