Stake Decentralization
Since in PoS the probability of being elected as a block proposer or as a validator is directly proportional to validators' stakes, the distribution of such stake into the system is the last fundamental factor of PoS Blockchains decentralization.
We will say that the stake is "completely decentralized" (\(d_S = 1\)) if and only if:
- All the stake in circulation is participating in the PoS validation;
- All the validators participating in PoS validation hold the same amount of stake.
The first condition is quantified by the Stake Participation Index (\(S_P\)) while the second conditions is quantified by the Stake Inequality Index (\(S_I\)).
One could still argue that even if all the participating stake is equally distributed among validators, a PoS consenus could still be centralized due to the concentrarion of stake in a few accounts. So an additional Account Participation Index (\(A_P\)) is introduced.
Therefore:
\[ d_S = S_P \cdot S_I \cdot A_P \]
Stake Participation Index
The Stake Participation Index (\(S_P\)) measures stake participation rate in PoS validation. It can be quantified as:
\[ S_P = S_{dyn} \cdot S_{p}\]
Where:
- Stake Dynamics \(S_{dyn}\): is the fraction of ciruclating supply (\(S_c\)) over the genesis supply (\( S_g\)). It only applies to fixed genesis supply Blockchains;
- Stake Participation \(S_{p}\): is the fraction of validation supply (\(S_v\)) over the circulating supply (\( S_c\)).
\(S_P\) can theoretically range from 0 (no participation in PoS validation) to 1 (complete participation in PoS validation).
Stake Inequality Index
The Stake Inequality Index (\(S_I\)) measures validators' inequality. We can borrow from Macroeconomics well-known wealth inequality or concentration indexes, such as: the Gini Index, the Theil Indexes or the Herfindahl–Hirschman Index. Those indexes are indicators of concentration, used mainly to measure the degree of inequality in a society or the degree of competition in a given market.
Gini Index
Gini Index is a measure of statistical dispersion intended to represent the wealth inequality within a nation or any other group of people.
Given a stake distribution \(S\) over a set of \(N\) validators, the Simple Gini Index \(GI\) can be calculated as:
\[ GI(S) = \frac{2}{N} \frac{\sum_{i=1}^{N} i y_i}{\sum_{i=1}^{N} y_i} - \frac{N+1}{N} \]
where \( y_1 \le y_2 \le ... \le y_N\) are the sorted validators' stake.
\(GI\) can theoretically range from 0 (complete equality) to 1 (complete inequality), therefore it one’s complement \((1 - GI)\) is a good candidate for the \(DEX\) index.
Theil Indexes
Theil's Indexs, called Theil's L and Theil's T, measure the inequality of a stake distribution \(S\) over a set of \(N\) validators, with different sensitivities:
- Theil's L is more sensitive to differences at the lower end of the distribution (small stakes):
\[ T_L = \frac{1}{N} \sum_{i=1}^{N} \frac{y_i}{\mu} \ln{\left(\frac{y_i}{\mu} \right)} \]
- Theil's T is more sensitive to differences at the top of the distribution (large stakes):
\[ T_T = \frac{1}{N} \sum_{i=1}^{N} \ln{\left(\frac{\mu}{y_i}\right)} \]
where \(y_i\) is the validator's stake and \(\mu\) is the mean stake:
\[ \mu = \frac{1}{N} \sum_{i=1}^{N} y_i\]
Both \(T_L\) and \(T_T\) can theoretically range from 0 (complete equality) to \( +\infty \) (complete inequality) and represent two different evaluations of inequality, depending on what we tend to consider worse: having even a few small stakes among many large ones or having even a few large stakes among many small ones.
Theil’s Indexes are not upper bounded, so are not good candidates for the \(DEX\) index. For sake of completeness we will evaluate them separately.
Herfindahl–Hirschman Index
Herfindahl–Hirschman Index is another indicator of concentration, mainly used to measure the degree of competition in a given market.
Given a stake distribution \(S\) over a set of \(N\) validators, the \(HHI\) index can be calculated as:
\[ HHI(S) = \sum_{i=1}^{N} {s_i}^2 \]
where \(s_i\) is the stake share of validator \(i\) in the total validating stake \(S\).
\(HHI\) can theoretically range from 0 (perfectly competitive market) to 1 (monopoly), therefore it one’s complement \((1 - HHI)\) is a good candidate for the \(DEX\) index.
The \(HHI\) implicitly considers both the stake distribution among the validators accounts and the absolute number of validators accounts. Therefore, in order to avoid double counting of the account participation factor, \(A_P = 1\) when \(S_I = 1 - HHI\).
Example 1
The largest validator holds 80% of the stake, the next 5 largest validators hold 2% each, the reminder is equally distributed among 10 validators:
\[ HHI = {0.80}^2 + 5 \cdot {0.02}^2 + 10 \cdot {0.01}^2 = 0.643 \]
Example 2
The 6 largest validators hold 15% of the stake each, the reminder is equally distributed among 10 validators:
\[ HHI = 6 \cdot {0.15}^2 + 10 \cdot {0.01}^2 = 0.136 \]
Example 3
All the stake is equally distributed among 20 validators:
\[ HHI = 20 \cdot {0.05}^2 = 0.005 \]
Example 4
All the stake is equally distributed among 100 validators:
\[ HHI = 100 \cdot {0.01}^2 = 0.001 \]
Account Participation Index
The Account Participation Index (\(A_P\)) represents the fraction of validator accounts (\(A_v\)) over the total accounts (\( A_{tot}\)):
\[ A_P = \frac{A_v}{A_{tot}} \]
It can theoretically range from 0 (no account participating in PoS validation) to 1 (all existing accounts participating in PoS validation).
In account-based Blockchains (like Algorand), accounts are identified by public keys. Permissionless and public networks are pseudonymous by design, therefore there is no way to assert if different public keys belong to different users. We assume that each public key belongs to a different participant in the PPoS consensus, with their own skin in the game.