We can learn a lot about voting systems from the existing literature.
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Edit: You can now see MolochDAO’s Banzhaf power index in real time. — JB 11/16/2019.
Introduction
Earlier this year, I published some analysis on a model measuring relative value of a voter’s stake in an on-chain governance system. Ultimately, analyzing the governance of MolochDAO made me realize that as an industry we still have a lot to learn about the mathematics of voting. Moloch is a great experiment with some nontrivial learnings and several important contributions, including ragequitting, which prevents the system from capturing a voter’s capital against their will. But even with these innovations, we are quite far from working voting systems. As for my model, it turns out I inadvertently re-created a version of the Banzhaf power index, which I cover below. Sadly, I implemented my index much less efficiently than Mr. Banzhaf— alas, if only I had read more of the literature!
These days in blockchain, discourse has shifted from valuing of governance tokens to debating whether on-chain plutocracy is acceptable and, now, even how 1T1V (one token, one vote) voting systems influence community psychology.
We are yet to acknowledge in blockchain that voting systems are a well-understood problem backed by accessible mathematics and several hundred years of academic research. Alas, if we only read more of the literature!
Weighted voting systems
Just about every blockchain-based voting system that we know about is a special case of a weighted voting system (WVS). It’s natural that blockchain systems have gravitated to weighted voting — it’s the simplest system to implement in the absence of a great Sybil resistance mechanism. Having such a mechanism would allow us to implement more elegant 1P1V (one person, one vote) systems.
WVS’s have been used as a mathematical construct extensively in the 20th Century to measure actual voting systems, such as the American Electoral College and many others. In such a system, each individual in a group of voters is given a number of votes and vote yea or nay on a proposal. The quota, usually denoted q in the literature, is the number of yea votes needed for the proposal to pass. The quota is somewhere between half the votes (majority rule) and the entire set of votes (total consensus).
To demonstrate WVS, we can turn to basic theory. (Here is a basic primer from Carleton College.) We can describe a weighted voting system using a notation that looks like this: [5; 2, 2, 2, 1]. In this WVS, there are 4 voters with voting weights 2, 2, 2, and 1 respectively and a total of 7 votes. Here, 5 is the quota. This notation captures all information necessary to analyze a weighted voting system.
To put this in ‘blockchain’ terms, here is the same notation for MolochDAO as of November 2, 2019:
[3522; 1000, 1000, 250, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 72, 50, 50, 35, 25, 16, 10, 10, 5, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
In Moloch, 70 members have a combined 7,042 votes and would need to hit a quota of 3,522 (7042/2+1) votes to pass a proposal. This assumes full voter turnout, and though Moloch prides itself on being quorumless — that is, there is no minimum voter turnout needed to pass proposals — this property actually tends to further disenfranchise individual voters in practice.
So how do we measure the voting power of an individual in a WVS, given a distribution of voting weights?
The Banzhaf power index
John F. Banzhaf, a mathematically-minded law professor then at Rutgers Law School, formalized this notion in the 1960s by considering the probability that a particular voter in a WVS will swing a vote. This measure is now known the Banzhaf power index and efficient algorithms for its computation have since been described.
To take a simple example, let’s consider the WVS from above described by the notation [5; 2, 2, 2, 1] and calculate its Banzhaf indices.
Normalized and absolute Banzhaf power indices for the system [5; 2, 2, 2, 1].
First, note that our calculations come in “normalized” and “absolute” form. The absolute Banzhaf index (also known as the Penrose index) gives us the probability of that voter swinging any given vote, while the “normalized” version is taken over all possible swings. This gives the neat property that normalized voting powers sum to 100% across all voters.
Overall, this simple calculation surfaces an interesting and non-trivial fact about WVS: voting power does not neatly correspond to the number of votes a voter has. In our example, the voter with 1 vote is just as powerful as a voter with 2 votes. This is because the 1-voter swings as many votes as everyone else.
Are blockchain voters ‘dummies’?
In voting theory parlance, a ‘dummy’ is a voter who is never ‘critical’ to any ‘winning coalition’ of voters — that is, their Banzhaf power index is 0 — or, in English, a dummy is someone whose vote simply doesn’t count.
Excerpt from Banzhaf power index calculation for MolochDAO. Full results here.
Are Moloch members dummies? To answer this question, we set out to calculate the Banzhaf power index of Moloch’s WVS. You can find the full result set here.
It turns out that in MolochDAO, even members with only 1 share might swing a vote 0.06% of the time. Strictly speaking, no one in Moloch today is a ‘dummy’. But the power index very clearly exposes how disproportionately voting power is distributed in the Moloch WVS. The top two shareholders — known to be Vitalik Buterin and Joe Lubin — each hold 1000 shares. If it weren’t for their apparent apathy for participating in the DAO’s votes, they each would be able to conclusively decide 49.74% of all proposals to come across the Moloch docket.
As the chart on the left shows, the relationship between share ownership and voting power is approximately (but not exactly) linear. This, on some level, seems “fair” — the more one buys into the system, the more influence they should have. After all, Joe and Vitalik are contributing proportionally more of their personal Ether capital to fund Moloch proposals. But consider this: if Joe and Vitalik formed a coalition that always voted together, they would be able to swing 99.48% of all Moloch proposals to whatever outcome they wanted while only holding 28.40% ownership in the DAO. This is the basic finding of a Banzhaf analysis on Moloch. (You can see this result in the second tab of the analysis.)
Conclusions
We should read more classic literature, especially as far as voting systems are concerned. There is much covered ground to be applied to on-chain governance. The Banzhaf power index is just one tool that might help us analyze the most predominant voting system in blockchain, the weighted voting system.
Moloch members are not ‘dummies’. But if our goal in Moloch is to delegate voting power effectively to a 70-member organization, we are very far from that goal. The same is true for Aragon votes and almost every other on-chain governance system today.
Most on-chain voting systems, by virtue of being WVS, disproportionally reward participants who can buy up significant stakes, even though those stakes might very well be minority stakes. I am not a detractor of plutocratic systems necessarily, and I recognize they may be advantageous in certain circumstances. But in systems that aim for enfranchising public participants and raising their average voting power, this type of system is highly suboptimal.
Finally, we probably need 1P1V systems in blockchain. The properties of voting power in these systems are much better suited to public goods and commons, which is what most decentralized systems strive for today. We can explore these kinds of systems in blockchain with solutions like 3Box identity, or Proof of Personhood, or even just good old KYC.
