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Efficient computation of the Banzhaf voting power index



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Cooperative game theory’s voting power indices will undoubtedly become important tools in blockchain-based governance systems and the pricing of governance tokens in the years to come.

In a previous post, we demonstrated the Banzhaf power index and performed a voting power analysis on the weighted voting system of MolochDAO. At the time of the analysis, Moloch’s voting system had fewer than 100 members and less than 8,000 units of voting power.

However, real tokens supplies have tens or hundreds of thousands of voters and millions or billions of units of voting power, making computations of their power indices intractable using naïve approaches.

In this post, we present practical strategies for computing Banzhaf voting power for realistic on-chain voting systems. We introduce an open source project for efficient Banzhaf calculations that can handle some real data, implemented in Go.

Computation strategies for the Banzhaf power index

This section is technical. If you are interested in practical applications and results, then skip to the last section.

We now examine approaches for efficient computations of the Banzhaf index. In the following discussion, let us denote by n the number of voters in the weighted voting system. Let t be the sum of all the voting weights of the voters — in other words, the number of units in the token supply. Finally, the quota of the system— the number of votes needed to pass a proposal — is denoted by q; we note that typically q=t/2+1 for systems with simple majority rule.

Computation using naïve enumeration of coalitions

Computing voting power entails counting coalitions (subsets) of a set of voters, and therefore the naïve implementation has complexity in O(2ⁿ). My original attempt at computing a Banzhaf index naïvely in [Brukhman 2019] would become intractable around n = 20. In light of the following much smarter methods, this method is not recommended for realistic computations; however, it may be useful for understanding the power index construct.

Computation using a special generating function

In [Heger 2013], a generating function approach is used to improve the complexity of the calculation. A product of a certain set of n binomials produces a polynomial P whose degree is t. Because of the special choice of binomials, the degree of each term of P represents a voting weight and the coefficient of each term counts the number of coalitions that achieved this weight.

Though Heger lists O(n²t) as the complexity, the algorithm presented can be improved my storing the polynomial across calculations for each voter. The polynomial may be computed in O(nt) time and O(t) space. Subsequently, we can traverse the polynomial and sum coefficients in order to count all the coalitions in which a particular voter was critical, also in O(nt).

Computation using dynamic programming

Both [Uno 2003] and [Keijzer 2008] discuss a dynamic programming method for precise calculations of Banzhaf indices. The method uses a recursively-defined function f to calculate the number of swings of voters and saves them in memory. A symmetric recursive function b is introduced, which calculates the same swings but in the backwards direction.

The mathematical intuition here is that f is closely related to the polynomial multiplication performed in the generating function approach. But by exploiting symmetries, we can get the complexity down from O(nt) to O(nq), which is up to a factor of 2 improvement.

Approximation using probabilistic sampling

The probabilistic sampling approach discussed in [Bachrach 2008] is only an approximation of the index, not a deterministic calculation. However, as the algorithm discussed invokes Hoeffding’s Inequality, we can approach the real value with an arbitrary precision.

The algorithm essentially goes through random trials where it considers a random voting outcome and measures in how many such trials a givˆen voter can swing the outcome. We can also calculate a number of trials k that we would need to perform in order to get an approximate Banzhaf value within ε of the actual one, with a confidence level of 1-δ.

For example, getting within 4 decimal places of the real value with a confidence of 99% would require at least k = 264,915,869 trials. Since the complexity of this algorithm is O(nk), the best use of this approximation technique is when the token supply is very large and high precision is required — our generating function algorithm above becomes too slow.

Approximation using re-weighting of the distribution

Banzhaf power is a discrete value but generally correlates with the voter’s percent ownership in the token supply. That means small perturbations and multiplicative operations on the entire set of voting weights (and quota) will generally result only in small deviations of the Banzhaf outputs.

This suggests the following strategy to make Banzhaf calculations more tractable on token supplies with a very large t: take the set of token weights W = (w_1, …, w_n) and a real divisor d > 0, and construct a new voting system W’ = (w_1/d, … w_n/d) whose Banzhaf indices will be approximately the same, but whose computation complexity will be O(nt/d) — an improvement on running time by a factor of d.

Contribute to open source for Banzhaf index computation

Github Repo:

If you are technical, please check out the go-banzhaf library on Github, which contains my implementation of the generating function algorithm above. I am looking for core contributors who can help to implement some of the dynamic programming and approximation approaches to Banzhaf calculations, especially those in [Uno 2003] and [Keijzer 2008].


  1. [Bachrach 2008] Approximating power indices
  2. [Brukhman 2019] Naïve implementation of a Banzhaf-like index
  3. [Heger 2013] Using generating functions to compute power indices
  4. [Uno 2003] Efficient computation of power indices for weighted majority games
  5. [Keijzer 2008] A survey on the computation of power indices

Efficient computation of the Banzhaf voting power index was originally published in The CoinFund Blog on Medium, where people are continuing the conversation by highlighting and responding to this story.



BTG Pactual becomes first bank in Brazil to participate directly in the crypto market



Financial institutions have been increasingly keen on being part of the booming $2 trillion cryptocurrency market, with over 13 of the world’s largest banks pushing roughly $3 billion in funding so far, into cryptocurrency. The latest major news came from Brazil.

One of the leading Brazilian investment banks BTG Pactual has been trending within the crypto community. It launched a new platform enabling customers to make direct investments in cryptocurrencies. Ergo, making it the first major financial institution in Brazil to participate directly in the crypto market.

According to local news outlet, The Rio Times,

“…the launch of Mynt, a platform that allows BTG Pactual Digital and BTG+ clients to trade directly, cryptocurrencies such as bitcoin and ether.”

Andre Porthilo, BTG Pactual’s head of digital assets, stated,

“At this first moment, we will have the two main assets of the market, but we will include other cryptos for trading over time. We will have a complete platform with blockchain-based assets.”

One thing that remained in common with different banks incorporating these tokens is the fact that they have been responding to “demand from our customers who wish to trade crypto.” This news is no different.


BTG Pactual CEO, Roberto Sallouti laid out his plan of action with this development. He opined, “As a new asset class, we will also have content to educate and inform our customers about these assets and the technology.”

Needless to say, Brazil has witnessed an immense amount of crypto traction. That said, regulators too have taken steps to curb illicit activities. As asserted by the CEO, the said bank will be regulated by both the Brazilian Securities Commission and the local central bank. Moreover,

“With the support of BTG Pactual, Mynt has fundamental differentials in security and credibility…”


The said bank had previously incorporated digital assets in previous operations as well. In 2019, it launched its real estate-backed security token ReitBZ. This step was executed post-examining digital assets for a couple of years. Just recently, the bank announced collaboration with Gemini, the U.S.-based crypto exchange. Gemini provided custody services to the first Bitcoin fund issued by a Brazilian investment bank.

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Basel Committee Crypto Rules Face Backlash




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Ethereum NFT Game Sorare Fantasy Soccer Raised $680 Million



Ethereum NFT game Sorare fantasy soccer just raised $680 million and it is now valued at $4.3 billion while launching a US office so let’s read more today in our latest Ethereum news.

The Ethereum NFT game Sorare fantasy soccer raised $680 million and the french startup now plans to expand into other sports as well as to onboard women’s soccer teams and US office. Sorare is an Ethereum-based fantasy soccer game based on NFT trading cards and grew dramatically in 2021 with a rising base and huge trading volume not to mention the new licensed teams in the mix. Today, however, the startup is using the momentum with the announcement of a new funding round.

NFT Game Founder, scammer, eth, cryptopunks

The $680 million Series B round values the company at $4.3 billion after a $50 million Series A round which was announced back in February and the new series was led by Japanese tech and investment conglomerate SoftBank that has a number of crypto industry dealings through SBI holdings financial services division. Sorare is focused on soccer with players from more than 175 licensed clubs like FC Barcelona, Juventus FC, and Paris Saint Germain FC as some of the most popular. Nicolas Julia, the co-founder and CEO of Sorare said that this week the company plans to expand into other sports and it will open a US-based office in the upcoming months to help “continue to grow with fans and sports leagues.” He continued:


“We have ambitious plans to keep growing Sorare into a sports entertainment giant. The new funding will be used to help us continue to grow a world-class team, expand to new leagues and new sports, and keep improving our experience for users, including shipping a mobile app.”

nba top shot

Dapper Labs’ NBA Top Shot is similar to Sorare, built around NFTs and based on professional athletes. An NFT acts like a deed of ownership to a scarce digital item so in the case of Sorare, it’s a digital trading card that can be purchased and resold with each available in limited quantity. One NFT card can even reach $290,000. Unlike Top Shot, however, Sorare has an ongoing game component and is similar to other fantasy sports in that you can choose your top players each week as well as the leagues with other users. Sorare also has close ties with Ubisoft as it took the accelerator program this year.


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