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Efficient computation of the Banzhaf voting power index

Republished by Plato



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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.


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Standard Custody takes new route to ‘qualified custodian’ status

It’s the first digital asset firm to receive approval on a de novo application for a New York trust license.

The post Standard Custody takes new route to ‘qualified custodian’ status appeared first on The Block.

Republished by Plato



Standard Custody received its license to operate as a New York state-chartered trust on May 4, and it’s already making a play to gate-crash the institutional custody space.

Just days after its licensing, the firm announced the close of a $53 million Series B round for its parent firm, PolySign.

Cowen Digital Asset Investment Company led the round with a $25 million strategic investment. The two will also partner, with PolySign providing digital asset custody solutions for Cowen clients through its newly licensed trust arm, Standard Custody. and Race Capital also participated in the round.

Through Standard Custody, PolySign is looking to fill a gap in the custody space. While many crypto firms are attempting to build all-in-one services, with exchange, brokerage and custody housed under the same roof,  CEO Jack McDonald says Standard Custody plans to differentiate itself by focusing solely on custody-based services for institutions.

Though Standard Custody plans to expand its range of services, McDonald says it will stop short of being an exchange unlike others in the custody space.

“We think that ultimately the institutions that are wading into the space, more and more of the traditional institutional asset managers, are going to want to see a segregation of duties there between exchange activity and custody activity,” he said.

That could mean hedge funds, family offices, endowments and exchanges could make up its client base going forward, but not retail-facing activities. Others serving the retail market have expressed interest in Standard Custody’s services, mostly due to its recent licensure. It’s the first to get approval for a de novo trust application in New York, and that’s positioning it to emerge as a favorable partner for a variety of clients, according to McDonald. 

To build out custody and escrow services, Standard Custody needed to be a qualified custodian. There’s more than one way to gain the distinction, but some fit better than others. To be a qualified custodian, firms can either become a registered broker-dealer with the Financial Industry Regulatory Authority (FINRA), a futures commodities merchant regulated by the Commodities Futures Trading Commission (CFTC) or you can be a federally or state-licensed trust bank.

For firms mainly looking to custody, it makes the most sense to become a trust but it’s recently become unclear how far a trust license extends outside state borders. The Securities and Exchange Commission (SEC) is currently seeking comment on how it should view state-licensed qualified custodians in the wake of a letter from Wyoming’s regulators. On the national level, Congress is still debating how much power the Office of the Comptroller of the Currency (OCC) should have to designate digital asset firms as national trusts and therefore qualified custodians.

Still, a New York trust license from the New York Start Department of Financial Service is the gold standard of state licenses. It’s the highest barrier of the state licensure frameworks, and also has more reciprocity than other states, meaning some other states recognize the New York trust charter and don’t require an additional license. Standard Custody is the first to receive a de novo approval, meaning it’s operating a new business as opposed to converting a previous entity like Gemini and Coinbase. That’s made it more attractive to businesses looking to set up shop in the U.S. without going through onerous regulatory frameworks.

“We do have a lot of interest in our technology from some of the more retail-oriented strategics out there and specifically wanting to tap our capabilities to business in New York and more broadly in the U.S.,” said McDonald.

© 2021 The Block Crypto, Inc. All Rights Reserved. This article is provided for informational purposes only. It is not offered or intended to be used as legal, tax, investment, financial, or other advice.

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How will ‘Ethereum killers’ affect the ETH rally?

Republished by Plato



Many in the crypto community didn’t anticipate this altcoin’s rally. Several questioned Ethereum’s move comparing it to its biggest competitor, Bitcoin. Looks like the evergreen debate of ETH vs BTC is going to resurface again. Can Ethereum actually flip Bitcoin anytime soon? This is the latest hot topic.

ETH, the world’s largest altcoin, has had an amazing start to this year. It commenced its journey at $720 on the charts to just under the $4000 mark at press time.

Ethereum’s market cap is now half the size of Bitcoin’s. The ratio between the assets is 0.08. Here’s a glimpse of the Flippening index.

The premier altcoin was the topic of discussion during a recent interview hosted by Unchained podcast with Jeff Dorman, the chief investment officer at Arca. The investor stated:

“With ETH almost hitting the $4k mark, other ‘Ethereum killers’ can’t just take its throne.”

He reiterated that the $4k mark directly pointed at the increasing institutional interest from different firms which were once only devoted to Bitcoin. He added:

“We are witnessing the early innings of institutional adoption beyond Bitcoin with Ethereum.”

In a note titled “The Rise of The Institutional Adoption of Ethereum,Eliézer Ndinga, a senior research associate at 21Shares expressed his optimism towards the surge in the Ethereum market. Furthermore, a recent discussion, by Su Zhu and Hasu with CryptoCobain too portrayed a prevalent bullish ETH mindset, indicating that the bull case for Ethereum is stronger than ever.

With respect to the upcoming Ethereum developments, the CIO remained critical of some of them. According to him ETH was still in ‘the event and catalyst-driven bucket’ stage. Talking about the EIP-1559 proposal, he stated:

“…..speaking about the transition, it actually hasn’t happened yet. things can go wrong, can be bugged. It’s too early to say about its effects, but ETH will still rise even if there are issues with this transition.”

However, Dorman also stated that:

“A lot of applications that are built on ETH are seen by traditional investors as better places to put their money than Ethereum itself, it made more sense.”

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Chainlink price prediction: Chainlink retests $41, set to move higher?

TL;DR Breakdown LINK tests $44 resistance overnight. Support retested at the $41 level over the past hours. Next support at $40. Today’s Chainlink price prediction is bearish as the market moved lower after setting a lower high around $44. Currently, LINK/USD retests the $41 support, once it is broken, we should see further downside over […]

Republished by Plato



TL;DR Breakdown

  • LINK tests $44 resistance overnight.
  • Support retested at the $41 level over the past hours.
  • Next support at $40.

Today’s Chainlink price prediction is bearish as the market moved lower after setting a lower high around $44. Currently, LINK/USD retests the $41 support, once it is broken, we should see further downside over the next 24 hours.

Chainlink price prediction: Chainlink retests $41, set to move higher? 1
Cryptocurrency heat map. Source: Coin360

The overall market trades with mixed results as Bitcoin trades flat around 0 percent, while Ethereum has lost almost 3 percent. Stellar (XLM) is among the best performers with a gain of 5 percent. 

LINK/USD opened at $41.5 after bearish close yesterday set a lower high at $48. Earlier today, another lower high was set around $44.5 after a retest of $41 support. Therefore, the market trades in an increasingly tighter range. Once the range is broken, we will see where the market is headed next week. 

Chainlink price movement in the last 24 hours

The LINK/USD price moved in a range of $41.08 – $44.61, indicating a moderate amount of volatility. 24 trading volume has decreased by 13.92 percent and stands at $2.2 billion. Meanwhile, the total market cap stands at $17.7 billion, ranking the cryptocurrency in 13th place overall.

LINK/USD 4-hour chart – LINK 

On the 4-hour chart, we can see the Chainlink price pushing to break the $41 mark once again. 

Chainlink price prediction: Chainlink retests $41, set to move higher?
LINK/USD 4-hour chart. Source: TradingView

Overall the market continues retracing from the all-time high set around the $53 mark on the 10th of May. The new all-time high was set due to a 70 percent upswing from the previous major support level around $31 set on the 23rd of April. Therefore, we could see similar performance over the upcoming weeks once the Chainlink price stops retracing.

Earlier this week, Chainlink made two separate waves lower, resulting in a total retracement of around 25 percent. The support around the $40-$41 mark has already been retested twice. Therefore, we could see the support break later today as bears continue pushing LINK/USD lower.

Once the support is broken, we could see LINK/USD move towards the next minor support, around $38. From there, the market could potentially start to reverse in a similar way as during the middle of April. 

Alternatively, if a further downside is rejected over the next hours and the $40-$41 support holds, we could see LINK/USD move sideways over the next 24 hours as it prepares a base from which to push higher early next week.  

Chainlink Price Prediction: Conclusion 

Chainlink price prediction is bearish as the market continues trading in a bearish momentum over the past days. Earlier today, another lower high was set around $44.5, indicating that bears are still in control, and we are likely to see LINK move below the $41-$40 support area early next week.

While waiting for further Chainlink price action development, read our latest guides on Litecoin mining software, Ethereum mining pools, as well as Ethereum mining software.

Disclaimer. The information provided is not trading advice. holds no liability for any investments made based on the information provided on this page. We strongly recommend independent research and/or consultation with a qualified professional before making any investment decisions.

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