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Efficient computation of the Banzhaf voting power index

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Introduction

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: https://github.com/jbrukh/go-banzhaf

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].

References

  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.

Source: https://blog.coinfund.io/efficient-computation-of-the-banzhaf-voting-power-index-b011ef4be55f?source=rss—-f5f136d48fc3—4

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3 reasons why Reef Finance, Bridge Mutual and Morpheus Network are rallying

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As new institutional and retail investors enter the cryptocurrency space on a daily basis, large-cap top performers like Bitcoin (BTC) and Ether (ETH) attract the lion’s share of investor’s attention as they are the well-known ‘secure’ blockchain projects. 

Once these new investors get a taste of the mainstay cryptocurrencies and how to navigate the volatile markets, their attention soon turns to smaller cap coins as they search for the up-and-coming projects that could be the next big thing.

Currently, CoinMarketCap shows that there are 8,475 tokens and more are added daily. This makes it difficult to keep up with the latest developments and find solid projects with real-world potential.

With that in mind, here are some interesting projects that have been gaining strength over the past few weeks. 

MRPH/USDT

Morpheus Network (MRPH) is a blockchain platform focused on logistics and supply chain optimization through the use of its SaaS middleware platform which is integrated with emerging technologies.

Supply chain managers are able to use the platform to create a digital representation of their network as information collected is transformed into actionable data, with all steps in the supply chain being notarized on the Morpheus blockchain.

MRPH was trading at a price of $0.412 on Jan.15 before an influx of trading activity lifted the token more than 920% to a high of $4.44 on Feb.8.

MRPH/USDT 4-hour chart. Source: TradingView

The rapid rise in price was due in part to the fresh attention the project received from several well-known YouTube influencers and recent verifiable MRPH partnerships, such as China’s Qingdao Maple Leaf International Trading Co. and the possibility of a partnership with Coca-Cola in Latin America.

Speculations aside, the Morpheus platform currently has more than 100 integrations with industry-leading service providers including DHL, FedEx, SWIFT, Oracle, and Salesforce. With significant real-world partnerships and the attention of cryptocurrency influencers, MRPH has strong fundamentals and is likely to gain more attention from investors.

BMI/USDT

Bridge Mutual (BMI) is a more recent arrival to the decentralized insurance space but it has quickly garnered the attention of investors.

The insurance platform offers coverage for stablecoins, centralized exchanges and smart contracts. It also allows users to provide insurance coverage, determine insurance payouts, and recie compensated for taking part in the ecosystem.

BMI’s initial decentralized exchange offering (IDO) was conducted on Jan. 30 with a token price of $0.125 and it was first listed on Uniswap for $1.03. Since listing, BMI has rallied by 540% to a high of $5.46 on Feb. 3. Currently, BMI trades at $3.24 following the downturn in the market that began on Feb. 21.

BMI/USD 1-hour chart. Source: CoinGecko

Decentralized insurance has thus far been dominated by Nexus Mutual (NXM), but BMI’s arrival offers a fresh challenger to a field with growing demand due to the risky nature of investing in DeFi platforms.

REEF/USDT

Reef (REEF) is a Polkadot-based DeFi platform that aims to offer cross-chain trading powered by a yield engine and smart liquidity aggregator that enables automation of the exchange process.

One issue Reef developers hope to provide a solution for is high gas fees on the Ethereum blockchain that are currently making DeFi unusable for many community participants. The team also hopes to help connect liquidity pools from separate networks, avoiding the need for multiple accounts which can be difficult to keep track of.

REEF/USDT 4-hour chart. Source: TradingView

Work on the project began in the second half of 2020 with the completion of its IDO on Sep.30. Following its listing on Binance and Uniswap in late December of 2020, REEF price bottomed out at $0.0067 on Jan.13 and has since increased more than 750% to a high of $0.054 on Feb.11.

DeFi remains one of the hottest growth areas in the cryptocurrency sector and Reef is well-positioned to capitalize on its continued growth. As the Polkadot ecosystem grows its user base and provides solutions that provide relief from high Ethereum transaction costs, cross-chain functionality projects like Reef stand ready to benefit as decentralized finance goes mainstream.

Source: https://cointelegraph.com/news/3-reasons-why-reef-finance-bridge-mutual-and-morpheus-network-are-rallying

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Bitcoin plunges, Ethereum suffers, Musk loses billions: Hodler’s Digest, Feb. 21–27

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Coming every Saturday, Hodler’s Digest will help you track every single important news story that happened this week. The best (and worst) quotes, adoption and regulation highlights, leading coins, predictions and much more — a week on Cointelegraph in one link.

Top Stories This Week

Increasing stock market volatility drags Bitcoin and altcoin prices lower

Bitcoin has had an exceptionally trying week, and it doesn’t bode well for March — a month that’s traditionally bearish for the world’s biggest cryptocurrency.

After hitting record highs of $58,300 last Sunday, Bitcoin suffered a dramatic reversal of fortunes — crashing to $46,000 on Tuesday. Elon Musk might not have helped matters… in the run-up to the correction, he had tweeted that BTC and ETH seemed high.

Analysts and investors alike breathed a sigh of relief on Wednesday when Bitcoin managed to retake $50,000 — with some proclaiming that the asset had undergone a “healthy correction.” But this narrative proved shaky when BTC plunged yet again on Friday to lows of $44,454.84.

All of this comes amid a backdrop of unease in the traditional markets, and this week’s price activity suggests BTC faces an uphill struggle if it’s going to appreciate further. Generally, analysts are looking for $50,000 to become an established support before expecting any bullish continuation.

MicroStrategy purchases another $1 billion worth of Bitcoin, now owns 90,000 BTC

A flurry of good news throughout the week may have prevented things from going bad to worse for Bitcoin. Early in the week, two institutions announced they were doubling down on their BTC buy-ins. 

MicroStrategy purchased an additional 19,452 coins, with CEO Michael Saylor declaring that his company has no intention of slowing down. It came after Square announced it had purchased 3,318 BTC for $170 million — following on from a $50-million spending spree in October 2020.

Bitfinex and Tether also announced that they had reached a settlement with the New York attorney general, linked to ongoing allegations that Tether misrepresented the degree to which USDT stablecoins were backed by fiat collateral. Under the terms of the deal, both companies will have to pay $18.5 million in damages, report on their reserves periodically, and stop serving customers in the state.

On Friday, JPMorgan helped to cheer up the markets by telling clients that allocating 1% of a portfolio to Bitcoin would serve as a hedge against fluctuations in stocks, bonds and commodities.

Cardano is now a top-three cryptocurrency as ADA price soars 27% in 24 hours

Moving beyond Bitcoin, there’s been a lot of movement in the altcoin markets. 

Last week, Binance Coin had stolen the show with a stunning triple-digit surge that helped it become the world’s No. 3 cryptocurrency. Fast forward to this week, and it’s now been overtaken by Cardano’s ADA.

A fresh wave of optimism and buying volume on Friday pushed its price to a new all-time high, and momentum for the project has been building throughout February. Open interest for ADA futures also rose to $580 million, surpassing Litecoin to become the third-largest derivatives market.

Despite NFTs entering into a bull market — with a report suggesting that they’ll explode in popularity even more as 2021 continues — it’s definitely been a week to forget for Ether. After touching new all-time highs of $2,000 last weekend, ETH has tumbled by more than 26% this week… taking it below $1,500 at times.

All of this comes as an exodus from the Ethereum blockchain continues, with 1inch becoming the latest DeFi project to expand to Binance Smart Chain.

Musk no longer world’s richest man after Tesla and Bitcoin slump

As the old saying goes: “The sun don’t shine on the same dog’s ass every day.”

The sun was certainly shining on Elon Musk when the week began. One analyst had suggested that Tesla had made $1 billion in profit since making its Bitcoin investment. That’s more than the profit generated by selling electric vehicles (what it’s known for) across the whole of 2020.

Alas, that was before the carnage seen on the crypto markets. To make matters worse, Tesla’s share price has dropped by more than 20% from the highs of $890 seen on Jan. 26. These joint factors prompted Musk to lose his crown as the world’s richest man. Some analysts wasted little time in attributing TSLA’s crash to its association with Bitcoin.

But there’s another threat on the horizon, with reports suggesting that the U.S. Securities and Exchange Commission could investigate Musk’s alleged impact on BTC and DOGE through his many, many tweets.

The billionaire made a concerted effort to shrug off these concerns, suggesting he would even welcome such a probe.

Coinbase has held Bitcoin on its balance sheets since 2012

We’ve been learning a lot more about Coinbase this week as it gears up to launch on the stock market. One particular hipster-ish announcement came when the exchange declared that it’s held Bitcoin and other cryptos on its balance sheet for nine years.

Coinbase sought to package this announcement as a paean to other corporations that might be considering a similar move — touting itself as an authority in advising institutions about how to deal with their own prospective investments.

In other news, the company submitted its S-1 report to the Securities and Exchange Commission this week. The filing revealed that the exchange generated revenues of $1.1 billion in 2020 — 96% of which came from transaction fees. Net income in 2020 came in at $327 million… a stark contrast to the $46 million loss seen the year before.

Winners and Losers

At the end of the week, Bitcoin is at $46,609.99, Ether at $1,470.17 and XRP at $0.43. The total market cap is at $1,429,222,267,885.

Among the biggest 100 cryptocurrencies, the top three altcoin gainers of the week are Fantom, Pundi X and Cardano. The top three altcoin losers of the week are Dodo, Horizen and Venus.

For more info on crypto prices, make sure to read Cointelegraph’s market analysis

Most Memorable Quotations

“As gas price stays too high, we see a lot of projects, tokens and users coming to BSC, and this is the right moment for 1inch to expand to other blockchains.”

Sergey Kunz, 1inch co-founder

“Since our founding in 2012, Coinbase has held bitcoin and other crypto assets on our balance sheet — and we plan to maintain an investment in crypto assets as we believe strongly in the long-term potential of the cryptoeconomy.”

Coinbase

“Incredible scale for a technology that critics claimed couldn’t scale.”

Ryan Watkins, Messari researcher

“It’s very rare to see pre-GPU era bitcoins move, it only happened dozens of times in the past few years. And no, it’s probably not Satoshi.”

Antoine Le Calvez

“The company now holds over 90,000 bitcoins, reaffirming our belief that bitcoin, as the world’s most widely-adopted cryptocurrency, can serve as a dependable store of value.”

Michael Saylor, MicroStrategy CEO

“[I’m] very positive on Bitcoin, very happy to see a healthy correction here.”

Cathie Wood, Ark Investment Management founder

“We are now sitting on 2.35x the previous cycle ATH OF 20k. WE ARE JUST GETTING STARTED.”

Bitcoin Archive

“Square believes that cryptocurrency is an instrument of economic empowerment, providing a way for individuals to participate in a global monetary system and secure their own financial future.”

Square

“I think you can expect that we’ll have a billion people storing their value — in essence, a savings account — on a mobile device within five years, and they’re going to want to use something like Bitcoin.”

Michael Saylor, MicroStrategy CEO 

“We’ve experienced 2018 & 2019. This is nothing.” 

Michaël van de Poppe, Cointelegraph Markets analyst

“I do think people get drawn into these manias who may not have as much money to spare. So, I’m not bullish on Bitcoin, and my general thought would be: If you have less money than Elon, you should probably watch out.”

Bill Gates, Microsoft founder

“But we’re now to the point where ETH 1.0 — oh, we need ETH 2.0 so soon, come on, Vitalik, get it going, man — ETH 1.0, most regular users are priced out of using the majority of applications on Ethereum.”

Lark Davis, crypto influencer

“I lost most of my life savings and haven’t received a response from a human. I’d think they would refund or they would lose all their customers. I’m sick to my stomach but will join the lawsuit with plenty of proof(screenshots) if not refunded.” 

u/dtk6802, Reddit user

“In our view, many institutional investors are entering with a buy-and-hold mentality given their understanding of Bitcoin as digital gold.”

Martin Gaspar, CrossTower research analyst

“I think Tesla is going to double down on its Bitcoin investment.”

Dan Ives, Wedbush analyst

Prediction of the Week

1 billion people will store life savings on their phone in Bitcoin by 2026 — MicroStrategy CEO

We love an outlandish prediction here at Hodler’s Digest… and Michael Saylor certainly delivered the goods this week.

The MicroStrategy CEO declared that Bitcoin will be the savings method of choice for a staggering 1 billion people in just five years’ time. That’s despite the fact that just 21 million BTC exist… and his company already owns 90,000 of it.

Saylor’s comments came after U.S. Treasury Secretary Janet Yellen launched her latest attack on Bitcoin, describing it as “inefficient.”

In a confident interview with CNBC, he declared that Bitcoin “is the dominant digital monetary network,” adding: “I think you can expect that we’ll have a billion people storing their value — in essence, a savings account — on a mobile device within five years, and they’re going to want to use something like Bitcoin.”

FUD of the Week 

Bill Gates warns Bitcoin buyers: If you have less money than Elon Musk, watch out

Microsoft founder Bill Gates had a big warning for Bitcoin buyers this week.

Speaking to Bloomberg, he warned: “Elon has tons of money, and he’s very sophisticated so, you know, I don’t worry that his Bitcoin would randomly go up or down.”

Gates said it would be a mistake for the average investor to blindly follow the mania of optimism surrounding Musk’s market moves, telling those who aren’t billionaires to “watch out.” 

Criticizing Bitcoin’s energy consumption, he added: “I do think people get drawn into these manias who may not have as much money to spare. So, I’m not bullish on Bitcoin, and my general thought would be: If you have less money than Elon, you should probably watch out.”

This isn’t to say that Gates thinks digital currencies are a bad thing. He just believes that they should be transparent, reversible and (essentially) centralized.

Whale who sold Bitcoin before 2020 crash cashed out $156 million before this week’s 20% dip

As you’d expect, a post-mortem is now fully underway after this week’s carnage in the crypto markets.

Curiously, data from Santiment suggests that the initial crash may have been linked to a huge transaction that took place after Sunday’s all-time high of $58,300. The transfer of 2,700 BTC — worth $156 million at the time — was the second-biggest transaction of 2021.

It’s possible that this whale cashing out contributed to unbearable selling pressure in the market, which snowballed into the largest one-hour candle in Bitcoin’s history. If enough alarm bells weren’t ringing, this self-same wallet also dumped 2,000 BTC just before last March’s infamous flash crash.

Crypto influencer warns Ethereum fees will drive users away

A prominent crypto influencer has warned that Ethereum’s competitors will continue to siphon away users should Eth2 fail to launch soon amid ever-increasing gas fees.

Lark Davis said Ethereum’s skyrocketing fees has meant that only “rich investors” can afford to use the network, prompting smaller users to switch to competitors like Binance Smart Chain. 

Describing the current gas fee prices as “totally loco,” Davis urged Ethereum developers to expedite the launch of Eth2 in response to the skyrocketing to prevent a further exodus of users to cheaper alternatives.He added: “We’re now to the point where ETH 1.0 — oh, we need ETH 2.0 so soon, come on, Vitalik, get it going, man — ETH 1.0, most regular users are priced out of using the majority of applications on Ethereum. […] A transaction on Uniswap costs $50 on average these days, and that is just crazy.”

Best Cointelegraph Features

Sam Bankman-Fried: The crypto whale who wants to give billions away

He’s just 28 years old, but Sam Bankman-Fried has already amassed a $10-billion fortune. But unlike most people in crypto, he’s building up this fortune to give half of it away.

Can’t beat ‘em? Join ‘em: Mastercard and Visa make a case for Bitcoin

Mastercard is set to open the shop doors to crypto as a means of payment in 2021, but it will likely be a challenge for the firm.

Bitcoin price flies solo? Institutional crypto push may be overrated

Bitcoin’s market cap broke the $1-trillion barrier without a final push from institutions — could their influence be overrated?

Source: https://cointelegraph.com/magazine/2021/02/27/bitcoin-plunge-ethereum-suffer-musk-billions-022127

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Nym: The World’s First Generic Incentivized Mixnet Releases its Whitepaper 

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[PRESS RELEASE – Please Read Disclaimer]

In a time when mass surveillance and data harvesting are ever present and not a day goes by without news of companies selling user data for profit, Nym Technologies is building a next generation privacy network that can change the way people use the internet.

Chelsea Manning, a famous whistleblower and technologist, says “As methods for network traffic analysis have dramatically improved in the last decade, I have frequently called for research (most notably in 2016) into alternative methods to Tor that avoid exposing the data within the network to such analysis. Nym is one such viable alternative worthy of research, and developmental implementation.”

Nym was conceived in 2017 and was the first privacy project to receive funding from Binance Labs in 2018, followed by a $2.5M raise from other well known investors. Today, the actual design of Nym has been made public after extensive review by technologists like Chelsea Manning, academics like Carmela Troncoso, and venture capital firms like Polychain Capital.

Carmela Troncoso (EPFL) notes “I spent a long part of my career working on improving mix-based anonymous communications systems. It is thrilling to see how the Nym team, a unique combination of expert software engineers and privacy experts, have made mixnets a reality.”

The Nym network is a generic, decentralized, and incentivized infrastructure that provides privacy to a broad range of applications and services, including any blockchain. A core component of Nym is a mixnet that protects the metadata of the internet packets sent to it with privacy superior to both VPNs and Tor.

Metadata is “data about data”, and includes IP addresses of the users, geolocations, information about who talked to who, when, and how often. All of this metadata can be monetized or used without users knowledge. Now it can be protected by Nym.

Anyone can join the network by running a node and get rewarded in NYM tokens for providing privacy to the network. Nodes do useful work anonymizing packets for users and services.

NYM tokens can be transformed into anonymous credentials that allow users to privately prove their “right to use” of services in a decentralized and verifiable manner. This allows users to be private at the network level as well as the application layer. Cosmos and the European Commission are amongst the many who have been supporting the use of Nym’s anonymous credentials.

The 3rd-party applications and services that can integrate their systems to the Nym network to protect their users from malicious actors and preserve their privacy range from crypto apps (wallets or DeFi projects) to messaging applications, IoT devices, or literally any data transfers over the internet that can leak metadata.

Currently, Nym is running an incentivized testnet with a 1500 capped number of nodes on the Liquid network, but this limit will raise in the next major release due to the high demand of people who want to join the network and test its features out.

Throughout human history, privacy has been considered a great asset and a prerequisite for freedom. However as privacy was not built into the fabric of the internet, power is now in the hands of a few powerful players. Nym is setting off to change this and give power back to users so they can decide if and how they reveal their data. To know more about the technicalities, read the whitepaper or join the Nym Telegram channel to stay up to date.

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Source: https://cryptopotato.com/nym-the-worlds-first-generic-incentivized-mixnet-releases-its-whitepaper/

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