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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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Crypto Price Analysis & Overview September 25th: Bitcoin, Ethereum, Ripple, Chainlink, and Tezos

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Bitcoin

Bitcoin dropped by a total of 3% over the past seven days of trading as it reached the $10,600 level today. The cryptocurrency briefly pushed above the $11,000 mark last Friday but could not sustain this level as it broke beneath it during the weekend. On Monday, BTC saw a precipitous 7.5% fall as it dropped from $10,910 to reach as low as $10,200.

Bitcoin continued to head lower on Wednesday as it reached $10,140 before the buyers regrouped to initiate a rebound, which happened yesterday. During the rebound, BTC managed to break back above the 100-days EMA and reach as high as $10,760 – where it ran into resistance at the bearish .382 Fib Retracement.

The coin has dropped slightly from there to trade at $10,600 today.

Looking ahead, if the bulls can continue above the $10,760 level, higher resistance lies at $10,900 $11,000, and $11,200. Added resistance is expected at $11,340 (bearish .618 Fib Retracement), $11,500, and $11,600.

On the other side, the first level of support lies at $10,430 (100-days EMA0. This is followed by support at $10,330, $10,140, and $10,000.

btcusd-sep25
BTC/USD. Source: TradingView

Ethereum

Ethereum suffered a steeper 11.3% price decline over the past week as it reached the $346 level today. The coin was trading above $380 last Friday but started to drop lower during the weekend. On Monday, Ethereum fell from $365 to reach as low as $335 (100-days EMA).

The price decline continued on Wednesday, which saw ETH drop as low as $320. Luckily, the bulls regrouped for a rebound on Thursday in which ETH managed to reach the $252 resistance (bearish .236 Fib Retracement). It also produced a bullish engulfing candle, which is a strong bullish signal.

Moving forward, if the buyers can break the $352 level, resistance lies at $364 (2019 high), $382, $390, and $400.

On the other side, support is first expected at $336 (100-days EMA). Beneath this, support lies at $320, $306, and $300.

ethusd-sep25
ETH/USD. Source: TradingView

Ethereum also struggled against Bitcoin this week as it fell from 0.035 BTC to reach as low as 0.0311 BTC yesterday. The coin had found support yesterday at the .618 Fib Retracement, which allowed it to bounce higher to the current 0.0326 BTC level.

Looking ahead, if the bulls push higher, the first level of resistance lies at 0.033 BTC. Above this, resistance is located at 0.0337 BTC (March 2019 Support), 0.0347 BTC, and 0.0352 BTC.

On the other side, the first level of support lies at 0.032 BTC. This is followed by support at 0.0315 BTC (100-days EMA), 0.0311 BTC (.618 Fib Retracement), and 0.030 BTC.

ethbtc-sep25
ETH/BTC. Source: TradingView

Ripple

XRP witnessed a 5.5% price fall this week as the coin dropped from $0.25 to reach as low as $0.22 yesterday. The cryptocurrency managed to rebound from there to get as high as $0.24 today. However, the market is facing resistance at the 100-days EMA and must pass this to continue higher.

If the bulls break $0.24, the first level of resistance lies at $0.251 (bearish .382 Fib Retracement). Following this, resistance lies at $0.261 (bearish .5 Fib Retracement), and $0.271 (bearish .618 Fib Retracement).

On the other side, the first level of support lies at $0.235 (200-days EMA). This is followed by support at $0.23, $0.22, and $0.217.

xrpusd-sep25
XRP/USD. Source: TradingView

XRP is also struggling against BTC as it posted a fresh 2-month low at 2165 SAT (downside 1.414 Fib Extension) yesterday. The coin had slipped from 2300 SAT last Friday and continued lower until it hit the 2165 SAT support.

XRP has since bounced higher to reach the 2250 SAT level today.

If the bulls can break 2250 SAT, resistance is first located at 2300 SAT (bearish .382 Fib Retracement). Above this, resistance lies at 2350 SAT (bearish .5 Fib Retracement), 2400 SAT, and 2460 SAT.

Alternatively, support lies at 2200 SAT, 2165 SAT, and 2111 SAT.

xrpbtc-sep25
XRP/BTC. Source: TradingView

Chainlink

LINK saw a substantial 10% price fall over the past seven days, which saw the coin breaking beneath the $10 level and hitting as low as $7.31. There, it found support at a downside 1.272 Fib Extension level, which allowed it to rebound yesterday to reach the $9.90 resistance level today.

If the bulls can break $10, the first level of resistance lies at $10.40. Above this, resistance is found at $11.37 (bearish .382 Fib Retracement), $12, and $12.63 (bearish .5 Fib Retracement).

On the other side, support first lies at $9.00. Beneath this, support is found at $8.67, $8.00, and $7.31.

linkusd-sep25
LINK/USD. Source: TradingView

Against BTC, LINK dropped as low as 0.00072 BTC during the week. It was trading above a 3-month-old rising trend last Friday, but it went on to collapse beneath this support over the weekend. After reaching 0.00072 BTC, LINK bounced higher to get to the current 0.000935 BTC level today. Notice that it has returned to the previous rising trend line, which is now acting as resistance.

Looking ahead, the first level of resistance lies at 0.00095 BTC. This is followed by resistance at 0.001 BTC, 0.00103 BTC (bearish .382 Fib Retracement), and 0.00112 BTC (bearish .5 Fib Retracement).

On the other side, support is first found at 0.00091 BTC. This is followed by support at 0.0082 BTC, 0.00072 BTC, and 0.0007 BTC.

linkbtc-sep25
LINK/BTC. Source: TradingView

Tezos

XTZ saw the steepest price fall on this list as it dropped by 13% over the past 7 days. The coin fell beneath the $2.32 (.618 Fib Retracement) support on Friday and continued lower to crate a new 5-month price low as it reached $1.91. This support held over the past few days, and Tezos rebounded from here yesterday to reach $2.15 today.

Looking ahead, if the bulls break $2.20, resistance is found at $2.32, $2.53 (bearish .382 Fib Retracement), and $2.72 (bearish .5 Fib Retracement).

On the other side, the first two levels of support lie at $2.00 and $1.91. Beneath this, support is expected at $1.74 (.786 Fib Retracement), $1.68, and $1.56.

xtzusd-sep25
XTZ/USD. Source: TradingView

Tezos is suffering further against BTC as it produced a 7-month price low this week after reaching 18,600 SAT. The coin has since bounced from here to break back above 20,000 SAT today.

Looking ahead, the first level of resistance lies at 21,000 SAT. This is followed by resistance at 22,000 SAT, 21,750 SAT, and 24,300 SAT.

On the other side, beneath 20,000 SAT, support lies at 19,380 SAT, 18,600 SATm ad 17,600 SAT (.886 Fib Retracement). Added support lies at 16,600 SAT and 15,600 SAT.

xtzbtc-sep25
XTZ/BTC. Source: TradingView
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Disclaimer: Information found on CryptoPotato is those of writers quoted. It does not represent the opinions of CryptoPotato on whether to buy, sell, or hold any investments. You are advised to conduct your own research before making any investment decisions. Use provided information at your own risk. See Disclaimer for more information.

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Source: https://cryptopotato.com/crypto-price-analysis-overview-september-25th-bitcoin-ethereum-ripple-chainlink-and-tezos/

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Bitcoin Maintains $10K As Crypto Market Lost $16 Billion in 7 Days: The Weekly Crypto Market Update

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Another action-packed week took place in the crypto field. Unfortunately, the market is currently in decline as the entire capitalization lost around $16 billion over the past seven days.

The good news for Bitcoin bulls is that BTC managed to maintain itself above the critical psychological and technical support of $10,000. Things were looking rather promising until Monday when the price shot up to about $11,000.

Unfortunately, it was then when Bitcoin’s price took a sharp turn in the wrong direction and tanked to about $10,400. From there, it was a couple of days of sideways action until Wednesday when it dipped even further, calling questions whether or not $10,000 will hold. In the late hours of Thursday, however, Bitcoin bulls woke up and pushed its price to where it currently rests around $10,650.

Elsewhere, on Sunday, Buzz Feed reported that major banking giants such as JP Morgan Chase, Bank of America, Standard Chartered, HSBC, and more, were knowingly facilitating the transfer of up to $2 trillion related to suspicious and even criminal activity. This further supports the narrative that cash is used for illicit activities way more than Bitcoin. After all, $2 trillion is roughly 10x Bitcoin’s total market capitalization.

In another interesting development, the world will finally have its first official Bitcoin exchange-traded fund, though it may not be quite where people expected. It’s a collaboration between Nasdaq and a regulated Brazilian fund manager and will be launched on the Bermuda Stock Exchange (BSX).

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DeFi markets continue to boom as the total value locked in lending protocols surpassed $10 billion this week.

In any case, it’s interesting to see whether or not the hype will continue or if we will soon witness the burst of what many consider to be a DeFi-fueled bubble.

Market Data

Market Cap: $337B | 24H Vol: 105B | BTC Dominance: 58.3%

BTC: $10,613 (+2.46%) | ETH: $344.46 (+2.61%) | XRP: $0.237(+5.3%)

cryptopost_friday_1

World’s First Bitcoin ETF Approved with Expected Launch in Bermuda by End of Year. Nasdaq has collaborated with a regulated Brazilian fund manager to launch the world’s first Bitcoin exchange-traded fund (ETF). It should go live by the end of the year on the Bermuda Stock Exchange (BSX).

CoinGecko: 23% Participate In Yield Farming, But 40% Can’t Read Smart Contracts. According to a recent report by CoinGecko, 23% of people involved in the cryptocurrency field invest in yield farming. However, 40% of them can’t read smart contracts, leaving them seriously exposed to inherent risks associated with failures in the code.

FEW Brings Out DeFi Risks: Ethereum Proponents Caught Planning to Dump on Investors. Leaked screenshots of a Telegram group chat that includes some of the most popular Ethereum proponents have sparked a tweetstorm in Crypto Twitter. It appeared as if the participants were planning to create a token, airdrop it to themselves, hype it up, and dump it on the community.

Social Capital CEO Chamath Palihapitiya: Bitcoin Is My Best Investment Bet. The popular venture capitalist Chamath Palihapitiya has said that his best investment bet so far has been on Bitcoin. This is despite him having invested in countless successful companies and startups.

Gemini Has Now Opened Doors to Crypto Investors in the UK. The popular US-based Bitcoin trading platform Gemini has officially launched operations for its entire range of services in the United Kingdom. This comes weeks after it received a license from the country’s Financial Conduct Authority (FCA).

Document Leak Suggests Major Banks Facilitated Transfer of $2 Trillion in Dirty Money – 10x Current Bitcoin’s Market Cap. Buzz Feed reported on a major document leak that suggests that major banks have been facilitating the transfer of up to $2 trillion associated with illicit and criminal activities. To put things in perspective, this is roughly 10x Bitcoin’s total market cap.

Charts

This week we have a chart analysis of Bitcoin, Ethereum, Ripple, Chainlink, and Tezos – click here for the full price analysis.

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Disclaimer: Information found on CryptoPotato is those of writers quoted. It does not represent the opinions of CryptoPotato on whether to buy, sell, or hold any investments. You are advised to conduct your own research before making any investment decisions. Use provided information at your own risk. See Disclaimer for more information.

Cryptocurrency charts by TradingView.


Source: https://cryptopotato.com/bitcoin-maintains-10k-as-crypto-market-lost-16-billion-in-7-days-the-weekly-crypto-market-update/

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Bitcoin Price Needs to Break This Key Level Before $11,000 is Possible (BTC Analysis)

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Bitcoin bulls appear to be recharging after successfully breaking back into the main channel (yellow) above $10,360 and surging to $10,790.

An influx of $5 billion into the global crypto market cap over the past 24 hours has really helped prop up BTC prices above $10,700 and is also giving several altcoins a moment in the sun.

However, while this is promising news, we still need to see the global crypto market set a new higher-high above $355 billion (total market cap) to break the current downtrend and confirm the market has flipped bullish.

Right now, the only thing standing in the way of Bitcoin returning to the psychological $11,000 level is the 50-EMA line (blue) on the following daily BTC/USD chart. Currently around $10,780.

Price Levels to Watch in the Short-term

The daily 50 EMA is the main target right now for bullish traders; however, just slightly above that at $10,787, we also have the 4-hour 200 EMA (red), which received a noticeable reaction from bearish traders yesterday during Bitcoin’s surging rally.

These two price points will need considerable buying pressure to break, but if bulls are successful again, it should be an easy ride to $11,000.

Looking above that, we have the $11,200, the $11,360, and the $11,530 as the next set of likely resistances if Bitcoin manages to sustain its current momentum.

If bearish traders manage to defend the daily 50-EMA line, however, then we should look to the $10,580 as the first area of support, followed by the $10,440 and the main channel support line at approximately $10,390.

On the daily RSI indicator, we can see that Bitcoin has still not broken above the long-standing resistance (yellow arrow) with a higher high since August 1, 2020. This will be a crucial factor to observe over the next week as the leading crypto asset attempts to retake $11,000. If the indicator line moves above this resistance it will provide a strong buy signal to traders and could help drive prices up towards a new monthly high.

Total market capital: $347 billion

Bitcoin market capital: $198 billion

Bitcoin dominance: 57.1%

*Data provided by Coingecko.

Bitstamp BTC/USD Daily Chart

bitcoin price chart
BTC/USD chart via Tradingview

Bitstamp BTC/USD 4-Hour Chart

bitcoin price
BTC/USD chart via Tradingview
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Disclaimer: Information found on CryptoPotato is those of writers quoted. It does not represent the opinions of CryptoPotato on whether to buy, sell, or hold any investments. You are advised to conduct your own research before making any investment decisions. Use provided information at your own risk. See Disclaimer for more information.

Cryptocurrency charts by TradingView.


Source: https://cryptopotato.com/bitcoin-price-needs-to-break-this-key-level-before-11000-is-possible-btc-analysis/

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