adding alpha derivation
This commit is contained in:
Andrew Clark
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2fc25888ea
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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"In this notebook, we are deriving what the alpha value in the @dev formula is based off of 1hive Sol [code](https://github.com/1Hive/conviction-voting-app/blob/bc81f4af266bab6bc4096a31966bd9fe8a89f4a2/contracts/ConvictionVoting.sol)\n",
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"\n",
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" /**\n",
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" * @dev Formula: ρ * supply / (1 - a) / (β - requestedAmount / total)**2\n",
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" * For the Solidity implementation we amplify ρ and β and simplify the formula:\n",
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" * weight = ρ * D\n",
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" * maxRatio = β * D\n",
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" * decay = a * D\n",
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" * threshold = weight * supply * D ** 2 * funds ** 2 / (D - decay) / (maxRatio * funds - requestedAmount * D) ** 2\n",
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" * @param _requestedAmount Requested amount of tokens on certain proposal\n",
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" * @return Threshold a proposal's conviction should surpass in order to be able to\n",
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" * executed it.\n",
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" */\n",
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" function calculateThreshold(uint256 _requestedAmount) public view returns (uint256 _threshold) {\n",
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" uint256 funds = vault.balance(requestToken);\n",
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" require(maxRatio.mul(funds) > _requestedAmount.mul(D), ERROR_AMOUNT_OVER_MAX_RATIO);\n",
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" uint256 supply = stakeToken.totalSupply();\n",
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" // denom = maxRatio * 2 ** 64 / D - requestedAmount * 2 ** 64 / funds\n",
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" uint256 denom = (maxRatio << 64).div(D).sub((_requestedAmount << 64).div(funds));\n",
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" // _threshold = (weight * 2 ** 128 / D) / (denom ** 2 / 2 ** 64) * supply * D / 2 ** 128\n",
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" _threshold = ((weight << 128).div(D).div(denom.mul(denom) >> 64)).mul(D).div(D.sub(decay)).mul(supply) >> 64;\n",
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" }"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"0.025"
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]
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},
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"execution_count": 1,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"import numpy as np\n",
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"# Z original code without alpha\n",
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"def trigger_threshold_original(requested, funds, supply, beta, rho):\n",
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" '''\n",
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" Function that determines threshold for proposals being accepted. \n",
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" '''\n",
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" share = requested/funds\n",
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" if share < beta:\n",
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" return rho*supply/(beta-share)**2\n",
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" else: \n",
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" return np.inf\n",
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" \n",
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"#Our new code code\n",
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"def trigger_threshold(requested, funds, supply, beta, rho, alpha):\n",
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" '''\n",
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" Function that determines threshold for proposals being accepted. \n",
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" '''\n",
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" share = requested/funds\n",
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" if share < beta:\n",
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" threshold = rho*supply/(beta-share)**2 * 1/(1-alpha)\n",
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" return threshold #* (1-alpha)\n",
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" else: \n",
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" return np.inf\n",
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" \n",
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"# Assumed values\n",
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"funds = 4000\n",
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"requested = 100\n",
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"supply = 21706\n",
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"share = requested/funds\n",
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"share"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"# Sol values\n",
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"D = 10000000\n",
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"decay = 9999599\n",
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"maxRatio = 2000000\n",
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"weight = 20000"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Based on the above equations, we can see that $\\rho$ is equal to:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"0.002"
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]
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},
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"execution_count": 3,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"rho = weight / D\n",
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"rho"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"We can derive $\\beta$ is equal to be:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"0.2"
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]
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},
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"beta = maxRatio/D\n",
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"beta"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"1417.5346938775506"
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]
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},
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"execution_count": 5,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"originalOutput = trigger_threshold_original(requested,funds,supply,beta,rho)\n",
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"originalOutput"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"sol code threshold:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 31,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"35349992.36602372"
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]
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},
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"execution_count": 31,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"solThreshold = weight * supply * D ** 2 * funds ** 2 / (D - decay) / (maxRatio * funds - requested * D) ** 2\n",
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"solThreshold"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Our new code threshold to match 1hive dev formulata "
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]
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},
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{
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"cell_type": "code",
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"execution_count": 15,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"'0.0000401'"
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]
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},
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"execution_count": 15,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"alpha = 1- 0.9999599\n",
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"\"{:.7f}\".format(alpha)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 19,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"1417.5915392982765"
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]
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},
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"execution_count": 19,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"newThreshold = trigger_threshold(requested, funds, supply, beta, rho, alpha)\n",
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"newThreshold"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.7.5"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}
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