From 3a4e78be3cf1bdc1e814969a8d7c898f584c3c2d Mon Sep 17 00:00:00 2001 From: Andrew Clark Date: Sun, 9 Aug 2020 14:01:03 -0400 Subject: [PATCH] adding alpha derivation --- Deriving Alpha and parameters.ipynb | 269 ++++++++++++++++++++++++++++ 1 file changed, 269 insertions(+) create mode 100644 Deriving Alpha and parameters.ipynb diff --git a/Deriving Alpha and parameters.ipynb b/Deriving Alpha and parameters.ipynb new file mode 100644 index 0000000..7a350f9 --- /dev/null +++ b/Deriving Alpha and parameters.ipynb @@ -0,0 +1,269 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "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", + "\n", + " /**\n", + " * @dev Formula: ρ * supply / (1 - a) / (β - requestedAmount / total)**2\n", + " * For the Solidity implementation we amplify ρ and β and simplify the formula:\n", + " * weight = ρ * D\n", + " * maxRatio = β * D\n", + " * decay = a * D\n", + " * threshold = weight * supply * D ** 2 * funds ** 2 / (D - decay) / (maxRatio * funds - requestedAmount * D) ** 2\n", + " * @param _requestedAmount Requested amount of tokens on certain proposal\n", + " * @return Threshold a proposal's conviction should surpass in order to be able to\n", + " * executed it.\n", + " */\n", + " function calculateThreshold(uint256 _requestedAmount) public view returns (uint256 _threshold) {\n", + " uint256 funds = vault.balance(requestToken);\n", + " require(maxRatio.mul(funds) > _requestedAmount.mul(D), ERROR_AMOUNT_OVER_MAX_RATIO);\n", + " uint256 supply = stakeToken.totalSupply();\n", + " // denom = maxRatio * 2 ** 64 / D - requestedAmount * 2 ** 64 / funds\n", + " uint256 denom = (maxRatio << 64).div(D).sub((_requestedAmount << 64).div(funds));\n", + " // _threshold = (weight * 2 ** 128 / D) / (denom ** 2 / 2 ** 64) * supply * D / 2 ** 128\n", + " _threshold = ((weight << 128).div(D).div(denom.mul(denom) >> 64)).mul(D).div(D.sub(decay)).mul(supply) >> 64;\n", + " }" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.025" + ] + }, + "execution_count": 1, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "import numpy as np\n", + "# Z original code without alpha\n", + "def trigger_threshold_original(requested, funds, supply, beta, rho):\n", + " '''\n", + " Function that determines threshold for proposals being accepted. \n", + " '''\n", + " share = requested/funds\n", + " if share < beta:\n", + " return rho*supply/(beta-share)**2\n", + " else: \n", + " return np.inf\n", + " \n", + "#Our new code code\n", + "def trigger_threshold(requested, funds, supply, beta, rho, alpha):\n", + " '''\n", + " Function that determines threshold for proposals being accepted. \n", + " '''\n", + " share = requested/funds\n", + " if share < beta:\n", + " threshold = rho*supply/(beta-share)**2 * 1/(1-alpha)\n", + " return threshold #* (1-alpha)\n", + " else: \n", + " return np.inf\n", + " \n", + "# Assumed values\n", + "funds = 4000\n", + "requested = 100\n", + "supply = 21706\n", + "share = requested/funds\n", + "share" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [], + "source": [ + "# Sol values\n", + "D = 10000000\n", + "decay = 9999599\n", + "maxRatio = 2000000\n", + "weight = 20000" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Based on the above equations, we can see that $\\rho$ is equal to:" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.002" + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "rho = weight / D\n", + "rho" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We can derive $\\beta$ is equal to be:" + ] + }, + { + "cell_type": "code", + "execution_count": 4, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0.2" + ] + }, + "execution_count": 4, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "beta = maxRatio/D\n", + "beta" + ] + }, + { + "cell_type": "code", + "execution_count": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "1417.5346938775506" + ] + }, + "execution_count": 5, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "originalOutput = trigger_threshold_original(requested,funds,supply,beta,rho)\n", + "originalOutput" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "sol code threshold:" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "35349992.36602372" + ] + }, + "execution_count": 31, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "solThreshold = weight * supply * D ** 2 * funds ** 2 / (D - decay) / (maxRatio * funds - requested * D) ** 2\n", + "solThreshold" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Our new code threshold to match 1hive dev formulata " + ] + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "'0.0000401'" + ] + }, + "execution_count": 15, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "alpha = 1- 0.9999599\n", + "\"{:.7f}\".format(alpha)" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "1417.5915392982765" + ] + }, + "execution_count": 19, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "newThreshold = trigger_threshold(requested, funds, supply, beta, rho, alpha)\n", + "newThreshold" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.7.5" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +}