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// Copyright 2019 Parity Technologies (UK) Ltd.
// This file is part of Substrate.
// Substrate is free software: you can redistribute it and/or modify
// it under the terms of the GNU General Public License as published by
// the Free Software Foundation, either version 3 of the License, or
// (at your option) any later version.
// Substrate is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// GNU General Public License for more details.
// You should have received a copy of the GNU General Public License
// along with Substrate. If not, see <>.
//! Rust implementation of the Phragmén election algorithm. This is used in several SRML modules to
//! optimally distribute the weight of a set of voters among an elected set of candidates. In the
//! context of staking this is mapped to validators and nominators.
//! The algorithm has two phases:
//! - Sequential phragmen: performed in [`elect`] function which is first pass of the distribution
//! The results are not optimal but the execution time is less.
//! - Equalize post-processing: tries to further distribute the weight fairly among candidates.
//! Incurs more execution time.
//! The main objective of the assignments done by phragmen is to maximize the minimum backed
//! candidate in the elected set.
//! Reference implementation:
//! Further details:
#![cfg_attr(not(feature = "std"), no_std)]
use rstd::{prelude::*, collections::btree_map::BTreeMap};
use sr_primitives::PerU128;
use sr_primitives::traits::{Zero, Convert, Member, SimpleArithmetic};
mod mock;
mod tests;
/// Type used as the fraction.
type Fraction = PerU128;
/// A type in which performing operations on balances and stakes of candidates and voters are safe.
/// This module's functions expect a `Convert` type to convert all balances to u64. Hence, u128 is
/// a safe type for arithmetic operations over them.
/// Balance types converted to `ExtendedBalance` are referred to as `Votes`.
pub type ExtendedBalance = u128;
// this is only used while creating the candidate score. Due to reasons explained below
// The more accurate this is, the less likely we choose a wrong candidate.
// TODO: can be removed with proper use of per-things #2908
const SCALE_FACTOR: ExtendedBalance = u32::max_value() as ExtendedBalance + 1;
/// These are used to expose a fixed accuracy to the caller function. The bigger they are,
/// the more accurate we get, but the more likely it is for us to overflow. The case of overflow
/// is handled but accuracy will be lost. 32 or 16 are reasonable values.
// TODO: can be removed with proper use of per-things #2908
pub const ACCURACY: ExtendedBalance = u32::max_value() as ExtendedBalance + 1;
/// A candidate entity for phragmen election.
#[derive(Clone, Default)]
#[cfg_attr(feature = "std", derive(Debug))]
pub struct Candidate<AccountId> {
/// Identifier.
pub who: AccountId,
/// Intermediary value used to sort candidates.
pub score: Fraction,
/// Sum of the stake of this candidate based on received votes.
approval_stake: ExtendedBalance,
/// Flag for being elected.
elected: bool,
/// A voter entity.
#[derive(Clone, Default)]
#[cfg_attr(feature = "std", derive(Debug))]
pub struct Voter<AccountId> {
/// Identifier.
who: AccountId,
/// List of candidates proposed by this voter.
edges: Vec<Edge<AccountId>>,
/// The stake of this voter.
budget: ExtendedBalance,
/// Incremented each time a candidate that this voter voted for has been elected.
load: Fraction,
/// A candidate being backed by a voter.
#[derive(Clone, Default)]
#[cfg_attr(feature = "std", derive(Debug))]
pub struct Edge<AccountId> {
/// Identifier.
who: AccountId,
/// Load of this vote.
load: Fraction,
/// Index of the candidate stored in the 'candidates' vector.
candidate_index: usize,
/// Means a particular `AccountId` was backed by a ratio of `ExtendedBalance / ACCURACY`.
pub type PhragmenAssignment<AccountId> = (AccountId, ExtendedBalance);
/// Final result of the phragmen election.
pub struct PhragmenResult<AccountId> {
/// Just winners.
pub winners: Vec<AccountId>,
/// Individual assignments. for each tuple, the first elements is a voter and the second
/// is the list of candidates that it supports.
pub assignments: Vec<(AccountId, Vec<PhragmenAssignment<AccountId>>)>
/// A structure to demonstrate the phragmen result from the perspective of the candidate, i.e. how
/// much support each candidate is receiving.
/// This complements the [`PhragmenResult`] and is needed to run the equalize post-processing.
/// This, at the current version, resembles the `Exposure` defined in the staking SRML module, yet
/// they do not necessarily have to be the same.
#[cfg_attr(feature = "std", derive(Debug))]
pub struct Support<AccountId> {
/// The amount of support as the effect of self-vote.
pub own: ExtendedBalance,
/// Total support.
pub total: ExtendedBalance,
/// Support from voters.
pub others: Vec<PhragmenAssignment<AccountId>>,
/// A linkage from a candidate and its [`Support`].
pub type SupportMap<A> = BTreeMap<A, Support<A>>;
/// Perform election based on Phragmén algorithm.
/// Returns an `Option` the set of winners and their detailed support ratio from each voter if
/// enough candidates are provided. Returns `None` otherwise.
/// * `candidate_count`: number of candidates to elect.
/// * `minimum_candidate_count`: minimum number of candidates to elect. If less candidates exist,
/// `None` is returned.
/// * `initial_candidates`: candidates list to be elected from.
/// * `initial_voters`: voters list.
/// * `stake_of`: something that can return the stake stake of a particular candidate or voter.
/// * `self_vote`. If true, then each candidate will automatically vote for themselves with the a
/// weight indicated by their stake. Note that when this is `true` candidates are filtered by
/// having at least some backed stake from themselves.
pub fn elect<AccountId, Balance, FS, C>(
candidate_count: usize,
minimum_candidate_count: usize,
initial_candidates: Vec<AccountId>,
initial_voters: Vec<(AccountId, Vec<AccountId>)>,
stake_of: FS,
self_vote: bool,
) -> Option<PhragmenResult<AccountId>> where
AccountId: Default + Ord + Member,
Balance: Default + Copy + SimpleArithmetic,
for<'r> FS: Fn(&'r AccountId) -> Balance,
C: Convert<Balance, u64> + Convert<u128, Balance>,
let to_votes = |b: Balance|
<C as Convert<Balance, u64>>::convert(b) as ExtendedBalance;
// return structures
let mut elected_candidates: Vec<AccountId>;
let mut assigned: Vec<(AccountId, Vec<PhragmenAssignment<AccountId>>)>;
// used to cache and access candidates index.
let mut c_idx_cache = BTreeMap::<AccountId, usize>::new();
// voters list.
let num_voters = initial_candidates.len() + initial_voters.len();
let mut voters: Vec<Voter<AccountId>> = Vec::with_capacity(num_voters);
// collect candidates. self vote or filter might apply
let mut candidates = if self_vote {
// self vote. filter.
initial_candidates.into_iter().map(|who| {
let stake = stake_of(&who);
Candidate { who, approval_stake: to_votes(stake), ..Default::default() }
.filter(|c| !c.approval_stake.is_zero())
.map(|(i, c)| {
voters.push(Voter {
who: c.who.clone(),
edges: vec![Edge { who: c.who.clone(), candidate_index: i, ..Default::default() }],
budget: c.approval_stake,
load: Fraction::zero(),
c_idx_cache.insert(c.who.clone(), i);
} else {
// no self vote. just collect.
.map(|(idx, who)| {
c_idx_cache.insert(who.clone(), idx);
Candidate { who, ..Default::default() }
// early return if we don't have enough candidates
if candidates.len() < minimum_candidate_count { return None; }
// collect voters. use `c_idx_cache` for fast access and aggregate `approval_stake` of
// candidates.
voters.extend(initial_voters.into_iter().map(|(who, votes)| {
let voter_stake = stake_of(&who);
let mut edges: Vec<Edge<AccountId>> = Vec::with_capacity(votes.len());
for v in votes {
if let Some(idx) = c_idx_cache.get(&v) {
// This candidate is valid + already cached.
candidates[*idx].approval_stake = candidates[*idx].approval_stake
edges.push(Edge { who: v.clone(), candidate_index: *idx, ..Default::default() });
} // else {} would be wrong votes. We don't really care about it.
Voter {
edges: edges,
budget: to_votes(voter_stake),
load: Fraction::zero(),
// we have already checked that we have more candidates than minimum_candidate_count.
// run phragmen.
let to_elect = candidate_count.min(candidates.len());
elected_candidates = Vec::with_capacity(candidate_count);
assigned = Vec::with_capacity(candidate_count);
// main election loop
for _round in 0..to_elect {
// loop 1: initialize score
for c in &mut candidates {
if !c.elected {
c.score = Fraction::from_xth(c.approval_stake);
// loop 2: increment score
for n in &voters {
for e in &n.edges {
let c = &mut candidates[e.candidate_index];
if !c.elected && !c.approval_stake.is_zero() {
// Basic fixed-point shifting by 32.
// `n.budget.saturating_mul(SCALE_FACTOR)` will never saturate
// since n.budget cannot exceed u64,despite being stored in u128. yet,
// `*n.load / SCALE_FACTOR` might collapse to zero. Hence, 32 or 16 bits are
// better scale factors. Note that left-associativity in operators precedence is
// crucially important here.
let temp =
n.budget.saturating_mul(SCALE_FACTOR) / c.approval_stake
* (*n.load / SCALE_FACTOR);
c.score = Fraction::from_parts((*c.score).saturating_add(temp));
// loop 3: find the best
if let Some(winner) = candidates
.filter(|c| !c.elected)
.min_by_key(|c| *c.score)
// loop 3: update voter and edge load
winner.elected = true;
for n in &mut voters {
for e in &mut n.edges {
if e.who == winner.who {
e.load = Fraction::from_parts(*winner.score - *n.load);
n.load = winner.score;
} else {
} // end of all rounds
// update backing stake of candidates and voters
for n in &mut voters {
let mut assignment = (n.who.clone(), vec![]);
for e in &mut n.edges {
if let Some(c) = elected_candidates.iter().cloned().find(|c| *c == e.who) {
if c != n.who {
let ratio = {
// Full support. No need to calculate.
if *n.load == *e.load { ACCURACY }
else {
// This should not saturate. Safest is to just check
if let Some(r) = ACCURACY.checked_mul(*e.load) {
r / n.load.max(1)
} else {
// Just a simple trick.
*e.load / (n.load.max(1) / ACCURACY)
assignment.1.push((e.who.clone(), ratio));
if assignment.1.len() > 0 {
// To ensure an assertion indicating: no stake from the voter going to waste, we add
// a minimal post-processing to equally assign all of the leftover stake ratios.
let vote_count = assignment.1.len() as ExtendedBalance;
let l = assignment.1.len();
let sum = assignment.1.iter().map(|a| a.1).sum::<ExtendedBalance>();
let diff = ACCURACY.checked_sub(sum).unwrap_or(0);
let diff_per_vote= diff / vote_count;
if diff_per_vote > 0 {
for i in 0..l {
assignment.1[i%l].1 =
// `remainder` is set to be less than maximum votes of a voter (currently 16).
// safe to cast it to usize.
let remainder = diff - diff_per_vote * vote_count;
for i in 0..remainder as usize {
assignment.1[i%l].1 =
Some(PhragmenResult {
winners: elected_candidates,
assignments: assigned,
/// Performs equalize post-processing to the output of the election algorithm. This happens in
/// rounds. The number of rounds and the maximum diff-per-round tolerance can be tuned through input
/// parameters.
/// No value is returned from the function and the `supports` parameter is updated.
/// * `assignments`: exactly the same is the output of phragmen.
/// * `supports`: mutable reference to s `SupportMap`. This parameter is updated.
/// * `tolerance`: maximum difference that can occur before an early quite happens.
/// * `iterations`: maximum number of iterations that will be processed.
/// * `stake_of`: something that can return the stake stake of a particular candidate or voter.
pub fn equalize<Balance, AccountId, FS, C>(
mut assignments: Vec<(AccountId, Vec<PhragmenAssignment<AccountId>>)>,
supports: &mut SupportMap<AccountId>,
tolerance: ExtendedBalance,
iterations: usize,
stake_of: FS,
) where
C: Convert<Balance, u64> + Convert<u128, Balance>,
for<'r> FS: Fn(&'r AccountId) -> Balance,
AccountId: Ord + Clone,
// prepare the data for equalise
for _i in 0..iterations {
let mut max_diff = 0;
for (voter, assignment) in assignments.iter_mut() {
let voter_budget = stake_of(&voter);
let diff = do_equalize::<_, _, C>(
if diff > max_diff { max_diff = diff; }
if max_diff < tolerance {
/// actually perform equalize. same interface is `equalize`. Just called in loops with a check for
/// maximum difference.
fn do_equalize<Balance, AccountId, C>(
voter: &AccountId,
budget_balance: Balance,
elected_edges: &mut Vec<(AccountId, ExtendedBalance)>,
support_map: &mut SupportMap<AccountId>,
tolerance: ExtendedBalance
) -> ExtendedBalance where
C: Convert<Balance, u64> + Convert<u128, Balance>,
AccountId: Ord + Clone,
let to_votes = |b: Balance|
<C as Convert<Balance, u64>>::convert(b) as ExtendedBalance;
let budget = to_votes(budget_balance);
// Nothing to do. This voter had nothing useful.
// Defensive only. Assignment list should always be populated.
if elected_edges.is_empty() { return 0; }
let stake_used = elected_edges
.fold(0 as ExtendedBalance, |s, e| s.saturating_add(e.1));
let backed_stakes_iter = elected_edges
.filter_map(|e| support_map.get(&e.0))
let backing_backed_stake = elected_edges
.filter(|e| e.1 > 0)
.filter_map(|e| support_map.get(&e.0))
let mut difference;
if backing_backed_stake.len() > 0 {
let max_stake = backing_backed_stake
.expect("vector with positive length will have a max; qed");
let min_stake = backed_stakes_iter
.expect("iterator with positive length will have a min; qed");
difference = max_stake.saturating_sub(min_stake);
difference = difference.saturating_add(budget.saturating_sub(stake_used));
if difference < tolerance {
return difference;
} else {
difference = budget;
// Undo updates to support
elected_edges.iter_mut().for_each(|e| {
if let Some(support) = support_map.get_mut(&e.0) { =;
support.others.retain(|i_support| i_support.0 != *voter);
e.1 = 0;
if let Some(e) = support_map.get(&e.0) { } else { Zero::zero() }
let mut cumulative_stake: ExtendedBalance = 0;
let mut last_index = elected_edges.len() - 1;
elected_edges.iter_mut().enumerate().for_each(|(idx, e)| {
if let Some(support) = support_map.get_mut(&e.0) {
let stake: ExtendedBalance =;
let stake_mul = stake.saturating_mul(idx as ExtendedBalance);
let stake_sub = stake_mul.saturating_sub(cumulative_stake);
if stake_sub > budget {
last_index = idx.checked_sub(1).unwrap_or(0);
cumulative_stake = cumulative_stake.saturating_add(stake);
let last_stake = elected_edges[last_index].1;
let split_ways = last_index + 1;
let excess = budget
.saturating_sub(last_stake.saturating_mul(split_ways as ExtendedBalance));
elected_edges.iter_mut().take(split_ways).for_each(|e| {
if let Some(support) = support_map.get_mut(&e.0) {
e.1 = (excess / split_ways as ExtendedBalance)
.saturating_sub(; =;
support.others.push((voter.clone(), e.1));
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