The Bader-Ofer election system in Israel

bader-ofer.ts

import _ from 'lodash'

type ExcessAgreements = [string, string][]

interface Inputs {
    readonly threshold: number
    readonly results: Record<string, number>
    readonly excessAgreements: ExcessAgreements
    readonly seats: number
}

interface VotesAndSeats {
    readonly votes: number
    seats: number
}

type Group = Record<string, VotesAndSeats>

function baderOfer({
    excessAgreements,
    threshold,
    results,
    seats,
}: Inputs): Record<string, number> {
    const sumVotes = sumValues(results)
    const minimalVotesNeeded = sumVotes * threshold
    const relevantResults = _.pickBy(results, v => v >= minimalVotesNeeded)

    return afterThresholdExclusion({
        excessAgreements: _.filter(excessAgreements, names =>
            _.every(names, name => _.has(relevantResults, name))
        ),
        results: relevantResults,
        seats,
    })
}

function afterThresholdExclusion({
    excessAgreements,
    results,
    seats,
}: Omit<Inputs, 'threshold'>): Record<string, number> {
    const sumVotes = sumValues(results)
    const pricePerSeat = sumVotes / seats
    const allocation: Record<string, VotesAndSeats> = _.mapValues(
        results,
        v => ({
            votes: v,
            seats: Math.floor(v / pricePerSeat),
        })
    )
    allocateExtraSeats(allocation, excessAgreements, seats)
    return _.mapValues(allocation, 'seats')
}

function allocateExtraSeats(
    initialAllocation: Record<string, VotesAndSeats>,
    excessAgreements: ExcessAgreements,
    seats: number
): void {
    const seatsAllocated = _(initialAllocation).values().sumBy('seats')
    const seatsRemaining = seats - seatsAllocated

    const groups = createGroups(initialAllocation, excessAgreements)
    _.times(seatsRemaining).forEach(() => {
        const winner = whichGroupGetsAnExtraSeat(groups)
        initialAllocation[winner].seats++
    })
}

function createGroups(
    results: Record<string, VotesAndSeats>,
    excessAgreements: ExcessAgreements
): Group[] {
    const groups = _.mapValues(results, (v, k) => ({ [k]: v }))
    excessAgreements.forEach(([a, b]) => {
        groups[a][b] = groups[b][b]
        delete groups[b]
    })
    return _.values(groups)
}

function whichGroupGetsAnExtraSeat(groups: Group[]): string {
    const highestPayingGroup = _.maxBy(groups, priceGroupWillPayForExtraSeat)
    return whichPartyGetsAnExtraSeat(highestPayingGroup!)
}

function whichPartyGetsAnExtraSeat(group: Group): string {
    const winningEntry = _(group)
        .entries()
        .maxBy(([_, vs]) => pricePartyWillPayForExtraSeat(vs))
    return winningEntry![0]
}

function priceGroupWillPayForExtraSeat(group: Group): number {
    const allSeats = _(group).values().sumBy('seats')
    const allVotes = _(group).values().sumBy('votes')
    return allVotes / (allSeats + 1)
}

function pricePartyWillPayForExtraSeat(vs: VotesAndSeats): number {
    return vs.votes / (vs.seats + 1)
}

function sumValues(obj: Record<string, number>): number {
    return _.sum(_.values(obj))
}

function print(seats: Record<string, number>) {
    console.table(seats)
}

function mainWikiExample() {
    // https://he.wikipedia.org/wiki/%D7%97%D7%95%D7%A7_%D7%91%D7%93%D7%A8-%D7%A2%D7%95%D7%A4%D7%A8
    const votes = {
        a: 501_000,
        b: 403_000,
        c: 204_000,
        d: 92_000,
    }
    print(
        baderOfer({
            results: votes,
            excessAgreements: [],
            threshold: 0,
            seats: 120,
        })
    )
}

function wikiExampleWithExcessAgreement() {
    // https://he.wikipedia.org/wiki/%D7%97%D7%95%D7%A7_%D7%91%D7%93%D7%A8-%D7%A2%D7%95%D7%A4%D7%A8
    const votes = {
        a: 501_000,
        b: 403_000,
        c: 204_000,
        d: 92_000,
    }
    print(
        baderOfer({
            results: votes,
            excessAgreements: [['c', 'd']],
            threshold: 0,
            seats: 120,
        })
    )
}

function knesset20Example() {
    // From the python implementation here: https://sourceforge.net/projects/baderofermethod/
    const results = {
        Likud: 985408,
        'Zionist Union': 786313,
        'Joint List': 446583,
        'Yesh Atid': 371602,
        Kulanu: 315360,
        'The Jewish Home': 283910,
        Shas: 241613,
        'Yisrael Beiteinu': 214906,
        'United Torah Judaism': 210143,
        Meretz: 165529,
        Yachad: 125158,
        'Ale Yarok': 47180,
        'Arab List': 4301,
        'The Greens': 2992,
        'We are all friends Na Nach': 2493,
        "U'Bizchutan": 1802,
        'Hope for Change': 1385,
        'Pirate Party of Israel': 895,
        'Flower Party': 823,
        'Brit Olam': 761,
        Or: 502,
        'Living with Dignity': 423,
        'Economy Party': 337,
        Democratura: 242,
        'Social Leadership': 223,
        'Defending Our Children - No More Feeding Them Pornography': 0,
    }
    const excessAgreements: ExcessAgreements = [
        ['Likud', 'The Jewish Home'],
        ['Zionist Union', 'Meretz'],
        ['Yisrael Beiteinu', 'Kulanu'],
        ['Shas', 'United Torah Judaism'],
    ]
    print(
        baderOfer({
            results,
            excessAgreements,
            threshold: 0.0325,
            seats: 120,
        })
    )
}

function knesset23Example() {
    // Results from https://votes23.bechirot.gov.il/
    const results = {
        likud: 1_352_449,
        kahol_lavan: 1_220_381,
        meshutefet: 581_507,
        shas: 352_853,
        yehadut_ha_tora: 274_437,
        avoda_gesher_merez: 267_480,
        israel_beytenu: 263_365,
        yemina: 249_689,
        ozma_yehudit: 19_402,
        ozam_liberalit: 3_781,
    }
    const excessAgreements: ExcessAgreements = [
        ['likud', 'yemina'],
        ['kahol_lavan', 'avoda_gesher_merez'],
        ['yehadut_ha_tora', 'shas'],
    ]
    print(
        baderOfer({
            results,
            excessAgreements,
            threshold: 0.0325,
            seats: 120,
        })
    )
}

function main() {
    console.log('wiki main example')
    mainWikiExample()

    console.log('\nwiki example with excess agreement')
    wikiExampleWithExcessAgreement()

    console.log('\nKnesset 20 (March 2015)')
    knesset20Example()

    console.log('\nKnesset 23 (March 2020)')
    knesset23Example()
}

main()