brackt/app/services/simulations/darts-simulator.ts

/**
 * PDC World Darts Championship Simulator
 *
 * Monte Carlo simulation of the PDC World Darts Championship.
 * The tournament is a 128-player single-elimination bracket with 7 rounds,
 * using best-of-sets formats that increase in length each round.
 *
 * Algorithm:
 *   1. Load all participants and their Elo + world ranking from participantExpectedValues.
 *   2. Two simulation paths:
 *      a. Bracket populated: simulate from actual draw, respecting completed matches.
 *      b. Pre-bracket: top 32 seeds placed into fixed balanced bracket positions;
 *         remaining 96 players randomly drawn into unseeded slots each simulation.
 *   3. Compute per-set win probability using the logistic sigmoid:
 *        p_set = 1 / (1 + e^(-(Elo1 - Elo2) / ELO_DIVISOR))
 *   4. Compute match win probability using the Bernoulli sets model:
 *        P(win) = sum_{w2=0}^{S-1} C(S-1+w2, w2) * p^S * (1-p)^w2
 *      where S = sets to win, which varies by round.
 *   5. Track integer placement counts per tier across 50,000 simulations.
 *   6. Convert to probability distributions using exact denominators (column sums = 1.0).
 *
 * Round format (PDC World Championship):
 *   R1  (R128): best-of-3  sets, first to 2
 *   R2  (R64):  best-of-5  sets, first to 3
 *   R3  (R32):  best-of-5  sets, first to 3
 *   R4  (R16):  best-of-7  sets, first to 4
 *   QF:         best-of-7  sets, first to 4
 *   SF:         best-of-11 sets, first to 6
 *   Final:      best-of-13 sets, first to 7
 *
 * Seeding (pre-bracket path):
 *   Top 32 players (by world ranking) are seeded into fixed bracket positions
 *   using the standard balanced bracket structure:
 *     R1 seeds: 1v32, 16v17, 9v24, 8v25 (top half) + 5v28, 12v21, 13v20, 4v29
 *               3v30, 14v19, 11v22, 6v27 (bottom half) + 7v26, 10v23, 15v18, 2v31
 *   Each seed's unseeded opponent slot is randomly filled from the 96 unseeded players
 *   in each simulation run — spreading the draw uncertainty across all simulations.
 *
 * Placement bucketing (8-slot probability model):
 *   probFirst          → Champion
 *   probSecond         → Finalist
 *   probThird/Fourth   → SF losers (2/sim)
 *   probFifth–Eighth   → QF losers (4/sim)
 *   Earlier rounds     → all 0
 */

import { database } from "~/database/context";
import { eq, and } from "drizzle-orm";
import * as schema from "~/database/schema";
import type { Simulator, SimulationResult } from "./types";

// ─── Simulation parameters ────────────────────────────────────────────────────

const NUM_SIMULATIONS = 50000;

/**
 * Controls how much Elo gaps affect per-set win probability.
 * Higher = softer probabilities (more randomness).
 * Lower  = sharper (Elo differences matter more).
 *
 * Standard chess uses 400. Snooker (more random than chess) uses 700.
 * Darts is moderately volatile; 500 gives:
 *   100-pt gap → ~55% per set
 *   300-pt gap → ~63% per set
 *   500-pt gap → ~73% per set
 */
const ELO_DIVISOR = 500;

/**
 * Sets needed to win per round, in bracket order (R1 first, Final last).
 * Index 0 = R1/R128  (64 matches, best-of-3,  need 2)
 * Index 1 = R2/R64   (32 matches, best-of-5,  need 3)
 * Index 2 = R3/R32   (16 matches, best-of-5,  need 3)
 * Index 3 = R4/R16   (8  matches, best-of-7,  need 4)
 * Index 4 = QF       (4  matches, best-of-7,  need 4)
 * Index 5 = SF       (2  matches, best-of-11, need 6)
 * Index 6 = Final    (1  match,   best-of-13, need 7)
 */
const SETS_TO_WIN = [2, 3, 3, 4, 4, 6, 7] as const;

/**
 * Number of seeds that get fixed bracket positions.
 * The remaining (128 - TOP_SEEDS) players are randomly drawn.
 */
const TOP_SEEDS = 32;

// ─── Math helpers ──────────────────────────────────────────────────────────────

/**
 * Per-set win probability for player 1 vs player 2 based on Elo.
 * Exported for unit testing.
 */
export function setWinProb(elo1: number, elo2: number): number {
  return 1 / (1 + Math.exp(-(elo1 - elo2) / ELO_DIVISOR));
}

/**
 * Match win probability for player 1 using the Bernoulli sets model.
 * For a best-of-(2S-1) match (first to S sets):
 *   P(win) = sum_{w2=0}^{S-1} C(S-1+w2, w2) * p^S * (1-p)^w2
 * Exported for unit testing.
 */
export function matchWinProb(p: number, setsToWin: number): number {
  const S = setsToWin;
  let prob = 0;
  for (let w2 = 0; w2 < S; w2++) {
    prob += binomialCoeff(S - 1 + w2, w2) * Math.pow(p, S) * Math.pow(1 - p, w2);
  }
  return prob;
}

/** Binomial coefficient C(n, k) via iterative multiplication. */
function binomialCoeff(n: number, k: number): number {
  if (k === 0) return 1;
  if (k > n - k) k = n - k;
  let result = 1;
  for (let i = 0; i < k; i++) {
    result = (result * (n - i)) / (i + 1);
  }
  return result;
}

/**
 * Returns the 128-player seeded bracket R1 pair list.
 * Each entry is [participantIdA, participantIdB].
 * Top 32 seeded players fill fixed positions; 96 unseeded players are randomly
 * shuffled and assigned to the remaining slots.
 *
 * Structure:
 *   - 32 seeded-vs-unseeded matches (seeds 1–32 each face a randomly drawn unseeded opponent)
 *   - 32 unseeded-vs-unseeded matches (remaining 64 unseeded players paired randomly)
 *   Total: 64 R1 matches ✓ (128 players)
 *
 * Note: in the hot simulation loop, seeded positions are pre-computed via getSeededMatchOrder
 * and inlined directly — this function is used for testing and bracket-draw path only.
 *
 * Exported for unit testing.
 */
export function buildR1Bracket(
  seededIds: string[],   // exactly 32, index 0 = seed 1
  unseededIds: string[]  // exactly 96, shuffled
): Array<[string, string]> {
  // The 32 seeded players each face one of the first 32 unseeded opponents.
  // Seeds are arranged in bracket order using the standard balanced structure
  // for 32 seeds (same algorithm as snooker's R32_BRACKET but generalised).
  const seededMatchOrder = getSeededMatchOrder(32); // returns 32 seed positions in bracket order

  const pairs: Array<[string, string]> = [];

  // 32 seeded-vs-unseeded R1 matches
  for (let i = 0; i < 32; i++) {
    const seedPos = seededMatchOrder[i] - 1; // 0-indexed
    pairs.push([seededIds[seedPos], unseededIds[i]]);
  }

  // 32 unseeded-vs-unseeded R1 matches (players 32–95)
  for (let i = 32; i < 96; i += 2) {
    pairs.push([unseededIds[i], unseededIds[i + 1]]);
  }

  return pairs;
}

/**
 * Returns seed positions in standard balanced bracket order for N seeds.
 * Guarantees seed 1 and seed 2 can only meet in the Final.
 * E.g. for N=4: [1, 4, 3, 2] → match order 1v4, 3v2 in the top/bottom halves.
 *
 * Algorithm: start with [1, 2], repeatedly interleave (n+1 - seed) complements.
 * Exported for unit testing.
 */
export function getSeededMatchOrder(n: number): number[] {
  let order = [1, 2];
  while (order.length < n) {
    const size = order.length;
    const newOrder: number[] = [];
    for (const seed of order) {
      newOrder.push(seed);
      newOrder.push(2 * size + 1 - seed);
    }
    order = newOrder;
  }
  return order;
}

/** Fisher-Yates shuffle (in-place, returns array). */
function shuffle<T>(arr: T[]): T[] {
  for (let i = arr.length - 1; i > 0; i--) {
    const j = Math.floor(Math.random() * (i + 1));
    [arr[i], arr[j]] = [arr[j], arr[i]];
  }
  return arr;
}

// ─── Simulator ────────────────────────────────────────────────────────────────

export class DartsSimulator implements Simulator {
  async simulate(sportsSeasonId: string): Promise<SimulationResult[]> {
    const db = database();

    // 1. Find the bracket scoring event (if it exists).
    const bracketEvent = await db.query.scoringEvents.findFirst({
      where: and(
        eq(schema.scoringEvents.sportsSeasonId, sportsSeasonId),
        eq(schema.scoringEvents.eventType, "playoff_game")
      ),
    });

    // 2. Load playoff matches (empty if bracket hasn't been drawn yet).
    const allMatches = bracketEvent
      ? await db.query.playoffMatches.findMany({
          where: eq(schema.playoffMatches.scoringEventId, bracketEvent.id),
          orderBy: (m, { asc }) => [asc(m.matchNumber)],
        })
      : [];

    // 3. Load Elo ratings and world rankings.
    const evRows = await db
      .select({
        participantId: schema.participantExpectedValues.participantId,
        sourceElo: schema.participantExpectedValues.sourceElo,
        worldRanking: schema.participantExpectedValues.worldRanking,
      })
      .from(schema.participantExpectedValues)
      .where(eq(schema.participantExpectedValues.sportsSeasonId, sportsSeasonId));

    const eloMap = new Map<string, number>();
    const rankingMap = new Map<string, number>();
    for (const r of evRows) {
      if (r.sourceElo !== null && r.sourceElo !== undefined) {
        eloMap.set(r.participantId, r.sourceElo);
      }
      if (r.worldRanking !== null && r.worldRanking !== undefined) {
        rankingMap.set(r.participantId, r.worldRanking);
      }
    }

    // Determine simulation path.
    const bracketPopulated = allMatches.some((m) => m.participant1Id && m.participant2Id);

    if (bracketPopulated) {
      return this.simulateBracket(allMatches, eloMap);
    } else {
      return this.simulatePreBracket(sportsSeasonId, eloMap, rankingMap, db);
    }
  }

  // ─── Path A: Bracket drawn ────────────────────────────────────────────────

  private async simulateBracket(
    allMatches: Awaited<ReturnType<ReturnType<typeof database>["query"]["playoffMatches"]["findMany"]>>,
    eloMap: Map<string, number>
  ): Promise<SimulationResult[]> {
    // Group matches by round, sorted by match count descending (R1 first = most matches).
    const byRound = new Map<string, typeof allMatches>();
    for (const m of allMatches) {
      if (!byRound.has(m.round)) byRound.set(m.round, []);
      byRound.get(m.round)?.push(m);
    }

    const sortedRounds = [...byRound.values()]
      .toSorted((a, b) => b.length - a.length)
      .map((matches) => matches.sort((a, b) => a.matchNumber - b.matchNumber));

    if (sortedRounds.length !== 7) {
      throw new Error(
        `Expected 7 rounds for PDC World Darts Championship, found ${sortedRounds.length}. ` +
          `Rounds: ${[...byRound.keys()].join(", ")}`
      );
    }

    const [r1Matches, r2Matches, r3Matches, r4Matches, qfMatches, sfMatches, finalMatches] = sortedRounds;

    if (r1Matches.length !== 64) {
      throw new Error(
        `Expected 64 R1 matches (128-player bracket), found ${r1Matches.length}.`
      );
    }

    // Collect all 128 participant IDs from R1.
    const participantIds: string[] = [];
    for (const m of r1Matches) {
      if (!m.participant1Id || !m.participant2Id) {
        throw new Error(
          `R1 match ${m.matchNumber} is missing participants. ` +
            `Assign all 128 players to the bracket before running simulation.`
        );
      }
      participantIds.push(m.participant1Id, m.participant2Id);
    }

    const fallbackElo = 1600;

    // Cache matchWinProb — Elo values are fixed across simulations.
    const matchProbCache = new Map<string, number>();
    const simMatch = (p1: string, p2: string, setsToWin: number): { winner: string; loser: string } => {
      const elo1 = eloMap.get(p1) ?? fallbackElo;
      const elo2 = eloMap.get(p2) ?? fallbackElo;
      const cacheKey = `${elo1},${elo2},${setsToWin}`;
      let winProb = matchProbCache.get(cacheKey);
      if (winProb === undefined) {
        winProb = matchWinProb(setWinProb(elo1, elo2), setsToWin);
        matchProbCache.set(cacheKey, winProb);
      }
      const winner = Math.random() < winProb ? p1 : p2;
      return { winner, loser: winner === p1 ? p2 : p1 };
    };

    const r1ByNum  = new Map(r1Matches.map((m) => [m.matchNumber, m]));
    const r2ByNum  = new Map(r2Matches.map((m) => [m.matchNumber, m]));
    const r3ByNum  = new Map(r3Matches.map((m) => [m.matchNumber, m]));
    const r4ByNum  = new Map(r4Matches.map((m) => [m.matchNumber, m]));
    const qfByNum  = new Map(qfMatches.map((m) => [m.matchNumber, m]));
    const sfByNum  = new Map(sfMatches.map((m) => [m.matchNumber, m]));
    const finalMatch = finalMatches[0];

    const championCounts  = new Map<string, number>(participantIds.map((id) => [id, 0]));
    const finalistCounts  = new Map<string, number>(participantIds.map((id) => [id, 0]));
    const sfLoserCounts   = new Map<string, number>(participantIds.map((id) => [id, 0]));
    const qfLoserCounts   = new Map<string, number>(participantIds.map((id) => [id, 0]));

    for (let s = 0; s < NUM_SIMULATIONS; s++) {
      // R1 (64 matches)
      const r1Winners: string[] = [];
      for (let i = 1; i <= 64; i++) {
        const m = r1ByNum.get(i);
        if (!m) continue;
        if (m.isComplete && m.winnerId) {
          r1Winners.push(m.winnerId);
        } else {
          const { winner } = simMatch(m.participant1Id ?? "", m.participant2Id ?? "", SETS_TO_WIN[0]);
          r1Winners.push(winner);
        }
      }

      // R2 (32 matches)
      const r2Winners: string[] = [];
      for (let i = 1; i <= 32; i++) {
        const dbMatch = r2ByNum.get(i);
        let winner: string;
        if (dbMatch?.isComplete && dbMatch.winnerId) {
          winner = dbMatch.winnerId;
        } else {
          const p1 = r1Winners[(i - 1) * 2];
          const p2 = r1Winners[(i - 1) * 2 + 1];
          ({ winner } = simMatch(p1, p2, SETS_TO_WIN[1]));
        }
        r2Winners.push(winner);
      }

      // R3 (16 matches)
      const r3Winners: string[] = [];
      for (let i = 1; i <= 16; i++) {
        const dbMatch = r3ByNum.get(i);
        let winner: string;
        if (dbMatch?.isComplete && dbMatch.winnerId) {
          winner = dbMatch.winnerId;
        } else {
          const p1 = r2Winners[(i - 1) * 2];
          const p2 = r2Winners[(i - 1) * 2 + 1];
          ({ winner } = simMatch(p1, p2, SETS_TO_WIN[2]));
        }
        r3Winners.push(winner);
      }

      // R4 (8 matches)
      const r4Winners: string[] = [];
      for (let i = 1; i <= 8; i++) {
        const dbMatch = r4ByNum.get(i);
        let winner: string;
        if (dbMatch?.isComplete && dbMatch.winnerId) {
          winner = dbMatch.winnerId;
        } else {
          const p1 = r3Winners[(i - 1) * 2];
          const p2 = r3Winners[(i - 1) * 2 + 1];
          ({ winner } = simMatch(p1, p2, SETS_TO_WIN[3]));
        }
        r4Winners.push(winner);
      }

      // QF (4 matches)
      const qfWinners: string[] = [];
      for (let i = 1; i <= 4; i++) {
        const dbMatch = qfByNum.get(i);
        let winner: string;
        let loser: string;
        if (dbMatch?.isComplete && dbMatch.winnerId && dbMatch.loserId) {
          winner = dbMatch.winnerId;
          loser = dbMatch.loserId;
        } else {
          const p1 = r4Winners[(i - 1) * 2];
          const p2 = r4Winners[(i - 1) * 2 + 1];
          ({ winner, loser } = simMatch(p1, p2, SETS_TO_WIN[4]));
        }
        qfWinners.push(winner);
        qfLoserCounts.set(loser, (qfLoserCounts.get(loser) ?? 0) + 1);
      }

      // SF (2 matches)
      const sfWinners: string[] = [];
      for (let i = 1; i <= 2; i++) {
        const dbMatch = sfByNum.get(i);
        let winner: string;
        let loser: string;
        if (dbMatch?.isComplete && dbMatch.winnerId && dbMatch.loserId) {
          winner = dbMatch.winnerId;
          loser = dbMatch.loserId;
        } else {
          const p1 = qfWinners[(i - 1) * 2];
          const p2 = qfWinners[(i - 1) * 2 + 1];
          ({ winner, loser } = simMatch(p1, p2, SETS_TO_WIN[5]));
        }
        sfWinners.push(winner);
        sfLoserCounts.set(loser, (sfLoserCounts.get(loser) ?? 0) + 1);
      }

      // Final
      let champion: string;
      let finalist: string;
      if (finalMatch?.isComplete && finalMatch.winnerId && finalMatch.loserId) {
        champion = finalMatch.winnerId;
        finalist = finalMatch.loserId;
      } else {
        ({ winner: champion, loser: finalist } = simMatch(sfWinners[0], sfWinners[1], SETS_TO_WIN[6]));
      }
      championCounts.set(champion, (championCounts.get(champion) ?? 0) + 1);
      finalistCounts.set(finalist, (finalistCounts.get(finalist) ?? 0) + 1);
    }

    return buildResults(participantIds, NUM_SIMULATIONS, {
      championCounts,
      finalistCounts,
      sfLoserCounts,
      qfLoserCounts,
    });
  }

  // ─── Path B: Pre-bracket simulation ──────────────────────────────────────────
  // Top 32 seeds are placed into fixed bracket positions.
  // Remaining 96 players are randomly drawn into unseeded slots each simulation.

  private async simulatePreBracket(
    sportsSeasonId: string,
    eloMap: Map<string, number>,
    rankingMap: Map<string, number>,
    db: ReturnType<typeof database>
  ): Promise<SimulationResult[]> {
    const allParticipants = await db
      .select({ id: schema.participants.id, name: schema.participants.name })
      .from(schema.participants)
      .where(eq(schema.participants.sportsSeasonId, sportsSeasonId));

    if (allParticipants.length < 2) {
      throw new Error(
        `Pre-bracket simulation requires at least 2 participants (got ${allParticipants.length}). ` +
          `Add players to this sports season first.`
      );
    }

    const fallbackElo = 1600;

    // Sort participants by world ranking (ascending). Fall back to Elo order (descending) for
    // any without a ranking, then alphabetical as a final tiebreak.
    const sorted = [...allParticipants].toSorted((a, b) => {
      const rankA = rankingMap.get(a.id);
      const rankB = rankingMap.get(b.id);
      if (rankA !== undefined && rankB !== undefined) return rankA - rankB;
      if (rankA !== undefined) return -1; // ranked before unranked
      if (rankB !== undefined) return 1;
      // Both unranked — sort by Elo descending
      return (eloMap.get(b.id) ?? fallbackElo) - (eloMap.get(a.id) ?? fallbackElo);
    });

    const topSeeds = sorted.slice(0, TOP_SEEDS).map((p) => p.id);   // seeds 1–32
    const unseeded = sorted.slice(TOP_SEEDS).map((p) => p.id);       // remaining players

    const allParticipantIds = allParticipants.map((p) => p.id);
    const championCounts  = new Map<string, number>(allParticipantIds.map((id) => [id, 0]));
    const finalistCounts  = new Map<string, number>(allParticipantIds.map((id) => [id, 0]));
    const sfLoserCounts   = new Map<string, number>(allParticipantIds.map((id) => [id, 0]));
    const qfLoserCounts   = new Map<string, number>(allParticipantIds.map((id) => [id, 0]));

    // Cache set-level probabilities — fixed across all simulations.
    const matchProbCache = new Map<string, number>();
    const simMatch = (p1Id: string, p2Id: string, setsToWin: number): string => {
      const elo1 = eloMap.get(p1Id) ?? fallbackElo;
      const elo2 = eloMap.get(p2Id) ?? fallbackElo;
      const cacheKey = `${elo1},${elo2},${setsToWin}`;
      let winProb = matchProbCache.get(cacheKey);
      if (winProb === undefined) {
        winProb = matchWinProb(setWinProb(elo1, elo2), setsToWin);
        matchProbCache.set(cacheKey, winProb);
      }
      return Math.random() < winProb ? p1Id : p2Id;
    };

    // Pre-compute fixed seeded bracket positions once — only the unseeded draw changes per sim.
    const seededMatchOrder = getSeededMatchOrder(TOP_SEEDS);
    const seededSlots = seededMatchOrder.map(seed => topSeeds[seed - 1]);

    // Pad unseeded pool to 96 once before the loop.
    // In practice the admin should always load 128 players; this guards against edge cases.
    const unseededPool = [...unseeded];
    while (unseededPool.length + topSeeds.length < 128) {
      unseededPool.push(`__bye_${unseededPool.length}`);
    }

    for (let s = 0; s < NUM_SIMULATIONS; s++) {
      // Draw: shuffle the unseeded pool — seeded positions are pre-computed.
      const drawnUnseeded = shuffle([...unseededPool]);

      // Build R1 pairs inline using pre-computed seeded slots.
      const r1Pairs: Array<[string, string]> = [];
      for (let i = 0; i < TOP_SEEDS; i++) {
        r1Pairs.push([seededSlots[i], drawnUnseeded[i]]);
      }
      for (let i = TOP_SEEDS; i < drawnUnseeded.length; i += 2) {
        r1Pairs.push([drawnUnseeded[i], drawnUnseeded[i + 1]]);
      }

      // R1 (64 matches)
      const r1Winners: string[] = [];
      for (const [p1, p2] of r1Pairs) {
        r1Winners.push(simMatch(p1, p2, SETS_TO_WIN[0]));
      }

      // R2–R4 (32 / 16 / 8 matches)
      const r2Winners: string[] = [];
      for (let i = 0; i < r1Winners.length; i += 2) {
        r2Winners.push(simMatch(r1Winners[i], r1Winners[i + 1], SETS_TO_WIN[1]));
      }
      const r3Winners: string[] = [];
      for (let i = 0; i < r2Winners.length; i += 2) {
        r3Winners.push(simMatch(r2Winners[i], r2Winners[i + 1], SETS_TO_WIN[2]));
      }
      const r4Winners: string[] = [];
      for (let i = 0; i < r3Winners.length; i += 2) {
        r4Winners.push(simMatch(r3Winners[i], r3Winners[i + 1], SETS_TO_WIN[3]));
      }

      // QF (4 matches)
      const qfWinners: string[] = [];
      for (let i = 0; i < r4Winners.length; i += 2) {
        const p1 = r4Winners[i], p2 = r4Winners[i + 1];
        const winner = simMatch(p1, p2, SETS_TO_WIN[4]);
        const loser = winner === p1 ? p2 : p1;
        qfWinners.push(winner);
        qfLoserCounts.set(loser, (qfLoserCounts.get(loser) ?? 0) + 1);
      }

      // SF (2 matches)
      const sfWinners: string[] = [];
      for (let i = 0; i < qfWinners.length; i += 2) {
        const p1 = qfWinners[i], p2 = qfWinners[i + 1];
        const winner = simMatch(p1, p2, SETS_TO_WIN[5]);
        const loser = winner === p1 ? p2 : p1;
        sfWinners.push(winner);
        sfLoserCounts.set(loser, (sfLoserCounts.get(loser) ?? 0) + 1);
      }

      // Final
      const champion = simMatch(sfWinners[0], sfWinners[1], SETS_TO_WIN[6]);
      const finalist = champion === sfWinners[0] ? sfWinners[1] : sfWinners[0];
      championCounts.set(champion, (championCounts.get(champion) ?? 0) + 1);
      finalistCounts.set(finalist, (finalistCounts.get(finalist) ?? 0) + 1);
    }

    return buildResults(allParticipantIds, NUM_SIMULATIONS, {
      championCounts,
      finalistCounts,
      sfLoserCounts,
      qfLoserCounts,
    });
  }
}

// ─── Shared result builder ─────────────────────────────────────────────────────

function buildResults(
  participantIds: string[],
  N: number,
  counts: {
    championCounts: Map<string, number>;
    finalistCounts: Map<string, number>;
    sfLoserCounts: Map<string, number>;
    qfLoserCounts: Map<string, number>;
  }
): SimulationResult[] {
  const { championCounts, finalistCounts, sfLoserCounts, qfLoserCounts } = counts;

  const results: SimulationResult[] = participantIds.map((participantId) => {
    const c  = championCounts.get(participantId)  ?? 0;
    const f  = finalistCounts.get(participantId)  ?? 0;
    const sf = sfLoserCounts.get(participantId)   ?? 0;
    const qf = qfLoserCounts.get(participantId)   ?? 0;
    return {
      participantId,
      probabilities: {
        probFirst:   c  / N,
        probSecond:  f  / N,
        probThird:   sf / (2 * N),
        probFourth:  sf / (2 * N),
        probFifth:   qf / (4 * N),
        probSixth:   qf / (4 * N),
        probSeventh: qf / (4 * N),
        probEighth:  qf / (4 * N),
      },
      source: "darts_world_championship_monte_carlo",
    };
  });

  // Per-position column normalisation — ensures sums are exactly 1.0.
  const positionKeys: Array<keyof typeof results[0]["probabilities"]> = [
    "probFirst", "probSecond", "probThird", "probFourth",
    "probFifth", "probSixth", "probSeventh", "probEighth",
  ];
  for (const key of positionKeys) {
    const colSum = results.reduce((s, r) => s + r.probabilities[key], 0);
    const residual = 1.0 - colSum;
    if (residual !== 0) {
      const maxResult = results.reduce((best, r) =>
        r.probabilities[key] > best.probabilities[key] ? r : best
      );
      maxResult.probabilities[key] += residual;
    }
  }

  return results;
}