Swift Model Training and Gameplay for Backgammon

AIPlayer.swift

/// Selects the best legal move for the current player using a trained NeuralNetwork.
public enum AIPlayer {

    /// Returns the move that maximises P(currentPlayer wins) according to the network.
    public static func bestMove(for state: GameState, network: NeuralNetwork) -> Move {
        let moves = uniqueByResult(MoveGenerator.legalMoves(for: state), from: state)
        let player = state.currentPlayer
        return moves.max { a, b in
            score(move: a, state: state, network: network, player: player)
            < score(move: b, state: state, network: network, player: player)
        } ?? Move([])
    }

    // MARK: - Private

    private static func score(
        move: Move, state: GameState, network: NeuralNetwork, player: Player
    ) -> Double {
        let result = GameEngine.applyUnchecked(move, to: state)
        if let winner = result.winner { return winner == player ? 1.0 : 0.0 }
        return network.evaluate(result, for: player)
    }

    /// Keeps one representative move per unique resulting board state.
    private static func uniqueByResult(_ moves: [Move], from state: GameState) -> [Move] {
        var seen = [GameState: Move]()
        for move in moves {
            let result = GameEngine.applyUnchecked(move, to: state)
            if seen[result] == nil { seen[result] = move }
        }
        return seen.isEmpty ? [Move([])] : Array(seen.values)
    }
}

backgammon_v2.json

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BoardEncoder.swift

/// Encodes a GameState into Tesauro's 198-feature vector.
///
/// Layout (per point, 24 points × 8 features = 192):
///   [0] 1 if white has ≥1 checker
///   [1] 1 if white has ≥2 checkers
///   [2] 1 if white has ≥3 checkers
///   [3] (n-3)/2 if white has ≥4 checkers, else 0
///   [4-7] same for black
/// [192] white bar / 2
/// [193] black bar / 2
/// [194] white borne off / 15
/// [195] black borne off / 15
/// [196] 1 if white to move
/// [197] 1 if black to move
public enum BoardEncoder {
    public static let inputSize = 198

    public static func encode(_ state: GameState) -> [Double] {
        var f = [Double](repeating: 0.0, count: inputSize)
        var offset = 0
        for point in 0..<24 {
            let raw = state.points[point]
            encode(count: max(0,  raw), into: &f, at: offset)
            encode(count: max(0, -raw), into: &f, at: offset + 4)
            offset += 8
        }
        f[192] = Double(state.whiteBar)      / 2.0
        f[193] = Double(state.blackBar)      / 2.0
        f[194] = Double(state.whiteBorneOff) / 15.0
        f[195] = Double(state.blackBorneOff) / 15.0
        f[196] = state.currentPlayer == .white ? 1.0 : 0.0
        f[197] = state.currentPlayer == .black ? 1.0 : 0.0
        return f
    }

    private static func encode(count n: Int, into f: inout [Double], at offset: Int) {
        f[offset]     = n >= 1 ? 1.0 : 0.0
        f[offset + 1] = n >= 2 ? 1.0 : 0.0
        f[offset + 2] = n >= 3 ? 1.0 : 0.0
        f[offset + 3] = n >= 4 ? Double(n - 3) / 2.0 : 0.0
    }
}

GameEngine.swift

import Foundation

/// Applies moves to game state and manages turn flow.
public enum GameEngine {

    // MARK: - Dice

    public static func roll() -> [Int] {
        let d1 = Int.random(in: 1...6)
        let d2 = Int.random(in: 1...6)
        return d1 == d2 ? [d1, d1, d1, d1] : [d1, d2]
    }

    // MARK: - Turn management

    /// Returns a new state with dice set, ready for move selection.
    public static func applyRoll(_ dice: [Int], to state: GameState) -> GameState {
        var next = state
        next.remainingDice = dice
        return next
    }

    /// Applies a full move (sequence of checker moves) and advances the turn.
    /// Returns nil if the move is not in the legal move list.
    public static func apply(_ move: Move, to state: GameState) -> GameState? {
        let legal = MoveGenerator.legalMoves(for: state)
        guard legal.contains(move) else { return nil }
        var next = state
        for cm in move.checkerMoves {
            applyCheckerMove(cm, player: state.currentPlayer, to: &next)
        }
        if next.winner == nil {
            next.currentPlayer = state.currentPlayer.opponent
            next.remainingDice = []
        }
        return next
    }

    /// Applies a move without legality checking (used internally by MoveGenerator and trainer).
    public static func applyUnchecked(_ move: Move, to state: GameState) -> GameState {
        var next = state
        for cm in move.checkerMoves {
            applyCheckerMove(cm, player: state.currentPlayer, to: &next)
        }
        if next.winner == nil {
            next.currentPlayer = state.currentPlayer.opponent
            next.remainingDice = []
        }
        return next
    }

    // MARK: - Internals

    public static func applyCheckerMove(_ cm: CheckerMove, player: Player, to state: inout GameState) {
        // Remove from source
        if cm.isFromBar {
            if player == .white { state.whiteBar -= 1 } else { state.blackBar -= 1 }
        } else {
            if player == .white { state.points[cm.from] -= 1 } else { state.points[cm.from] += 1 }
        }

        if cm.isBearOff {
            if player == .white { state.whiteBorneOff += 1 } else { state.blackBorneOff += 1 }
            return
        }

        // Hit opponent blot?
        let opponentCount = state.checkerCount(at: cm.to, for: player.opponent)
        if opponentCount == 1 {
            if player == .white { state.points[cm.to] = 0; state.blackBar += 1 }
            else                { state.points[cm.to] = 0; state.whiteBar += 1 }
        }

        // Place on destination
        if player == .white { state.points[cm.to] += 1 } else { state.points[cm.to] -= 1 }
    }
}

GameState.swift

/// Represents the complete state of a backgammon game at a single point in time.
///
/// Board layout (from White's perspective):
///   Points 1-24, where White bears off at point 0 and Black bears off at point 25.
///   Positive counts = White checkers, negative counts = Black checkers.
///
/// Bar: whitebar holds White checkers on the bar, blackBar holds Black checkers.
public struct GameState: Sendable, Hashable {

    // MARK: - Board

    /// 24 points. Positive = White checkers, negative = Black checkers.
    public var points: [Int]

    /// White checkers on the bar.
    public var whiteBar: Int

    /// Black checkers on the bar.
    public var blackBar: Int

    /// White checkers already borne off.
    public var whiteBorneOff: Int

    /// Black checkers already borne off.
    public var blackBorneOff: Int

    // MARK: - Turn

    public var currentPlayer: Player

    /// Remaining dice to use this turn. Empty when waiting for a roll.
    public var remainingDice: [Int]

    // MARK: - Init

    public init() {
        points = Array(repeating: 0, count: 24)
        whiteBar = 0
        blackBar = 0
        whiteBorneOff = 0
        blackBorneOff = 0
        currentPlayer = .white
        remainingDice = []
        setupInitialPosition()
    }

    public init(
        points: [Int],
        whiteBar: Int = 0,
        blackBar: Int = 0,
        whiteBorneOff: Int = 0,
        blackBorneOff: Int = 0,
        currentPlayer: Player = .white,
        remainingDice: [Int] = []
    ) {
        self.points = points
        self.whiteBar = whiteBar
        self.blackBar = blackBar
        self.whiteBorneOff = whiteBorneOff
        self.blackBorneOff = blackBorneOff
        self.currentPlayer = currentPlayer
        self.remainingDice = remainingDice
    }

    // MARK: - Derived state

    public var winner: Player? {
        if whiteBorneOff == 15 { return .white }
        if blackBorneOff == 15 { return .black }
        return nil
    }

    public var isGameOver: Bool { winner != nil }

    /// Checker count on a point from the current player's perspective (positive = owned).
    public func checkerCount(at point: Int, for player: Player) -> Int {
        let raw = points[point]
        return player == .white ? raw : -raw
    }

    // MARK: - Private setup

    private mutating func setupInitialPosition() {
        // Standard backgammon starting position (White's perspective, 1-indexed → 0-indexed)
        // White: 2 on point 24, 5 on point 13, 3 on point 8, 5 on point 6
        // Black: mirror image (negative values)
        points[23] =  2   // White: point 24
        points[12] =  5   // White: point 13
        points[ 7] =  3   // White: point 8
        points[ 5] =  5   // White: point 6

        points[ 0] = -2   // Black: point 1  (mirror of White point 24)
        points[11] = -5   // Black: point 12 (mirror of White point 13)
        points[16] = -3   // Black: point 17 (mirror of White point 8)
        points[18] = -5   // Black: point 19 (mirror of White point 6)
    }
}

Move.swift

/// A single checker movement using one die.
public struct CheckerMove: Sendable, Hashable {
    /// Source point (0-23), or 24 meaning "from the bar".
    public let from: Int
    /// Destination point (0-23), or -1 meaning "bear off".
    public let to: Int
    /// The die value used.
    public let die: Int

    public static let barIndex = 24
    public static let bearOffIndex = -1

    public var isFromBar: Bool { from == CheckerMove.barIndex }
    public var isBearOff: Bool { to == CheckerMove.bearOffIndex }

    public init(from: Int, to: Int, die: Int) {
        self.from = from
        self.to = to
        self.die = die
    }
}

/// A full turn: an ordered sequence of checker moves that consumes some or all remaining dice.
public struct Move: Sendable, Hashable {
    public let checkerMoves: [CheckerMove]

    public init(_ checkerMoves: [CheckerMove]) {
        self.checkerMoves = checkerMoves
    }

    public var isEmpty: Bool { checkerMoves.isEmpty }
}

MoveGenerator.swift

/// Generates all legal moves for the current player given a rolled set of dice.
///
/// Backgammon rules enforced:
///   - Bar checkers must enter before any other move.
///   - Must use both dice if possible; if only one can be used, must use the higher.
///   - Bearing off requires all checkers in the home board; exact or higher die allowed.
///   - Cannot land on a point holding 2+ opponent checkers (a "made point").
public enum MoveGenerator {

    public static func legalMoves(for state: GameState) -> [Move] {
        let dice = state.remainingDice
        guard !dice.isEmpty, state.winner == nil else { return [] }

        let player = state.currentPlayer
        let uniqueDice = uniqueCombinations(of: dice)

        var allMoves: [Move] = []
        for diceSet in uniqueDice {
            let moves = generateMoves(state: state, player: player, dice: diceSet, path: [])
            allMoves.append(contentsOf: moves)
        }

        // Deduplicate by checker move sequence
        let unique = Array(Set(allMoves))

        // Must maximise dice usage: prefer moves that use more dice
        return filterMaximalMoves(unique)
    }

    // MARK: - Recursive move building

    private static func generateMoves(
        state: GameState,
        player: Player,
        dice: [Int],
        path: [CheckerMove]
    ) -> [Move] {
        if dice.isEmpty { return [Move(path)] }

        let bar = player == .white ? state.whiteBar : state.blackBar
        var results: [Move] = []

        if bar > 0 {
            // Must enter from bar
            for (i, die) in dice.enumerated() {
                if let move = enterMove(from: bar > 0, die: die, player: player, state: state) {
                    var nextState = state
                    GameEngine.applyCheckerMove(move, player: player, to: &nextState)
                    var remaining = dice
                    remaining.remove(at: i)
                    let sub = generateMoves(state: nextState, player: player, dice: remaining, path: path + [move])
                    results.append(contentsOf: sub)
                }
            }
            if results.isEmpty { return [Move(path)] }
            return results
        }

        let canBearOff = allCheckersInHomeBoard(state: state, player: player)

        for (i, die) in dice.enumerated() {
            // Skip duplicate dice values already processed at this level
            if i > 0 && dice[i] == dice[i - 1] { continue }

            let sources = sourcesForNormalMove(state: state, player: player, die: die, canBearOff: canBearOff)
            for from in sources {
                let to = destination(from: from, die: die, player: player, canBearOff: canBearOff, state: state)
                let move = CheckerMove(from: from, to: to, die: die)
                var nextState = state
                GameEngine.applyCheckerMove(move, player: player, to: &nextState)
                var remaining = dice
                remaining.remove(at: i)
                let sub = generateMoves(state: nextState, player: player, dice: remaining, path: path + [move])
                results.append(contentsOf: sub)
            }
        }

        if results.isEmpty { return [Move(path)] }
        return results
    }

    // MARK: - Enter from bar

    private static func enterMove(from _: Bool, die: Int, player: Player, state: GameState) -> CheckerMove? {
        let target: Int
        if player == .white {
            // White enters on points 19-24 (indices 18-23), die 1 → index 23, die 6 → index 18
            target = 24 - die  // index
        } else {
            // Black enters on points 1-6 (indices 0-5), die 1 → index 0, die 6 → index 5
            target = die - 1   // index
        }
        guard target >= 0, target < 24 else { return nil }
        guard canLand(at: target, player: player, state: state) else { return nil }
        return CheckerMove(from: CheckerMove.barIndex, to: target, die: die)
    }

    // MARK: - Normal move sources

    private static func sourcesForNormalMove(
        state: GameState,
        player: Player,
        die: Int,
        canBearOff: Bool
    ) -> [Int] {
        var sources: [Int] = []
        for point in 0..<24 {
            let count = state.checkerCount(at: point, for: player)
            guard count > 0 else { continue }

            let rawDest = pointIndex(from: point, die: die, player: player)

            if rawDest >= 0 && rawDest < 24 {
                if canLand(at: rawDest, player: player, state: state) {
                    sources.append(point)
                }
            } else if canBearOff {
                // Bear off: destination is off the board
                if bearOffAllowed(from: point, die: die, player: player, state: state) {
                    sources.append(point)
                }
            }
        }
        return sources
    }

    private static func destination(
        from point: Int,
        die: Int,
        player: Player,
        canBearOff: Bool,
        state: GameState
    ) -> Int {
        let raw = pointIndex(from: point, die: die, player: player)
        if raw >= 0 && raw < 24 { return raw }
        if canBearOff { return CheckerMove.bearOffIndex }
        return raw // shouldn't reach
    }

    // MARK: - Bear off helpers

    private static func allCheckersInHomeBoard(state: GameState, player: Player) -> Bool {
        let bar = player == .white ? state.whiteBar : state.blackBar
        if bar > 0 { return false }

        // White home board: points 1-6 (indices 0-5)
        // Black home board: points 19-24 (indices 18-23)
        let homeRange: Range<Int> = player == .white ? 0..<6 : 18..<24
        for point in 0..<24 {
            guard !homeRange.contains(point) else { continue }
            if state.checkerCount(at: point, for: player) > 0 { return false }
        }
        return true
    }

    private static func bearOffAllowed(from point: Int, die: Int, player: Player, state: GameState) -> Bool {
        let raw = pointIndex(from: point, die: die, player: player)
        if player == .white {
            if raw == -1 { return true } // exact
            if raw < -1 {
                // Die overshoots; only allowed if no checker is on a higher point
                let higherRange = (point + 1)..<6
                return higherRange.allSatisfy { state.checkerCount(at: $0, for: player) == 0 }
            }
        } else {
            if raw == 24 { return true }
            if raw > 24 {
                let lowerRange = 18..<point
                return lowerRange.allSatisfy { state.checkerCount(at: $0, for: player) == 0 }
            }
        }
        return false
    }

    // MARK: - Landing rules

    private static func canLand(at point: Int, player: Player, state: GameState) -> Bool {
        let opponentCount = state.checkerCount(at: point, for: player.opponent)
        return opponentCount <= 1  // 0 = empty/own, 1 = blot (can hit), 2+ = made point
    }

    // MARK: - Geometry

    private static func pointIndex(from point: Int, die: Int, player: Player) -> Int {
        point + player.direction * die
    }

    // MARK: - Maximise dice usage

    private static func filterMaximalMoves(_ moves: [Move]) -> [Move] {
        guard !moves.isEmpty else { return [] }
        let maxUsed = moves.map { $0.checkerMoves.count }.max()!
        let maxMoves = moves.filter { $0.checkerMoves.count == maxUsed }

        // If only one die can be used, must use the higher value
        if maxUsed == 1 {
            let maxDie = maxMoves.map { $0.checkerMoves[0].die }.max()!
            return maxMoves.filter { $0.checkerMoves[0].die == maxDie }
        }
        return maxMoves
    }

    // MARK: - Dice combinations

    private static func uniqueCombinations(of dice: [Int]) -> [[Int]] {
        // For a standard roll of 2 dice (or 4 for doubles), permutations matter for
        // move ordering but not legality — return the sorted unique set.
        guard !dice.isEmpty else { return [[]] }
        return [dice.sorted()]
    }
}

NeuralNetwork.swift

import Foundation

/// Feedforward network: 198 → sigmoid(40) → sigmoid(1).
/// Output is P(White wins) from any board position.
///
/// Weight layout (row-major):
///   w1[i * hiddenSize + j]  — input i to hidden j
///   w2[j]                   — hidden j to output
public final class NeuralNetwork: @unchecked Sendable {
    public static let inputSize  = 198
    public static let hiddenSize =  40

    public var w1: [Double]   // inputSize × hiddenSize
    public var b1: [Double]   // hiddenSize
    public var w2: [Double]   // hiddenSize
    public var b2: [Double]   // 1

    public init() {
        w1 = Self.rand(count: Self.inputSize * Self.hiddenSize)
        b1 = [Double](repeating: 0.0, count: Self.hiddenSize)
        w2 = Self.rand(count: Self.hiddenSize)
        b2 = [0.0]
    }

    public init(w1: [Double], b1: [Double], w2: [Double], b2: [Double]) {
        self.w1 = w1; self.b1 = b1; self.w2 = w2; self.b2 = b2
    }

    // MARK: - Inference

    /// Returns P(White wins) ∈ (0,1).
    public func evaluate(_ state: GameState) -> Double {
        forward(BoardEncoder.encode(state)).output
    }

    /// Returns P(player wins) ∈ (0,1).
    public func evaluate(_ state: GameState, for player: Player) -> Double {
        let v = evaluate(state)
        return player == .white ? v : 1.0 - v
    }

    // MARK: - Forward pass (cache-friendly row-major accumulation)

    public func forward(_ input: [Double]) -> (hidden: [Double], output: Double) {
        let H = Self.hiddenSize
        var hidden = b1                           // start with biases
        for i in 0..<Self.inputSize {
            let xi = input[i]
            guard xi != 0 else { continue }       // ~60% of features are 0 at start
            let base = i * H
            for j in 0..<H { hidden[j] += xi * w1[base + j] }
        }
        for j in 0..<H { hidden[j] = sigmoid(hidden[j]) }

        var out = b2[0]
        for j in 0..<H { out += hidden[j] * w2[j] }
        return (hidden, sigmoid(out))
    }

    // MARK: - Gradients + forward in one pass (avoids double forward in TD update)

    /// Computes ∂output/∂weights given pre-computed forward activations.
    /// Call `forward` once, pass results here to avoid a second forward pass.
    public func gradients(input: [Double], hidden: [Double], output: Double) -> NetworkGradients {
        let H   = Self.hiddenSize
        let dOut = sigmoidDeriv(output)

        var gW2 = [Double](repeating: 0.0, count: H)
        let gB2 = [dOut]
        for j in 0..<H { gW2[j] = hidden[j] * dOut }

        var dH  = [Double](repeating: 0.0, count: H)
        var gB1 = [Double](repeating: 0.0, count: H)
        for j in 0..<H {
            let d = sigmoidDeriv(hidden[j]) * w2[j] * dOut
            dH[j]  = d
            gB1[j] = d
        }

        var gW1 = [Double](repeating: 0.0, count: Self.inputSize * H)
        for i in 0..<Self.inputSize {
            let xi   = input[i]
            let base = i * H
            for j in 0..<H { gW1[base + j] = xi * dH[j] }
        }
        return NetworkGradients(w1: gW1, b1: gB1, w2: gW2, b2: gB2)
    }

    // MARK: - Private

    private func sigmoid(_ x: Double) -> Double { 1.0 / (1.0 + exp(-x)) }
    private func sigmoidDeriv(_ y: Double) -> Double { y * (1.0 - y) }

    private static func rand(count: Int) -> [Double] {
        (0..<count).map { _ in Double.random(in: -0.1...0.1) }
    }
}

// MARK: - Gradient / eligibility-trace arrays (same shape as network weights)

public struct NetworkGradients {
    public var w1: [Double]
    public var b1: [Double]
    public var w2: [Double]
    public var b2: [Double]

    public static func zero() -> NetworkGradients {
        NetworkGradients(
            w1: [Double](repeating: 0.0, count: NeuralNetwork.inputSize * NeuralNetwork.hiddenSize),
            b1: [Double](repeating: 0.0, count: NeuralNetwork.hiddenSize),
            w2: [Double](repeating: 0.0, count: NeuralNetwork.hiddenSize),
            b2: [0.0]
        )
    }
}

Player.swift

public enum Player: Int, Sendable, Hashable, CaseIterable {
    case white = 1
    case black = -1

    public var opponent: Player {
        self == .white ? .black : .white
    }

    /// Direction of movement on the points array. White moves from high to low index.
    public var direction: Int {
        self == .white ? -1 : 1
    }
}

swift_backgammon_readme.md

A backgammon game with a trained AI opponent.

Here's what has been built:

BackgammonEngine — a complete Swift game engine with legal move generation, all backgammon rules, and the TD(λ) neural network
BackgammonTrainer — a self-play training harness that ran 500,000 episodes in under 50 minutes on your M4 Macbook Pro
BackgammonApp — a SwiftUI iPhone game where you play against the trained AI

Set which developer toolchain runs by default.
sudo xcode-select --switch /Applications/Xcode.app/Contents/Developer

Training the model

cd /Users/gregh/dev/src/xcode/Backgammon
swift run -c release BackgammonTrainer ~/downloads/models/backgammon_v2.json

Running the game in the iPhone simulator
Select an iPhone simulator from the toolbar and hit Cmd+R.

The game begins with a prompt to load the trained model so the AI actually plays well:

After the app launches in the simulator, tap Load Model
Navigate to ~/downloads/models/backgammon_v1.json
Then tap Roll Dice to start your first game