Program.fs

open Domain

let awardPoint player state =
    let result = Domain.awardPoint state player
    printfn "Point to %A. Game is now %A" player result.Score
    result

[<EntryPoint>]
let main argv =
    let player1 = "Foo"
    let player2 = "Bar"

    GameState.NewGame player1 player2
    |> awardPoint player1
    |> awardPoint player2
    |> awardPoint player1
    |> awardPoint player2
    |> awardPoint player2
    |> awardPoint player2
    |> awardPoint player1
    |> awardPoint player2
    |> awardPoint player2
    |> awardPoint player2
    |> ignore
    0 // return an integer exit code

Tennis.fs

module Domain

type Point =
    | Love
    | Fifteen
    | Thirty
    | Forty

type Player = string

type Score =
    | LoveGame
    | Points of Point * Point
    | Deuce
    | Advantage of Player
    | Game of Player

type GameState =
    { Player1: Player
      Player2: Player
      Score: Score }

module GameState =
    let NewGame p1 p2 =
        { Player1 = p1
          Player2 = p2
          Score = LoveGame }


let updateScore (originalScore: Score) (toPlayer: Player) =
    match originalScore with
    | LoveGame -> Points(Fifteen, Point.Love)
    | Game _ -> originalScore
    | Points (Point.Love, b) -> Points(Fifteen, b)
    | Points (Point.Fifteen, b) -> Points(Thirty, b)
    | Points (Thirty, Forty) -> Deuce
    | Points (Point.Thirty, b) -> Points(Forty, b)
    | Points (Forty, _) -> Game toPlayer
    | Deuce -> Advantage toPlayer
    | Advantage player when player <> toPlayer -> Deuce
    | Advantage player when player = toPlayer -> Game player

let flip (originalScore: Score) =
    match originalScore with
    | Points (a, b) -> Points(b, a)
    | _ -> originalScore

let awardPoint (currentScore: GameState) (toPlayer: Player): GameState =
    match currentScore with
    | { Player1 = player } when player = toPlayer ->
        { currentScore with
              Score = updateScore currentScore.Score toPlayer }
    | { Player2 = player } when player = toPlayer ->
        { currentScore with
              Score = flip (updateScore (flip currentScore.Score) toPlayer) }

No no.. I won't cheat.. I'll take on the comments.. only way to learn.. and attempt to make 40-40 unrepresentable..

Also, just noticed Player could be better represented