Keeping score over three rounds
The three-round rule, scoring on paper, and the rematch key that gives a set a clear ending.
The hard part of a two-person evening is not the playing, it is knowing when to stop. No scoreboard on the screen tells you the set is over, and by the fourth round the two of you remember the previous ones differently. One sheet of paper and a pen settle this more cheaply than any app. It also settles the question of who plays first, which is the other argument an evening usually contains.
Agree the rule before the first round, and say it as a full sentence: three rounds, first to two wins ends the set, a draw means one extra round. Write that sentence at the top of the sheet. Once the rule is on paper nobody has to remember it, and nobody can reinterpret it in their own favour during the third round. Both people should be able to repeat it back before the first round starts.
The simplest way to keep score is a tally mark per round won, in two separate columns. Do not record the in-game score, only who won each round. The in-game number moves constantly and says nothing about the set, while the tally marks can be counted at a glance in one second, even while both people are talking loudly. Two columns, one name at the top of each, and the marks go left to right.
Record the mark the moment a round ends, before pressing play again. That habit matters more than it sounds: leave it until after the next round and the two of you will argue about who won the previous one, and that argument always lasts longer than the round did. A mark made in two seconds ends the dispute before it starts. Keep the pen on the sheet, because hunting for a pen is its own way of postponing the next round.
For the rematch, use exactly one key and agree which one it is, either R or the play-again button on the result screen. The person who just lost presses it, because that is the person who wants the next round immediately. Do not reload the whole page to start a new round: reloading loses the choices already made and sometimes loses the round the other player is in the middle of.
If the game has a clock, put a limit on the whole set rather than on each round. A set of three usually fits inside fifteen minutes, and if it runs past that, stop at the end of the current round instead of squeezing in one more. The limit is not there to cut the fun short, it is there so the evening still has room for something else. A clock on the wall works as well as one on the screen, as long as both people can see it.
When the gap between the two of you is obvious, change the handicap rather than the game. The fairest version gives the weaker player one small advantage: they choose the first game, or they choose their side, or they start one round ahead. Changing the handicap keeps both people in the same game, while changing the game just lets the stronger player keep picking whatever they are best at. The stronger player will resist at first, and will accept it once the third round is close.
Never use the in-game score to decide a set. Plenty of games add points for time survived, lives left or streaks, so two people can each win three rounds and still end with different numbers. The paper that records wins and losses is the only thing both of you see the same way and read the same way. The paper is the only record both people can point at without opening anything.
Last, keep the sheet for next time. A page with a date and a few dozen marks says more than any on-screen board: which evenings ran long, which rounds kept getting demanded back, and who won more often that week. It is the cheapest souvenir an evening can leave behind. Keep it in a drawer with the pen, and the next evening starts with the rules already agreed and the score already visible.