The math of Rock Paper Scissors

One throw

Suppose two players each choose a gesture independently, with Rock, Paper, and Scissors equally likely. Then the first player wins with probability 1/3, loses with probability 1/3, and ties with probability 1/3. Each gesture beats one of the other two, loses to the other, and ties with itself, so whatever one player picks, the other player’s three equally likely choices produce one win, one loss, and one tie.

How long ties take

A throw settles the game with probability 2/3. When tied throws are replayed, as in the official rules, the number of throws until one is decisive averages 3/2 (1.5), which is one divided by 2/3.

RPS Royale does not replay a tied throw. A tie goes to a tap race instead, so these counts describe the official rules rather than a match in the game.

The tap race is explained in the How to play guide.

Best of three and best of five

Call two players equally matched when every decided throw is a fair coin flip between them. Then each side wins a best-of-three with probability 1/2, and the same holds for a best-of-five: the format changes how long the match lasts, not who is favored. A best-of-three ends after 2 or 3 decided throws, and a best-of-five after 3, 4, or 5.

Replayed ties lengthen the match. Each decided throw takes 3/2 throws on average, so the total is the decided throws times 3/2.

Why random play cannot be exploited

Score a win as one point for you, a loss as one point for your opponent, and a tie as nothing. A player who picks each gesture with probability 1/3 has an expected score of 0 against any strategy the opponent uses, including always choosing Rock, because every gesture the opponent can pick wins, loses, and ties equally often against a uniform choice. An opponent therefore cannot gain an edge over a player who really chooses uniformly at random, and has nothing to read in that player’s choices.

The guarantee is also a limit. Random play scores zero on average against every opponent, so it removes the risk of being outguessed but does not create an edge.

What one study found about how people actually choose is in the strategy guide.

Brackets: how many matches and stages

In single elimination every match removes exactly one player, so N players need N − 1 matches in total, however the pairings fall. Each stage pairs the players still in, so the number of stages is the number of times the player count can be halved, rounding up, before one player is left. When the count is odd, one player has no opponent in that stage and takes a bye. In RPS Royale that player advances without a match and is paired in the next stage.

For example, 8 players need 3 stages with 4, 2, 1 matches, and 5 players need 3 stages with 2, 1, 1 matches and 2 byes. The table lists more sizes; it shows how brackets scale, and it is not a statement of how many players a room allows.

Room sizes and settings are on the How to play page.

Your chance of winning a round

If the pairings are shuffled and every match is an even contest, each of N players has the same chance of winning the whole bracket: 1/N. With 8 players that is 1/8, and with 5 players it is 1/5. A bye gives a player one match fewer to win, but the bye goes to a random player each time, so it does not change the chance; working through the stages one at a time gives exactly 1/N for every size.

This is a baseline for evenly matched players, not a prediction for a real game. In RPS Royale a tied throw goes to a tap race decided by button presses, which is a speed contest rather than a coin flip.

Try it with a group

Counting throws by hand is the quickest way to watch these numbers appear. The group activities guide sets out a probability experiment and a way to predict the bracket before a game starts.

Activities for a class, a team, or a game night are in the group activities guide.