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Part I · The rules of the game

What is a game?

Rock, Paper, Scissors is a real game, and so is a surprising amount of life. Before you can win a game you have to be able to write it down. This chapter shows you how. Then it asks a question with a surprising answer: what is the best way to play Rock, Paper, Scissors?

Every game has three parts. Mathematicians use these exact words, so learn them now:

  • Players: whoever is choosing. Here it is you and Robo.
  • Strategies: the complete list of choices each player could make. Rock, Paper, or Scissors.
  • Payoffs: what each player gets at the end, for every possible combination of choices. A win, a loss, or a tie.

Play ten rounds against Robo, who picks at random. Watch the grid while you play. Your choice picks the row, Robo's choice picks the column, and the box where they meet is the outcome. The grid is not a scoreboard. It is the whole game, written down.

Rock, Paper, Scissors

Robo picks at random.
Robo's brain
Pick one to play.
Rounds0
Wins0
Losses0
Ties0

After ten rounds, compare your number of ties with the math below.

The grid is the game

You have 3 strategies and Robo has 3 strategies, so there are 3 × 3 = 9 possible rounds. That is why the grid has 9 boxes. Three of them, the diagonal, are ties. Of the other six, you win three and lose three. The game is symmetric: swap the players and nothing changes.

Here is a bigger version that some people play: Rock, Paper, Scissors, Lizard, Spock. Scissors cuts Paper, Paper covers Rock, Rock crushes Lizard, Lizard poisons Spock, Spock smashes Scissors, Scissors decapitates Lizard, Lizard eats Paper, Paper disproves Spock, Spock vaporizes Rock, Rock crushes Scissors. Five strategies each, so 5 × 5 = 25 boxes. Look at the grid and count: every weapon beats exactly two others and loses to exactly two. That is what makes it fair.

Rock, Paper, Scissors, Lizard, Spock

Rows are you, columns are Robo. Hover a row to count its wins.

25 boxes: 5 ties, 10 wins for the row player, 10 losses.

Now design your own. Below is an empty 5-weapon grid. Click a box to decide who wins that matchup (the mirror box fills in by itself, because if A beats B then B loses to A). Your job: make every weapon beat exactly two of the others. Then think about why you could never do this with four weapons.

Build a fair game

Green = row beats column. Red = row loses. Click to flip.
Weapon names
Fill in all ten matchups.

Can you be smarter than random?

Against a Robo that picks at random, the honest answer is no. Whatever you do, each of Robo's three choices is equally likely, so you win a third of the time, lose a third, and tie a third. No plan beats a coin flip, and no plan loses to one either.

But most opponents are not random. Switch Robo's brain to Pattern Robo: it assumes you will repeat your last move, and plays whatever beats that. Figure out its pattern and you can beat it almost every round. Then try Learning Robo, which counts what you have played most often and plays whatever beats your favorite. The only way to beat a learner is to have no favorite at all. That idea, being deliberately unpredictable, is the whole subject of Chapter 4.

Math corner

Counting boxes. If you have a strategies and Robo has b, the grid has a × b boxes. Two players each choosing a number from 1 to 10 make a 100-box grid, and a game of chess makes a grid too big to write down, but it is a grid all the same.

Chance. Random Robo lands on each of the 9 boxes with probability 1/9. Three boxes are ties, so the chance of a tie in one round is 3/9 = 1/3. Over ten rounds you should expect about 10 × 1/3 ≈ 3.3 ties. You will not get exactly that, and the more rounds you play, the closer your fraction of ties gets to one third. Mathematicians call that the law of large numbers.

Why five works and four does not. With n weapons there are n(n − 1)/2 matchups between different weapons (for 5 weapons, 10 matchups). In a fair game every weapon wins the same number of them, so each wins (n − 1)/2. For that to be a whole number, n − 1 must be even, so n must be odd. Three works. Five works. Four cannot.

Big idea

A game is players, strategies, and payoffs. Once you write it down as a grid, you can reason about it: count its boxes, spot its symmetries, and find the best way to play it. Everything else on this site is a way of reading grids.

Try it on paper

1. Two players each secretly pick a number from 1 to 4. How many boxes does the grid have? How many of them are ties?

Answer

4 × 4 = 16 boxes, and 4 ties (the diagonal, where both picked the same number).

2. In ordinary Rock, Paper, Scissors, in how many of the 9 boxes does the row player win? What fraction is that?

Answer

3 boxes: Rock over Scissors, Paper over Rock, Scissors over Paper. That is 3/9 = 1/3 of the grid.

3. With 7 weapons in a fair game, how many others does each weapon beat, and how many matchups are there in total?

Answer

Each beats (7 − 1)/2 = 3 others. Total matchups: 7 × 6 / 2 = 21.

Challenge

Learning Robo plays whatever beats your most common move so far. Suppose you know that. Can you design a sequence of moves that beats Learning Robo every single round after the first few? Try to work it out on paper, then test it on the board.

Answer

Yes. Learning Robo's next move is completely determined by your history, so you can always predict it and play the counter. Suppose you have played Rock more than anything else. Robo will play Paper. You play Scissors and win. Keep your counts arranged so that you always know the leader, and you always know Robo's move. The lesson is not about Robo. It is that any opponent who follows a fixed rule can be beaten by anyone who learns the rule.

True story

Game theory was invented by John von Neumann, a Hungarian mathematician who could multiply eight-digit numbers in his head as a child. In 1928 he published a paper on "parlor games" that proved every two-player game like Rock, Paper, Scissors has a best way to play it, even if that best way is to roll a die. In 1944 he and the economist Oskar Morgenstern turned the idea into a 600-page book, and a new branch of mathematics was born. Rock, Paper, Scissors itself is much older: it comes from a family of Japanese hand games called janken and reached Europe in the early 1900s.

Stars in this chapter

Earn them by doing the clever thing, not by clicking around.