Game Theory & Rock Paper Scissors

RPS: The Textbook Example (Literally)

Rock Paper Scissors is a finite, two-player, zero-sum, simultaneous game. If you've ever taken an introductory game theory course, this is one of the first things they make you study. It's also one of the last things you fully understand. The game looks simple. The math behind it would like a word.

The Payoff Matrix (Don't Panic)

Every RPS matchup can be written as a 3x3 grid. Win gets +1, loss gets -1, tie gets 0:

This is a symmetric game. Both players have the same options and the same payoffs. There's no pure-strategy Nash equilibrium, which is a fancy way of saying: if you have a pattern, someone can exploit it.

Nash Equilibrium: The Mathematically Perfect Strategy That's No Fun at All

John Nash (yes, the A Beautiful Mind guy) proved that every finite game has at least one equilibrium in mixed strategies. For RPS, the Nash equilibrium is beautifully boring: play each gesture with exactly 1/3 probability. Rock 33.3%. Paper 33.3%. Scissors 33.3%.

If you do this perfectly, no opponent can gain an advantage against you, ever. Your expected payoff is zero forever. You can never be beaten, and you can never win. It's the game theory equivalent of eating plain oatmeal for every meal. Optimal. Joyless.

Why Humans Can't Pull This Off

Here's where it gets interesting. Research consistently shows that humans are absolutely terrible at being random. A study from Zhejiang University (2014) tracked 360 students through 300 rounds each and found:

• Players repeat winning throws (win-stay bias)
• Players switch to the throw that would have beaten their last loss (lose-shift)
• People throw Rock about 36% of the time, well above the expected 33%

These are not random. These are patterns. And patterns are what competitive RPS players live for. See our psychology guide for the full breakdown.

Bayesian Reasoning: Watching and Learning

Advanced players use Bayesian inference, which means they update their predictions about what you'll throw based on what you've already thrown. This is fundamentally different from Nash:

• Nash says: Assume your opponent is a perfectly rational robot. Play randomly.
• Bayes says: Your opponent is a human being with feelings and habits. Watch them. Exploit them.

In a single round, Bayesian reasoning doesn't help much. But in a best-of-three or longer series (standard in WRPSA tournaments), every throw gives you new data. It's like poker, except your tells are your fingers.

Levels of Thinking (It Goes Deep)

Game theorists describe competitive RPS strategy in levels:

• Level 0: Random play. You're a coin flip with thumbs.
• Level 1: You respond to what your opponent just threw.
• Level 2: You anticipate what your opponent thinks you'll throw, and counter it.
• Level 3: You anticipate their anticipation of your anticipation, and now everyone has a headache.

Elite players operate at Level 3 or higher while simultaneously tracking what level their opponent is playing at. It's like chess, but faster and with more hand signals.

RPS in Evolutionary Game Theory

The side-blotched lizard (Uta stansburiana) has three male variants whose mating strategies follow an RPS cycle: aggressive orange-throated males beat blue, blue beat yellow, yellow beat orange. This isn't a metaphor. This is actual lizard behavior empirically documented by biologists.

Similar dynamics show up in bacteria, coral reef fish, and political competition. RPS isn't just a game. It's a fundamental pattern woven into how living things compete. Nature invented this before we did.

What This Means for You

Understanding game theory won't make you unbeatable at RPS. But it'll help you understand whycertain strategies work and when to deploy them. The key insight: Nash equilibrium is your safe fallback when you have no read. Exploiting the fact that your opponent is a predictable human being? That's how you win tournaments.