Game Theory & Rock Paper Scissors
Nash equilibrium, mixed strategies, Bayesian reasoning. The math that proves the simplest game is quietly very complicated.
In the standard zero-sum Rock Paper Scissors game, every pure strategy can be beaten by another. The unique Nash equilibrium is to choose each gesture independently with probability 1/3. That mix gives an expected payoff of zero against any opponent strategy; it does not guarantee a tie in a short match or an exactly even realized record. An advantage is possible only when an opponent departs from the equilibrium in a way that can be estimated and exploited before it changes.
RPS: The Textbook Example (Literally)
Rock Paper Scissors is a finite, two-player, zero-sum, simultaneous-move game. Its small payoff matrix makes pure strategies, mixed strategies, best responses, and equilibrium visible without hiding the assumptions.
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:
| Rock | Paper | Scissors | |
|---|---|---|---|
| Rock | 0 | -1 | +1 |
| Paper | +1 | 0 | -1 |
| Scissors | -1 | +1 | 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
Nash’s 1950 existence result shows that every finite game has at least one equilibrium in mixed strategies. For the standard RPS payoff matrix, solving the indifference conditions gives a unique mix: choose Rock, Paper, and Scissors independently with probability 1/3 each.
This strategy guarantees expected payoff zero against any opposing mix. Realized results still vary, sometimes sharply, over a finite match. “Unexploitable” describes expected value under the stated payoff model; it does not mean a player can never lose.
What Repeated-Play Experiments Found
Human RPS results depend on the experimental setting. Two often-cited studies measured different populations and protocols:
- Wang et al. (2014): 360 students played 300 monetarily incentivized rounds in randomly paired six-person groups. Population cycling was explained by a model of outcome-conditioned stay and rotation probabilities.
- Dyson et al. (2016): 31 undergraduates played 225 rounds against an equal-random computer. Switching increased after losses and draws, but staying after wins was not statistically significant.
These studies supply candidate features for an opponent model, not a universal recipe. See the psychology review for populations, protocols, and limits.
Bayesian Reasoning: A Model for Updating a Read
Bayesian inference offers a disciplined way to update uncertainty about an opponent’s throw distribution. It requires a stated prior, a likelihood model, and observed data; merely changing your mind after a throw is not automatically Bayesian analysis:
- Equilibrium baseline: With no reliable evidence, use independent equal randomization.
- Update: As comparable throws accumulate, estimate how likely the observations are under competing opponent models and keep uncertainty visible.
One round usually carries little information, and short matches are noisy. A strong prior claim or a complex model can create false confidence unless it is tested on later throws.
Iterated Reasoning as an Example
Players can reason about how an opponent responds to a prediction. The following ladder is an illustration, not a measured ranking system for competitors:
- Baseline: Choose independently with equal probability.
- One response: Predict a documented opponent tendency and choose its counter.
- Second order: Consider whether the opponent expects that counter and will adjust.
- Stop rule: Avoid an unsupported chain of “they know that I know”; return to the baseline when observations cannot distinguish the models.
Deeper stories are not automatically better predictions. Prefer the simplest model that performs on later, unseen throws.
RPS in Evolutionary Game Theory
Sinervo and Lively (1996) modeled three male side-blotched-lizard morph strategies with non-transitive, frequency-dependent relationships. Six years of field observations showed oscillating morph frequencies consistent with the model’s prediction.
Biologists call this an RPS dynamic because each strategy has an advantage over one and a disadvantage against another. It is a mathematical analogy about selection among morphs, not evidence that animals play the human hand game or that every competitive system follows the same cycle.
What This Means for You
Game theory identifies the safe baseline and the exact assumptions behind it. An opponent-specific strategy can improve expected value only if a departure from that baseline is estimated well enough, persists, and is tested without ignoring misses.