Rock Paper Scissors Statistics

The Mathematical Baseline

In the one-round symmetric zero-sum model, choosing Rock, Paper, and Scissors independently with probability 1/3 each is a Nash equilibrium. Against that mixed strategy, no fixed alternative has a higher expected payoff. This is an expected-value statement under a model, not a promise about a finite match.

Human datasets can depart from independence or uniform choice, but the result depends on the population, interface, incentives, information, and protocol. No reviewed study licenses the claim that every player is predictable or exploitable.

Observed Action Frequencies Are Dataset-Specific

The retained participants in the 2014 Zhejiang experiment averaged 0.36 Rock, 0.33 Paper, and 0.32 Scissors, with reported standard deviations of 0.08, 0.07, and 0.06 across players. These were repeated, incentivized laboratory sessions, not a representative sample of all players.

A frequency estimate becomes useful only with its source, sample, observation window, interface, and uncertainty. It cannot by itself prove a universal opener, a physical tell, or a stable advantage against a new opponent.

Two Different Study Populations

1. Repeated laboratory play (Zhejiang University, 2014)

The experiment recruited 360 Zhejiang University students into 60 groups of six for 300 randomly paired rounds with cash-linked payoffs. After excluding one group, the retained 354 players had mean action frequencies of 0.36 Rock, 0.33 Paper, and 0.32 Scissors, with substantial between-player variation. The authors modeled population cycling with win–lose–tie conditional responses. This protocol does not establish a universal next-throw rule.

Wang, Xu & Zhou (2014), Scientific Reports, doi:10.1038/srep05830

2. A 2007 Facebook application dataset (published 2019)

The paper analyzed 2,636,417 matches recorded from 23 May through 14 August 2007 in the Roshambull Facebook app. Players saw a scouting sheet with opponent history. The authors reported that players used that information and that more experienced users did so more effectively and were more likely to win. Those results belong to that interface, incentive system, period, and observed user population.

Batzilis et al. (2019), Games, doi:10.3390/g10020018

An earlier experiment reported an automatic-imitation effect in face-to-face play, but a larger replication did not support that conclusion. The imitation result should therefore be treated as disputed, not as a settled universal rule.

How a Hypothetical Stable Edge Compounds

Assume each decisive round is independent, the same player has a constant 55% chance of winning each decisive round, and ties are replayed outside the calculation. Under that model:

• Single round: you win 55% of the time.
• Best of 3: your match win probability rises to about 57.5%.
• Best of 5: about 59.3%.

At a constant 60% decisive-round probability, the same independent-round model gives a 68.3% chance of winning a first-to-three series. These are binomial-series calculations, not evidence that longer formats measure only skill or remove luck. The tournament guide treats format as a declared event rule.

There are exactly 27 ordered three-action sequences, because multiplying three choices across three positions produces 27 sequences. The names in The 27 Gambits are WRPSA editorial mnemonics; the count does not give any sequence an inherent advantage.

What the Numbers Do and Do Not Support

• No reviewed dataset establishes a universally best blind opener against a new opponent.
• A win-conditioned repeat appeared in the Zhejiang experiment, but the paper does not guarantee that a particular opponent will repeat. See the strategy guide for opponent-specific testing.
• A loss-conditioned shift was part of one laboratory model. Treat it as a population-and-protocol result, not a next-throw instruction for an unknown opponent.
• Opponent modeling adds assumptions and uncertainty. The psychology review keeps laboratory findings tied to their samples and procedures.

How WRPSA Publishes Its Own Numbers

The panel above reads WRPSA's official match records and labels every count separately. The ways to play record each match under versioned counting rules; World RPS Day does not create a separate population estimate. A research release needs a frozen, reviewed edition with its methods, exclusions, dates, and correction history.

Sources

• Wang, Z., Xu, B., & Zhou, H.-J. (2014). Social cycling and conditional responses in the Rock-Paper-Scissors game. Scientific Reports 4:5830 (arXiv:1404.5199).
• Batzilis, D., Jaffe, S., Levitt, S., List, J. A., & Picel, J. (2019). Behavior in Strategic Settings: Evidence from a Million Rock-Paper-Scissors Games. Games 10(2):18.
• Cook, R., Bird, G., Lünser, G., Huck, S., & Heyes, C. (2011). Automatic imitation in a strategic context: players of rock-paper-scissors imitate opponents' gestures. Proceedings of the Royal Society B. See also the published replication and critique.