This week I learned about the World Rock Paper Scissors Association, which turns out to be a research library with a membership tier and online gameplay. It maintains a strategy section, a glossary, versioned counting rules for its own match data, a catalogue of all 27 three-throw sequences (each with a name), and a list of peer-reviewed papers in which every summary links the original study and states its limits. That is a standard I wish more medical societies met. I had never considered the game strategic or interesting. Under it sits a small and surprisingly disciplined literature, and I went and read it.
What follows is from the primary papers. Where the association's summaries and the papers differed in emphasis, the paper won.
Rock Paper Scissors is a finite, two-player, zero-sum, simultaneous-move game. Each throw beats exactly one other and loses to exactly one other. That is the whole structure, and it is why the game shows up in textbooks (and, as it turns out, in lizards).
| Rock | Paper | Scissors | |
|---|---|---|---|
| Rock | 0 | −1 | +1 |
| Paper | +1 | 0 | −1 |
| Scissors | −1 | +1 | 0 |
Payoff to the row player. Win +1, loss −1, tie 0. The matrix is symmetric and no pure strategy survives. Any fixed choice has a counter.
The unique mixed-strategy Nash equilibrium is to throw each option independently with probability one third. That strategy has expected payoff zero against anything an opponent can do. It is unexploitable. It is also unexploiting. A player who randomizes perfectly cannot lose on average and cannot win on average either, which the association's game theory page summarizes as "the mathematically perfect strategy that's no fun at all."
A non-transitive cycle. Biologists borrow the name whenever three strategies each beat one and lose to another, which is why the same diagram describes lizard throat colors and toxin-producing bacteria.
Fairness follows from symmetry. Between two players who both randomize, any best-of format is a coin flip. Skill only enters if one player has a stable edge per decisive throw, and then it compounds the way a binomial series does. A 55% edge per throw becomes 57.5% in a best-of-three and 59.3% in a best-of-five. A 60% edge wins a first-to-three series 68.3% of the time. Longer formats reward an edge. They do not create one.
Every empirical finding about human play is a finding about departures from one third. The departures are real, they are small, and they depend heavily on who was playing, against what, and for what stakes. The association's research library is unusually insistent on this point, and it is right to be.
360 students in 60 groups of six, 300 rounds each, random pairing within the group, cash payoffs scaled by a parameter from 1.1 to 100. One group was excluded, leaving 354 players in the frequency analysis.
What they foundAggregate frequencies sat near equilibrium (Rock 0.36, Paper 0.33, Scissors 0.32). Underneath, players repeated after wins and shifted after losses, and when they shifted they went in a consistent direction. The population as a whole cycled counterclockwise through the three throws at a small but persistent rate (about 0.02 to 0.03 per round, all significantly above zero). A six-parameter conditional-response model reproduced the cycling. An independent-decision model could not. Optimized conditional strategies beat the Nash player by roughly 10% at high stakes.
LimitsUniversity students, repeated opponents inside a six-person group, lab incentives. The authors describe the result as bounded rationality, with win-stay lose-shift as a special case of their model, and are careful not to promote it to a universal rule.
The Zhejiang result is the one that reached the newspapers as "how to win at Rock Paper Scissors." The paper's own point is more interesting. Marginal distributions can look like equilibrium while the system underneath is in constant motion. The authors call the game "an intrinsic non-equilibrium system." An aggregate can be balanced and still be cycling.
31 undergraduates (26 women), 225 trials in three blocks against a computer playing exactly one third each, no money at stake.
What they foundThe outcome-by-strategy interaction was significant (p < 0.001). After a loss, players switched rather than stayed. After a draw, they switched specifically to the throw that would have beaten their last one (an "upgrade"). After a win, nothing reached significance; staying was only numerically favored. Rock was thrown 35.6% of the time, which was not significantly above one third (p = 0.20). The shifts carried over. An upgrade tended to be followed by another upgrade.
LimitsSmall, mostly female, unpaid, playing a machine. The authors call the effects small and attribute part of that to the lack of stakes. Dyson's reading is that the irrational choices are driven by an emotional reaction to a negative outcome, the Rock Paper Scissors version of poker tilt.
2,636,417 best-of-three matches in Roshambull, a 2007 Facebook app, 23 May to 14 August 2007, about 340,000 users. Before each match a "scouting sheet" showed the opponent's first-throw distribution, overall distribution, last five matches and record. The analysis centers on 7,758 experienced players with at least 100 clean matches.
What they found82% of experienced players had first-throw distributions consistent with uniform, but 47% responded significantly to opponent history (more rock after an opponent's scissors-heavy history, less rock after paper-heavy). Experienced players won 34.7% and lost 22.2% of first throws, a net of about 1.5 cents per throw in the app's currency. A best-responder to the same predictable patterns would have netted about 14.3 cents. Level-k estimates put 74% of throws at level 0, 18.5% at level 1 and 7.7% at level 2.
LimitsAn interface that hands you a scouting sheet, prestige points rather than money, and a self-selected 2007 Facebook population. The most durable finding is the gap between the edge available and the edge taken.
WRPSA published its first annual data report in September 2026 under a CC BY 4.0 license, with the aggregate counts as a CSV, a SHA-256 commitment on the computer's throw so the machine cannot peek, and a list of limits longer than the findings. The headline is that people avoid scissors.
Share of throws recorded on wrpsa.com, 29 July to 27 September 2026 (3,035 throws from 947 accounts; 1,830 first throws). The dashed line marks one third. A chi-square test against an even split gives about 36 on two degrees of freedom, p < 0.001. Source: WRPSA, State of Rock Paper Scissors 2026, CC BY 4.0.
Next throw after each outcome, same dataset (302, 311 and 533 transitions; each percentage is good to roughly five points). After a win, only one in five players repeated, well below the one third chance would give and the opposite of the Zhejiang win-stay result. WRPSA's working explanation is the opponent. Most of these rounds were against the computer, and people expect a machine to adapt.
Taken together, the three human studies and the association's dataset agree on two things and disagree on a third. People open with rock more than scissors. People react to the previous outcome rather than drawing a fresh throw. Whether winners stay or switch depends on whom they think they are playing, which is a reminder that a "bias" measured in one setting is a hypothesis in the next.
Paired a sighted player with a blindfolded one and asked whether the sighted player involuntarily copies the hand they can see forming, raising the draw rate.
What they foundCook and colleagues reported draws at chance in blind-blind pairs and significantly elevated in blind-sighted pairs, and billed it as the first evidence of automatic imitation in a strategic context. Aczel and colleagues reviewed the analyses and ran the experiment with double the sample. Their data did not support the original conclusion.
StatusWRPSA files this as disputed rather than settled, which is the correct filing. It also notes that in a refereed match, responding to a forming hand is a late throw, not a tell.
15 healthy volunteers, NIRS over prefrontal cortex, playing a computer task either to win or to lose on purpose.
What they foundThe play-to-lose condition produced larger, laterally dominant prefrontal oxygenation changes than play-to-win. Inhibiting the obvious response costs more frontal effort than producing it.
LimitsFifteen people and a modified task. It says nothing about tells, and WRPSA says so in the same paragraph.
The University of Tokyo's Ishikawa group built a janken robot that wins 100% of the time. It does this by recognizing the human hand's shape in about one millisecond with a high-speed camera and forming the counter in roughly another millisecond. Version 1 finished its throw about 20 ms after the human's. Version 2 closed that gap to near simultaneity. Version 3 added pan-tilt tracking so the hand can be anywhere in view. It is a beautiful piece of vision and actuation research. Under WRPSA rules it is also a foul on every round, because it reacts after the reveal begins. The association's framing is the useful one. The robot demonstrates reaction, not prediction, and a great many systems described as predictive are in fact reacting to information that leaked early.
Prediction proper has its own history. Darse Billings ran the First International RoShamBo Programming Competition in 1999. The winner, Dan Egnor's Iocaine Powder, did not try to be random. It ran several predictors at once (history matching, frequency analysis, a random fallback), layered meta-strategies on top that assumed the opponent was trying to predict it back, and played whichever layer was currently winning. Kaggle revived the format in 2020 with 1,000-round bot matches. None of these can beat a truly random opponent. All of them beat people.
This is the crossover into animals, and it is the part where the mathematics does its best work. Three systems, three very different organisms, one cycle.
Male Uta stansburiana come in three throat colors with three mating strategies. Orange males are aggressive and hold large territories with many females. Blue males hold small territories and guard their mates closely. Yellow males have no territory and mimic females to sneak matings. Orange beats blue by force. Blue beats yellow because a guarded female is hard to sneak. Yellow beats orange because an orange male cannot guard a harem that large.
What they foundEach morph can invade when rare and is invaded when common. Over six years of field observation the morph frequencies oscillated as the model predicted, with the fitness of each morph tracking year-to-year frequency changes.
LimitsAn evolutionary analogy to the game's structure. The lizards are not throwing anything.
A colicin-producing strain kills a colicin-sensitive strain. The sensitive strain outgrows a colicin-resistant strain, because resistance has a metabolic cost. The resistant strain outgrows the producer, because producing the toxin costs more than resisting it.
What they foundOn a static agar plate, where cells interact only with neighbors, all three strains coexisted. In a well-mixed flask, or on a plate where cells were redistributed at each transfer, diversity collapsed quickly. Local interaction and dispersal preserve a non-transitive community. Global mixing destroys it.
Why it mattersA result about ecology that reads as a result about organizational design. Communities with a dominance loop survive when interaction is local.
Seven chimpanzees at Kyoto University's Primate Research Institute learned a touchscreen task. Could they learn a circular relationship (A beats B, B beats C, C beats A) rather than the linear rankings animals usually learn? 38 children aged 35 to 71 months were tested with real hand gestures for comparison.
What they foundFive of seven chimpanzees learned the full loop after a mean of 307 sessions. The first pair (paper beats rock) took 1.7 sessions and the second (rock beats scissors) 3.1. The third pair, scissors beats paper, which closes the circle, took 14.3. The chimpanzees' mixed-pair performance matched that of four-year-old children. The children's accuracy rose with age and exceeded chance from about 50 months.
LimitsThe chimpanzees never played against anyone. They chose the winning shape from two on a screen for food. The finding is about learning a loop, not about strategy.
One more human system belongs here because it is the field test the laboratory studies lack. Palacios-Huerta (2003) analyzed 1,417 professional penalty kicks and found that kickers and goalkeepers mixed their choices so that success rates were equal across options and the sequences were serially independent. Professionals, with money and careers at stake, play minimax. Undergraduates in a lab with no stakes do not. Stakes and practice appear to buy randomness.
These are my readings of the papers, not claims the papers make. Six things carry over.
- Know whether the other side is random. Against a random process, no strategy beats one third and the correct move is to stop looking for one. Against an adapting opponent, the question changes entirely. Most wasted effort in strategy is applying the second mode to the first kind of problem.
- Protect first, exploit second. The mixed baseline guarantees you cannot be exploited. WRPSA's "test and reset" routine is a decent general protocol. State the read before acting on it, count hits and misses, and return to baseline when the evidence weakens. Do not keep paying for an attractive story after the data stop supporting it.
- Losses distort the next decision more than wins. Dyson's players switched after losses and draws in predictable directions and were indifferent after wins. The practical version is that the decision made right after a bad outcome is the one most worth slowing down.
- Aggregates can hide cycling. The Zhejiang population looked like equilibrium in its marginals and was cycling underneath. A balanced dashboard is not evidence of a stable system.
- Local structure preserves diversity. Kerr's bacteria coexisted when interaction was local and collapsed when it was global. Non-transitive ecosystems (three approaches that each beat one of the others) survive segmentation and die under forced mixing.
- Most "prediction" is reaction to leaked information. The Tokyo robot is the cleanest example in the literature. Before crediting a system with foresight, check what it saw and when.
The 27 gambits. WRPSA catalogues every ordered three-throw sequence with a mnemonic and states, in the same sentence, that the mnemonic "carries no measured advantage." A few of the names are worth having anyway.
A pre-committed sequence does reduce reactive switching, which the Dyson data suggest is the main leak. It does not make the sequence random, and a repeated script is a pattern like any other.
WRPSA runs a free tier (match history, free entry to official Opens, community access, no ads) and a Supporter membership at $10 a year. Supporters get a badge in rankings and brackets, their name on a public wall, 48-hour early registration for Opens, 15% off merchandise, and the Throw Pattern Lab, a private view of their own throw mix and record. The Lab is the gift that keeps on humbling. The pricing page does not mention gift memberships. The shop has a 2026 gift guide of stickers, pins, scorebooks and hats, with a US order-by date of December 8 for Christmas delivery. The site also hosts a research and citation policy that tells readers to cite the paper, not the summary. Few professional societies manage that.
Against a random opponent nothing works, and against a person something does. The interesting science is not in the hand game but in the loop underneath it, which keeps three lizard morphs, three bacterial strains, and three arbitrary throws alive together as long as nobody can see the other's hand and nobody mixes the plate.
Sources
WRPSA: Research Library, Science, Game Theory, Strategy, Psychology, Tells, The 27 Gambits, Statistics, Probability, Pricing, Gift Guide 2026.
WRPSA data: World Rock Paper Scissors Association (2026, September 27). State of Rock Paper Scissors 2026: What 3,035 throws show. Data CC BY 4.0; chart data above reproduced with attribution.
Zhejiang: Wang, Z., Xu, B. and Zhou, H.-J. (2014). Social cycling and conditional responses in the Rock-Paper-Scissors game. Scientific Reports 4, 5830. doi:10.1038/srep05830
Losses: Dyson, B. J., Wilbiks, J. M. P., Sandhu, R., Papanicolaou, G. and Lintag, J. (2016). Negative outcomes evoke cyclic irrational decisions in Rock, Paper, Scissors. Scientific Reports 6, 20479. doi:10.1038/srep20479
Million games: Batzilis, D., Jaffe, S., Levitt, S., List, J. A. and Picel, J. (2019). Behavior in Strategic Settings: Evidence from a Million Rock-Paper-Scissors Games. Games 10(2), 18. doi:10.3390/g10020018
Imitation: Cook, R., Bird, G., Lünser, G., Huck, S. and Heyes, C. (2011). Automatic imitation in a strategic context. Proc. R. Soc. B 279, 780. doi:10.1098/rspb.2011.1024. Aczel, B., Bago, B. and Foldes, A. (2012). Is there evidence for automatic imitation in a strategic context? Proc. R. Soc. B 279, 3231. doi:10.1098/rspb.2012.0500
NIRS: Kikuchi, S. et al. (2007). Prefrontal cerebral activity during a simple "rock, paper, scissors" task measured by NIRS. Psychiatry Research: Neuroimaging 156, 199. doi:10.1016/j.pscychresns.2007.01.002
Lizards: Sinervo, B. and Lively, C. M. (1996). The rock-paper-scissors game and the evolution of alternative male strategies. Nature 380, 240. doi:10.1038/380240a0
Bacteria: Kerr, B., Riley, M. A., Feldman, M. W. and Bohannan, B. J. M. (2002). Local dispersal promotes biodiversity in a real-life game of rock-paper-scissors. Nature 418, 171. doi:10.1038/nature00823
Chimpanzees: Gao, J., Su, Y., Tomonaga, M. and Matsuzawa, T. (2018). Learning the rules of the rock-paper-scissors game: chimpanzees versus children. Primates 59, 7. doi:10.1007/s10329-017-0620-0
Penalty kicks: Palacios-Huerta, I. (2003). Professionals Play Minimax. Review of Economic Studies 70, 395. Chiappori, P.-A., Levitt, S. and Groseclose, T. (2002). Testing Mixed-Strategy Equilibria When Players Are Heterogeneous. American Economic Review 92, 1138.
Robot: Ishikawa Group Laboratory, University of Tokyo. Janken (rock-paper-scissors) robot with 100% winning rate.