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arxiv: 2605.18616 · v1 · pith:JGP3X733new · submitted 2026-05-18 · ⚛️ physics.soc-ph · cs.GT· q-bio.NC

Toward an Origin of Human Randomness: Interaction-Driven Enhancement in the Rock-Paper-Scissors Game

Song-Ju Kim , Shoma Ohara , Hiroaki Kurokawa This is my paper

Pith reviewed 2026-05-20 01:52 UTC · model grok-4.3

classification ⚛️ physics.soc-ph cs.GTq-bio.NC
keywords rock-paper-scissorshuman randomnessLempel-Ziv complexityinteraction effectssequence entropysensitivity measurefrequency bias
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The pith

Sensitivity to an opponent's recent move bias can raise the entropy of that opponent's future moves in rock-paper-scissors.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Human players produce rock-paper-scissors sequences whose complexity sometimes exceeds the maximum observed when facing a random-number generator. The study defines a sensitivity measure that tracks whether a player counters the opponent's most frequent recent move. Partial regression shows this sensitivity predicts higher entropy in the opponent's next moves after controlling for the opponent's current entropy. The link appears strongest when the opponent currently displays low entropy and a clear frequency bias. Circular-shift surrogates indicate the effect is tied to the actual interaction rather than fixed player traits.

Core claim

The central claim is that a focal player's sensitivity to the opponent's recent frequency bias positively predicts an increase in the opponent's future move-sequence entropy, with the relation clearest in low-entropy opponent states. This interaction-driven mechanism produces a high-complexity tail in human-human matches that is absent when the same players face an RNG opponent, suggesting that human randomness can be locally enhanced by responsive play rather than remaining an isolated individual property.

What carries the argument

The sensitivity measure, which registers whether a player selects the move that beats the opponent's most frequent recent choice and thereby links that choice to a subsequent rise in opponent entropy.

If this is right

  • High sensitivity tends to raise the entropy of the opponent's subsequent moves.
  • The entropy boost is most evident when the opponent currently occupies a low-entropy state containing a detectable frequency bias.
  • A small fraction of human-human sequences exceed the highest complexity seen against an RNG opponent.
  • Human randomness is shaped by dynamic interaction rather than fixed solely by individual cognitive or motor limits.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same sensitivity mechanism could be tested in other repeated-choice games to see whether bias detection reliably increases sequence entropy.
  • Generative models of human behavior might add an explicit interaction term that raises entropy when one agent detects and counters the other's recent bias.
  • Social settings that reward counter-bias play could be examined for their effect on overall randomness in decision sequences.

Load-bearing premise

The circular-shift surrogate analyses isolate an interaction-specific causal component in the sensitivity-entropy link rather than reflecting residual individual differences.

What would settle it

A controlled experiment in which sensitivity is measured yet shows no positive association with future opponent entropy, or in which surrogate tests fail to show a stronger relation under real interaction than under shuffled controls.

read the original abstract

Human-generated randomness is constrained by cognitive, motor, and strategic biases. This study examines how these constraints appear in individual behavior and how they may be modified through interaction with another human. We analyzed repeated rock-paper-scissors data from 9 participants, yielding 108 human-human matches and 216 individual player sequences. Using Lempel-Ziv complexity (LZC), we compared human-human sequences with the RNG-opponent condition. In the RNG-opponent condition, the maximum human LZC value was 84, which we used as an empirical reference. In the human-human condition, most sequences remained below this value, but a small number exceeded it, producing a small high-complexity tail that was not present in the RNG-opponent condition. We introduced a sensitivity measure that captures whether a player responds to the opponent's recent frequency bias by choosing the move that beats the opponent's most frequent recent move. Partial regression showed that focal-player sensitivity positively predicted future entropy in the opponent's move sequence after controlling for the opponent's current entropy. Circular-shift surrogate analyses indicated that this relation was most clearly interaction-specific when the opponent was in a low-entropy state, where the recent move distribution contained a clear frequency bias. These results suggest that human randomness is not only an isolated individual capacity, but can be shaped by interaction in a state-dependent manner. The findings identify a local mechanism by which interaction may destabilize biased behavior and increase entropy, providing a concrete basis for future causal experiments and generative models of high-complexity human behavior.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes rock-paper-scissors sequences from 9 participants (108 human-human matches, 216 sequences) to argue that interaction enhances human randomness. It reports a high-complexity tail in Lempel-Ziv complexity (LZC) for human-human play absent in the RNG-opponent condition (max LZC=84 used as reference), introduces a sensitivity measure for responding to opponent's recent frequency bias, and uses partial regression to show that focal sensitivity positively predicts future opponent entropy after controlling for current entropy, clearest in low-entropy opponent states. Circular-shift surrogates are presented as evidence that the relation is interaction-specific rather than artifactual.

Significance. If the central regression and surrogate results hold after addressing confounds, the work identifies a plausible local mechanism—state-dependent sensitivity to bias—that can destabilize low-entropy behavior through interaction, offering a concrete basis for generative models of human randomness beyond isolated cognitive biases. The empirical LZC reference and surrogate controls are positive features, though the small N limits claims about generalizability.

major comments (2)
  1. [Partial regression and surrogate analyses] The partial regression (abstract and results) controls only for the opponent's current entropy and does not include player fixed effects, random intercepts, or match-level clustering. With N=9 and multiple sequences per player, the reported positive coefficient on focal sensitivity may partly reflect stable between-player variance rather than within-match dynamic influence. The circular-shift surrogates preserve player-level marginals and autocorrelations, so they do not rule out this alternative; a mixed-effects reanalysis is needed to support the interaction-specific claim.
  2. [LZC comparison and sensitivity definition] The high-complexity tail is identified post-hoc by comparing against the maximum LZC (84) observed in the RNG-opponent condition. No sensitivity analysis varying this threshold, no effect sizes or confidence intervals for the regression, and no report of sequence lengths or exact window sizes for the sensitivity measure are provided, weakening the robustness of both the tail observation and the state-dependent entropy prediction.
minor comments (2)
  1. [Methods] Clarify the exact operational definition of 'recent frequency bias' and the temporal window used for the sensitivity measure, as this is central to reproducibility.
  2. [Data description] Report participant demographics, exact sequence lengths, and any preprocessing steps for the 216 sequences to allow assessment of data quality.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback, which identifies key opportunities to strengthen the statistical robustness and transparency of our analyses. We respond to each major comment below and outline the revisions we will implement.

read point-by-point responses

  1. Referee: [Partial regression and surrogate analyses] The partial regression (abstract and results) controls only for the opponent's current entropy and does not include player fixed effects, random intercepts, or match-level clustering. With N=9 and multiple sequences per player, the reported positive coefficient on focal sensitivity may partly reflect stable between-player variance rather than within-match dynamic influence. The circular-shift surrogates preserve player-level marginals and autocorrelations, so they do not rule out this alternative; a mixed-effects reanalysis is needed to support the interaction-specific claim.

    Authors: We agree that the current partial regression does not explicitly model player-level heterogeneity and that the circular-shift surrogates, while disrupting temporal alignment between players, preserve individual marginal distributions and thus cannot fully isolate within-match dynamics from stable between-player differences. To address this directly, we will reanalyze the data using linear mixed-effects models with player identity as a random intercept (and, where appropriate, random slopes for sensitivity). This will allow us to estimate the within-player effect of focal sensitivity on subsequent opponent entropy while accounting for between-player variance. We will report both the mixed-effects results and a comparison with the original partial regression to demonstrate that the key positive association holds after these controls. revision: yes

  2. Referee: [LZC comparison and sensitivity definition] The high-complexity tail is identified post-hoc by comparing against the maximum LZC (84) observed in the RNG-opponent condition. No sensitivity analysis varying this threshold, no effect sizes or confidence intervals for the regression, and no report of sequence lengths or exact window sizes for the sensitivity measure are provided, weakening the robustness of both the tail observation and the state-dependent entropy prediction.

    Authors: We acknowledge that the choice of the empirical LZC threshold of 84 was presented without accompanying robustness checks and that several quantitative details were omitted. In the revised manuscript we will (i) conduct and report a sensitivity analysis that varies the LZC threshold in a range around 84 (e.g., 80–88) and shows that the presence of the high-complexity tail in the human–human condition is stable across reasonable cut-offs; (ii) report standardized effect sizes together with 95% confidence intervals for all regression coefficients; and (iii) explicitly state the sequence lengths (fixed by match duration) and the precise window size (number of preceding moves) used to compute the frequency-bias sensitivity measure. These additions will improve reproducibility and allow readers to evaluate the robustness of both the tail observation and the state-dependent prediction. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical partial regression on observed sequences

full rationale

The paper's central result is a partial regression relating a directly observed sensitivity measure (response to opponent's recent frequency bias) to future opponent entropy, after controlling for current entropy, with circular-shift surrogates applied to the same empirical sequences. No equations, fitted parameters, or self-citations are invoked that reduce the reported association to a definitional identity or construction. The analysis operates on raw move sequences from 9 participants; sensitivity and entropy are computed independently from the data, and the regression tests a statistical relation rather than deriving one quantity from another by algebraic rearrangement or renaming. This is a standard empirical finding with explicit controls and surrogate checks, self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The analysis rests on standard assumptions about complexity measures and regression controls plus one newly defined sensitivity construct; no external benchmarks or machine-checked derivations are referenced.

free parameters (1)
  • LZC reference threshold = 84
    Maximum LZC value of 84 observed in the RNG-opponent condition is used as the empirical cutoff for identifying the high-complexity tail.
axioms (2)
  • domain assumption Lempel-Ziv complexity is an appropriate metric for quantifying randomness in short finite sequences of RPS moves
    Used to compare human-human and RNG-opponent conditions throughout the analysis.
  • domain assumption Partial regression with current-entropy control isolates the unique contribution of sensitivity to future opponent entropy
    Central to the reported positive prediction result.
invented entities (1)
  • sensitivity measure no independent evidence
    purpose: Quantifies whether a player responds to the opponent's recent frequency bias by selecting the counter-move
    Newly introduced construct used to predict future entropy changes.

pith-pipeline@v0.9.0 · 5814 in / 1579 out tokens · 106298 ms · 2026-05-20T01:52:10.218019+00:00 · methodology

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Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    Information and Control 9(6), 602–619 (1966) https://doi.org/10.1016/S0019-9958(66)80018-9

    Martin-L¨ of, P.: The definition of random sequences. Information and Control 9(6), 602–619 (1966) https://doi.org/10.1016/S0019-9958(66)80018-9

    work page doi:10.1016/s0019-9958(66)80018-9 1966

  2. [2]

    In: Proceedings of the 23rd Annual Symposium on Foundations of Computer Science, pp

    Yao, A.C.-C.: Theory and applications of trapdoor functions. In: Proceedings of the 23rd Annual Symposium on Foundations of Computer Science, pp. 80–

    work page

  3. [3]

    https://doi.org/10.1109/ SFCS.1982.45

    IEEE Computer Society, Los Alamitos, CA (1982). https://doi.org/10.1109/ SFCS.1982.45

    work page 1982

  4. [4]

    Springer, New York (2008)

    Li, M., Vit´ anyi, P.: An Introduction to Kolmogorov Complexity and Its Applications, 3rd edn. Springer, New York (2008). https://doi.org/10.1007/ 978-0-387-49820-1

    work page 2008

  5. [5]

    NIST Special Publication 800-22 Revision 1a, National Institute of Standards and Technology, Gaithersburg, MD (2010)

    Rukhin, A., Soto, J., Nechvatal, J., Smid, M., Barker, E., Leigh, S., Levenson, M., Vangel, M., Banks, D., Heckert, A., Dray, J., Vo, S.: A statistical test suite for random and pseudorandom number generators for cryptographic applications. NIST Special Publication 800-22 Revision 1a, National Institute of Standards and Technology, Gaithersburg, MD (2010)

    work page 2010

  6. [6]

    Technical Report ISEC2003-87, IEICE (2003)

    Kim, S.-J., Umeno, K., Hasegawa, A.: On the NIST statistical test suite for randomness. Technical Report ISEC2003-87, IEICE (2003). https://eprint.iacr. org/2004/018

    work page 2003

  7. [7]

    IEEE Transactions on Information Theory22(1), 75–81 (1976) https://doi.org/10.1109/TIT.1976

    Lempel, A., Ziv, J.: On the complexity of finite sequences. IEEE Transactions on Information Theory22(1), 75–81 (1976) https://doi.org/10.1109/TIT.1976. 1055501

    work page doi:10.1109/tit.1976 1976

  8. [8]

    Psychological Bulletin77(1), 65–72 (1972) https://doi.org/ 10.1037/h0032060

    Wagenaar, W.A.: Generation of random sequences by human subjects: A critical survey of literature. Psychological Bulletin77(1), 65–72 (1972) https://doi.org/ 10.1037/h0032060

    work page doi:10.1037/h0032060 1972

  9. [9]

    Psychological Review109(2), 330–357 (2002) https://doi.org/10.1037/0033-295X.109.2.330

    Nickerson, R.S.: The production and perception of randomness. Psychological Review109(2), 330–357 (2002) https://doi.org/10.1037/0033-295X.109.2.330

    work page doi:10.1037/0033-295x.109.2.330 2002

  10. [10]

    Connectionist models of recognition memory: Constraints imposed by learning and forgetting functions.Psychological Review, 97(2):285–308, 1990

    Rapoport, A., Budescu, D.V.: Randomization in individual choice behavior. Psy- chological Review104(3), 603–617 (1997) https://doi.org/10.1037/0033-295X. 104.3.603

    work page doi:10.1037/0033-295x 1997

  11. [11]

    randomly?

    Neuringer, A.: Can people behave “randomly?” the role of feedback. Journal of Experimental Psychology: General115(1), 62–75 (1986) https://doi.org/10. 1037/0096-3445.115.1.62

    work page 1986

  12. [12]

    Quarterly Journal of Experimental Psychology Section A51(4), 819–852 (1998) https://doi.org/10.1080/713755788 28

    Baddeley, A.D., Emslie, H., Kolodny, J., Duncan, J.: Random generation and the executive control of working memory. Quarterly Journal of Experimental Psychology Section A51(4), 819–852 (1998) https://doi.org/10.1080/713755788 28

    work page doi:10.1080/713755788 1998

  13. [13]

    Cambridge University Press, Cambridge (1975)

    Hacking, I.: The Emergence of Probability: A Philosophical Study of Early Ideas About Probability, Induction and Statistical Inference. Cambridge University Press, Cambridge (1975)

    work page 1975

  14. [14]

    MIT Press, Cambridge, MA (1991)

    Varela, F.J., Thompson, E., Rosch, E.: The Embodied Mind: Cognitive Science and Human Experience. MIT Press, Cambridge, MA (1991)

    work page 1991

  15. [15]

    MIT Press, Cambridge, MA (1997)

    Clark, A.: Being There: Putting Brain, Body, and World Together Again. MIT Press, Cambridge, MA (1997)

    work page 1997

  16. [16]

    MIT Press, Cambridge, MA (1995)

    Hutchins, E.: Cognition in the Wild. MIT Press, Cambridge, MA (1995)

    work page 1995

  17. [17]

    Scientific Reports4, 5830 (2014) https://doi.org/10

    Wang, Z., Xu, B., Zhou, H.-J.: Social cycling and conditional responses in the rock-paper-scissors game. Scientific Reports4, 5830 (2014) https://doi.org/10. 1038/srep05830

    work page 2014

  18. [18]

    Contemporary Physics57(2), 151– 163 (2016) https://doi.org/10.1080/00107514.2015.1026556

    Zhou, H.-J.: The rock–paper–scissors game. Contemporary Physics57(2), 151– 163 (2016) https://doi.org/10.1080/00107514.2015.1026556

    work page doi:10.1080/00107514.2015.1026556 2016

  19. [19]

    Games12(3), 70 (2021) https://doi.org/10.3390/g12030070

    Brockbank, E., Vul, E.: Formalizing opponent modeling with the rock, paper, scissors game. Games12(3), 70 (2021) https://doi.org/10.3390/g12030070

    work page doi:10.3390/g12030070 2021

  20. [20]

    Cognitive Psychology151, 101654 (2024) https: //doi.org/10.1016/j.cogpsych.2024.101654

    Brockbank, E., Vul, E.: Repeated rock, paper, scissors play reveals limits in adaptive sequential behavior. Cognitive Psychology151, 101654 (2024) https: //doi.org/10.1016/j.cogpsych.2024.101654

    work page doi:10.1016/j.cogpsych.2024.101654 2024

  21. [21]

    In: Proceedings of the 2018 RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing, Honolulu, HI, USA, pp

    Komai, T., Kim, S.-J., Kurokawa, H.: Characteristic extraction method of human’s strategy in the rock-paper-scissors game. In: Proceedings of the 2018 RISP International Workshop on Nonlinear Circuits, Communications and Signal Processing, Honolulu, HI, USA, pp. 592–595 (2018)

    work page 2018

  22. [22]

    Journal of Signal Processing23(4), 177–180 (2019)

    Komai, T., Kim, S.-J., Kousaka, T., Kurokawa, H.: A human behavior strategy estimation method using homology search for rock-scissors-paper game. Journal of Signal Processing23(4), 177–180 (2019)

    work page 2019

  23. [23]

    Applied Sciences12(23), 12192 (2022) https://doi.org/10.3390/ app122312192

    Komai, T., Kurokawa, H., Kim, S.-J.: Human randomness in the rock-paper- scissors game. Applied Sciences12(23), 12192 (2022) https://doi.org/10.3390/ app122312192

    work page 2022

  24. [24]

    Physical Review Letters 55(5), 449–452 (1985) https://doi.org/10.1103/PhysRevLett.55.449

    Wolfram, S.: Origins of randomness in physical systems. Physical Review Letters 55(5), 449–452 (1985) https://doi.org/10.1103/PhysRevLett.55.449

    work page doi:10.1103/physrevlett.55.449 1985

  25. [25]

    Wolfram Media, Champaign, IL (2002)

    Wolfram, S.: A New Kind of Science. Wolfram Media, Champaign, IL (2002)

    work page 2002

  26. [26]

    SIAM Review49(2), 211–235 (2007) https://doi.org/10.1137/S0036144504446436 29

    Diaconis, P., Holmes, S., Montgomery, R.: Dynamical bias in the coin toss. SIAM Review49(2), 211–235 (2007) https://doi.org/10.1137/S0036144504446436 29

    work page doi:10.1137/s0036144504446436 2007

  27. [27]

    ACM Transactions on Modeling and Computer Simulation8(1), 3–30 (1998) https://doi.org/10.1145/ 272991.272995

    Matsumoto, M., Nishimura, T.: Mersenne twister: A 623-dimensionally equidis- tributed uniform pseudo-random number generator. ACM Transactions on Modeling and Computer Simulation8(1), 3–30 (1998) https://doi.org/10.1145/ 272991.272995

    work page arXiv 1998

  28. [28]

    URLhttps://doi.org/10.1016/0167-2789(92)90242-f

    Theiler, J., Eubank, S., Longtin, A., Galdrikian, B., Farmer, J.D.: Testing for nonlinearity in time series: The method of surrogate data. Physica D: Non- linear Phenomena58(1–4), 77–94 (1992) https://doi.org/10.1016/0167-2789(92) 90102-S

    work page doi:10.1016/0167-2789(92 1992

  29. [29]

    Wolfram Language & System Documentation Center

    Wolfram Research: Random Number Generation. Wolfram Language & System Documentation Center. Accessed 2026. https://reference.wolfram.com/language/ tutorial/RandomNumberGeneration.html

    work page 2026

  30. [30]

    Games12(3), 52 (2021) https://doi.org/10.3390/ g12030052

    Zhang, H., Moisan, F., Gonzalez, C.: Rock-paper-scissors play: Beyond the win-stay/lose-change strategy. Games12(3), 52 (2021) https://doi.org/10.3390/ g12030052

    work page 2021

  31. [31]

    Games10(2), 18 (2019) https://doi.org/10.3390/g10020018

    Batzilis, D., Jaffe, S., Levitt, S.D., List, J.A., Picel, J.: Behavior in strategic settings: Evidence from a million rock-paper-scissors games. Games10(2), 18 (2019) https://doi.org/10.3390/g10020018

    work page doi:10.3390/g10020018 2019

  32. [32]

    Discover Artificial Intelligence5, 86 (2025) https://doi.org/10.1007/s44163-025-00283-z

    Tsuji, K., Takashima, K., Sato, Y., Akimoto, T.: Universality in game-driven random walks with strategies generated by a genetic algorithm. Discover Artificial Intelligence5, 86 (2025) https://doi.org/10.1007/s44163-025-00283-z

    work page doi:10.1007/s44163-025-00283-z 2025

  33. [33]

    Scientific Reports6, 38634 (2016) https://doi.org/10

    Kim, S.-J., Naruse, M., Aono, M., Hori, H., Akimoto, T.: Random walk with chaotically driven bias. Scientific Reports6, 38634 (2016) https://doi.org/10. 1038/srep38634

    work page 2016

  34. [34]

    PLOS ONE 13, e0203657

    Takahashi, H., Izuma, K., Matsumoto, M., Matsumoto, K., Omori, T.: The anterior insula tracks behavioral entropy during an interpersonal competitive game. PLOS ONE10(6), 0123329 (2015) https://doi.org/10.1371/journal.pone. 0123329 30

    work page doi:10.1371/journal.pone 2015