The Illusion of Randomness: How LLM s Fail to Emulate Stochastic Decision-Making in Rock-Paper-Scissors Games? (2025)

Abstract: Prior research indicates that although large language models (LLMs) can precisely articulate the theoretical probability distributions associated with optimal strategic choices, their actual decision-making systematically diverges from these prescriptions—a phenomenon we define as the cognition–behaviour gap in LLMs. For example, in a Rock–Paper–Scissors (RPS) game, LLMs correctly identify the strategy of Nash equilibrium as selecting each action (Rock, Paper, Scissors) with equal probability \frac{1}{3} , but their observed choices systematically deviate from this uniform distribution. Through a comprehensive evaluation of 20 state-of-the-art LLMs, we identify two critical insights: (1) we demonstrate that intrinsic biases inherited from pre-training corpora alone are insufficient to explain the observed deviations; (2) we introduce a semantic-free paradigm that strips away intrinsic biases to isolate pure positional bias-LLMs exhibit distinct position preferences—for example, o1 favours the first option, DeepSeek-V3 peaks the middle and DeepSeek-R1 shows a bimodal bias toward first and last positions. Our findings advocate innovation to bridge the gap between strategic reasoning and decision-making in LLMs.

## introduction large language models (llms) have demonstrated remarkable capabilities in strategic reasoning tasks, from solving mathematical games @xcite to simulating negotiations @xcite . however, existing research shows that the stochastic decision-making of llms remains deficient @xcite , and llms deviate from ideal randomness in tasks as simple as coin-flipping @xcite , dice-rolling @xcite , and pseudorandom number generation @xcite . in particular, such deviations persist when llms explicitly attempt to approximate mixedstrategy nash equilibrium @xcite . for instance, in the rock-paper-scissors (rps) game, @xcite reports that gpt-4 correctly identifies the nash equilibrium as uniform randomization (33.3% per action) yet selects rock in 67% of 100 independent trials-approximately twice the theoretical frequency. this limitation may pose an obstacle to realworld deployments, for instance, llm-driven agents in embodied intelligence. consider a simplified, illustrative scenario involving two robots engaged in a penalty shootout, as depicted in figure 1 . in this hypothetical situation, an llm-controlled goalkeeper correctly identifies the nash equilibrium strategy, which is diving left, center, or right with equal probability. but if its actual decisions skew, say, 70% to the right, an opposing kicker can learn and exploit that bias, dramatically undermining the goalkeeper's effectiveness. we call this mismatch between what the llms know and what they do as the "cognition-behaviour gap". initially, @xcite attributes such gaps to failures in probabilistic sampling despite correct distributional reasoning. a prevailing explanation further links these failures to intrinsic biases inherited from pre-training corpora @xcite . however, the evidence remains tentative: existing results cover only a narrow set of models and experimental designs, and no systematic, cross-model analysis has cleanly disentangled corpus-level effects from other confounders, which motivates the present investigation. in this work, we center our analysis on rps, whose unique mixed-strategy nash equilibrium prescribes uniform random play. we comprehensively evaluate 20 state-of-the-art llms to examine how llms fail to emulate this stochastic decision-making. we find that nearly all advanced llms (e.g., gpt-4.5 @xcite , deepseek-r1 @xcite ) exhibit systematic biases, with empirical action distributions deviating from the probability distributions they explicitly derive. we establish two insightful findings: the remainder of this paper is structured as follows: section 2 synthesises related work on the gap of randomness behaviour in llms. section 3 introduces the background of our research. section 4 presents empirical results to declare our findings. section 5 discusses the positional-bias failure in stochastic choices and outlines practical mitigations. section 6 concludes our empirical findings. ## related work recent work has highlighted significant challenges in llms' abilities to generate truly random samples. for instance, when gpt-4 "rolls" a virtual die, it outputs some faces far more often than the theoretical 1 6 frequency. in longer sequences, the marginal distribution becomes nearly uniform, yet adjacent numbers repeat less often than chance would predict @xcite . even the ostensibly simpler task of producing binary sequences is affected: gpt-4 and llama 3 reproduce human-like cognitive biases in simulated coin flips, whereas gpt-3.5 exhibits behaviour that more closely approximates randomness @xcite . these errors persist even when models are asked to sample from arbitrary target distributions @xcite , suggesting that the difficulty is agnostic to output space. the gap is also evident in game-theoretic settings. in rps, for example, @xcite show that gpt-4 correctly states the uniform mixed-strategy nash equilibrium yet still chooses rock in 67% of independent rounds-roughly twice the prescribed 33% frequency. a similar mismatch has been documented in other game scenarios like matching pennies, chicken games, etc. across gpt-3.5 and gpt-4 @xcite , indicating a consistent failure to translate theoretical distributions into action. @xcite attributes the cognitionbehaviour gap, in which the mismatch between llms' stated understanding of probability and their generated outputs to intrinsic bias inherited from the pre-training corpora. several studies document a "frequency effect" as the factor of intrinsic bias, where token prevalence in pre-training corpora systematically skews llm outputs @xcite @xcite . building upon this claim, we want to uncover how more frequent words can dominate stochastic generation even when uniform randomness is prescribed. recent work has brought attention to positional bias-the tendency of llms to favour answer options based on their placement within the prompt (pezeshkpour and hruschka, 2023). for example, @xcite show that when llms are asked to choose between two colours, gpt-4o-mini consistently selects the first option, while llama-3.1-8b favours the second. however, such colour preference questions inherently engage corpus-driven biases about common colour associations. to further study it, we disentangle the source of biases and accurately measure the impact of positional bias on llm decision-making. prior works have identified and mitigated llm selection biases in multiple-choice question (mcq) tasks (pezeshkpour and hruschka, 2023). for example, @xcite proposed grey-box probability calibration and a black-box two-hop querying strategy to counter option-order biases. @xcite further introduced a black-box bias node pruning method to remove internal bias parameters, as well as an auxiliary option injection technique that adds a dummy "i don't know" choice to flatten the selection distribution. however, these approaches assume an objectively correct answer and often require numerous model queries or direct model access, making them less applicable to our scenarios, where all options are equally valid (e.g., rps) and efficiency is paramount. ## experiment we begin with a case study to visualize action distributions of 20 llms to confirm the presence of a cognition-behaviour gap. then, we examine its underlying determinants by systematically analysing two factors: (1) intrinsic bias inherited from pre-training corpora, and (2) positional bias arising from the ordering of answer options within prompts. more experiments, discussions, and explanations can be found in appendix a. 2025 ), grok-3 (xai, 2025), and qwen-2.5-max @xcite . configurations. all model evaluations were conducted in a vanilla state using the official api interface 1 . to ensure statistical significance, each experimental scenario was tested on 100 independent games, with a new session established for each call to eliminate any contextual interference. we conduct an initial experiment involving 1,000 independent generations from the llms in appendix a.1. the results show that 100 runs are sufficient to approximate the model's output distribution. the experimental parameters were fixed at @xmath0, top-@xmath1, and repetition penalties were disabled, ensuring nondeterministic generation and consistent experimental settings. ## discussion our study identifies a failure mode in llms' stochastic decision-making: models can state the uniform mixed-strategy nash equilibrium yet still choose systematically non-uniform actions, and these choices are strongly affected by positional bias in the prompt. we next examine two mcq debiasing techniques in this setting, namely grey-box probability weighting (gbp) @xcite and black-box auxiliary option injection (aoi) @xcite , to test whether methods developed for mcq tasks transfer to label-free, uniformtarget stochastic decision making. both methods are adapted to a uniform-target, label-free rps protocol; the experimental details appear in appendix a.9. in brief, neither method moves the empirical distribution closer to the uniform mixedstrategy target; deviations persist, and for several models the tvd even increases under aoi or gbp. to mitigate the failure in llms' stochastic decision-making, we outline two possible directions aimed at reducing this pathology. (1) verbalized probability calibration (vpc). apply vpc to uniform target games: elicit a probability vector over the n options and apply a light post-hoc calibration (e.g., temperature or isotonic regression with simplex projection) so the expectation leans toward 1/n @xcite . (2) test-time positional retrieval. preparing a small annotated reference set could guide option ordering, potentially damping order sensitivity with a single model call @xcite . resolving these basic failures in stochastic choice may translate into more reliable llm agents in complex, high-stakes settings, where robust randomization decisions are required. ## conclusion after the evaluation of twenty contemporary llms, two sources of bias emerge that limit truly random choice. intrinsic bias from pre-training regularities, which pulls probability mass toward high-frequency or semantically salient options. in addition, positional bias occurs where the placement of options in the prompt alters choice probabilities. together, they produce a persistent cognitionbehaviour gap: models can state the uniform mixed-strategy nash equilibrium, yet their samples remain non-uniform, and in many cases, position alone is sufficient to shift the distribution away from uniform. how to make llms achieve robust randomization for stochastic decisions, especially under black-box constraints and realistic cost budgets, remains an open challenge.