Fair Coin Research Articles

ChatGPT- versus human-generated answers to frequently asked questions about diabetes: A Turing test-inspired survey among employees of a Danish diabetes center.

Large language models have received enormous attention recently with some studies demonstrating their potential clinical value, despite not being trained specifically for this domain. We aimed to investigate whether ChatGPT, a language model optimized for dialogue, can answer frequently asked questions about diabetes. We conducted a closed e-survey among employees of a large Danish diabetes center. The study design was inspired by the Turing test and non-inferiority trials. Our survey included ten questions with two answers each. One of these was written by a human expert, while the other was generated by ChatGPT. Participants had the task to identify the ChatGPT-generated answer. Data was analyzed at the question-level using logistic regression with robust variance estimation with clustering at participant level. In secondary analyses, we investigated the effect of participant characteristics on the outcome. A 55% non-inferiority margin was pre-defined based on precision simulations and had been published as part of the study protocol before data collection began. Among 311 invited individuals, 183 participated in the survey (59% response rate). 64% had heard of ChatGPT before, and 19% had tried it. Overall, participants could identify ChatGPT-generated answers 59.5% (95% CI: 57.0, 62.0) of the time, which was outside of the non-inferiority zone. Among participant characteristics, previous ChatGPT use had the strongest association with the outcome (odds ratio: 1.52 (1.16, 2.00), p = 0.003). Previous users answered 67.4% (61.7, 72.7) of the questions correctly, versus non-users' 57.6% (54.9, 60.3). Participants could distinguish between ChatGPT-generated and human-written answers somewhat better than flipping a fair coin, which was against our initial hypothesis. Rigorously planned studies are needed to elucidate the risks and benefits of integrating such technologies in routine clinical practice.

Memory optimal dispersion by anonymous mobile robots

Consider a team of k≤n autonomous mobile robots initially placed at a node of an arbitrary graph G with n nodes. The dispersion problem asks for a distributed algorithm that allows the robots to reach a configuration in which each robot is at a distinct node of the graph. If the robots are anonymous, i.e., they do not have any unique identifiers, then the problem is not solvable by any deterministic algorithm. However, the problem can be solved even by anonymous robots if each robot is given access to a fair coin which they can use to generate random bits. In this setting, it is known that the robots require Ω(logΔ) bits of memory to achieve dispersion, where Δ is the maximum degree of G. On the other hand, the best known memory upper bound is min{Δ,max{logΔ,logD}} (D = diameter of G), which can be ω(logΔ), depending on the values of Δ and D. In this paper, we close this gap by presenting an optimal algorithm requiring O(logΔ) bits of memory.

Investment Skill and Consistent Long-Term Alpha

How should an active investor calibrate their investment decisions, when the goal is to reach an outperformance target over the long term, without succumbing to large drawdowns in the interim? Closed-form expressions for the probability of consistent long-term alpha are derived and show that maximizing this probability requires a thoughtful calibration of skill, volatility, decision frequency, and costs. The article derives the minimum skill level for which the probability of consistent long-term alpha trends to its maximum for long investment horizons. This minimum skill level is not typically large, at only marginally above a fair coin toss. Unfortunately, the maximum probability of consistent long-term alpha is also not very large, even when skill is high and the investment horizon is long. The reason is that, although the probability of reaching the long-term goal rises with the time horizon for sufficiently high skill, so does the probability of experiencing the intertemporal loss that can force a stop-out in the interim.

Embodied nonlinear dynamics of cognitive performance.

Study participants are typically unable to generate binary button-press sequences that pass as classically random sequences, such as from successive "fair coin" flips. Instead, their sequences repeat or alternate between responses too often. These deviations from randomness are commonly explained in terms of limitations or idiosyncrasies in cognitive processing. This article tests a novel hypothesis that randomness departures in participant-generated binary sequences are driven by coordination dynamics; alternating and repeating sequences are related to bimanual coordination attractors. Participants (N = 128) were asked to generate sequences that were representative of a random sequence, by successively pressing either of two buttons across 1,600 trials. Statistical analyses identify the binary button-press dynamics with a discrete sine-circle version of the Haken, Kelso, Bunz bimanual coordination model. Permutation analyses revealed the most common one- to five-trial subsequences were identified with the most dynamically stable coordinative relationships, consistent with bimanual coordination predictions. The sequences were consistent with scaling noises. Thus, participants' sequences departed from classical randomness by virtue of membership in a more inclusive category of variability that subsumes classical randomness. Recurrence quantification analysis revealed the mixture of stochasticity and determinism in the sequences was better approximated by the sine-circle model than by phase-randomized surrogate data sets that preserved both the power spectral densities and distributions of each participant's sequence. A relationship between randomness production and two-alternative forced-choice performance is established that constrains response time distribution models. The article's organization illustrates a nonreductive approach to inference for cognitive systems, inspired by statistical physics concepts such as renormalization group theory and universality. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

The Modal Future

<i>The Modal Future</i>

LETRNG—A Lightweight and Efficient True Random Number Generator for GNU/Linux Systems

Unpredictable and irreproducible digital keys are required to modulate security-related information in secure communication systems. True random number generators (TRNGs) rather than pseudorandom number generators (PRNGs) are required for the highest level of security. TRNG is a significant component in the digital security realm for extracting unpredictable binary bitstreams. Presently, most TRNGs extract high-quality “noise” from unpredictable physical random phenomena. Thus, these applications must be equipped with external hardware for collecting entropy and converting them into a random digital sequence. This study introduces a lightweight and efficient true random number generator (LETRNG) that uses the inherent randomness of a central processing unit (CPU) and an operating system (OS) as the source of entropy. We then utilize a lightweight post-processing method based on XOR and fair coin operation to generate an unbiased random binary sequence. Evaluations based on two famous test suites (NIST and ENT) show that LETRNG is perfectly capable of generating high-quality random numbers suitable for various GNU/Linux systems.

Rationality Now!

Rationality Now!

Gambler’s Ruin and the ICM

Consider gambler’s ruin with three players, 1, 2, and 3, having initial capitals A, B, and C units. At each round a pair of players is chosen (uniformly at random) and a fair coin flip is made resulting in the transfer of one unit between these two players. Eventually, one of the players is eliminated and play continues with the remaining two. Let σ∈S3 be the elimination order (e.g., σ=132 means player 1 is eliminated first and player 3 is eliminated second, leaving player 2 with A+B+C units). We seek approximations (and exact formulas) for the elimination order probabilities PA,B,C(σ). Exact, as well as arbitrarily precise, computation of these probabilities is possible when N:=A+B+C is not too large. Linear interpolation can then give reasonable approximations for large N. One frequently used approximation, the independent chip model (ICM), is shown to be inadequate. A regression adjustment is proposed, which seems to give good approximations to the elimination order probabilities.

Is The Probability of Tossing a Coin Really 50–50%? Part 2: Dynamic Model with Rebounds

In the first part of this paper, we considered several theoretical models, a static and four dynamic models without rebounds, of the throw of a fair coin landing on its edge, to demonstrate that the probability of heads or tails is less than 50%, depending on the initial toss conditions, the coin geometry and conditions of the coin and landing surfaces. For the dynamic model with rebounds that is the subject of this second part of the paper, it is found that the probability that a 50 Euro cent coin thrown from a normal height with common initial velocity conditions and appropriate surface conditions will end up on its edge is in the order of one against several thousand.

Is the Probability of Tossing a Coin Really 50–50%? Part 1: Static Model and Dynamic Models without Rebounds

Considering that a fair coin has two sides and a cylindrical edge, the probability that it would fall on its edge is calculated, yielding the probability of heads or tails of less than 50%. In this first part, the theoretical models for a static case and for five dynamic cases, without rebounds, show that there is a small probability that the coin does not fall on its head or tail, depending on the initial toss conditions, the coin geometry and conditions of the coin and landing surfaces. It is found that the probability that a 50 Eurocent coin thrown from a normal height with common initial velocity conditions and appropriate surface conditions will end up on its edge is in the order of one against several thousand. In the second part of the paper, the dynamic model with rebounds is investigated.

On the Sn/n problem

AbstractThe Chow–Robbins game is a classical, still partly unsolved, stopping problem introduced by Chow and Robbins in 1965. You repeatedly toss a fair coin. After each toss, you decide whether you take the fraction of heads up to now as a payoff, otherwise you continue. As a more general stopping problem this reads $V(n,x) = \sup_{\tau }\mathbb{E} \left [ \frac{x + S_\tau}{n+\tau}\right]$ , where S is a random walk. We give a tight upper bound for V when S has sub-Gaussian increments by using the analogous time-continuous problem with a standard Brownian motion as the driving process. For the Chow–Robbins game we also give a tight lower bound and use these to calculate, on the integers, the complete continuation and the stopping set of the problem for $n\leq 489\,241$ .

Casting New Light on Statistical Power: An Illuminating Analogy and Strategies to Avoid Underpowered Trials.

Current standards for methodological rigor and trial reporting underscore the critical issue of statistical power. Still, the chance of detecting most effects reported in randomized controlled trials in medicine and other disciplines is currently lower than winning a toss of a fair coin. Here we propose that investigators who retain a practical understanding of how statistical power works can proactively avoid the potentially devastating consequences of underpowered trials. We first offer a vivid, carefully constructed analogy that illuminates the underlying relationships among 3 of the 5 essential parameters-namely, statistical power, effect size, and sample size-while holding the remaining 2 parameters constant (type of statistical test and significance level). Second, we extend the analogy to a set of critical scenarios in which investigators commonly miss detecting intervention effects due to insufficient statistical power. Third, we highlight effective pragmatic strategies for the design and conduct of sufficiently powered trials, without increasing sample size.

Children Consider Procedures, Outcomes, and Emotions When Judging the Fairness of Inequality.

Children tend to view equal resource distributions as more fair than unequal ones, but will sometimes view even unequal distributions as fair. However, less is known about how children form judgments about inequality when different procedures are used. In the present study, we investigated children’s consideration of procedures (i.e., resource-distributing processes), outcomes (i.e., the distributions themselves), and emotions (i.e., the emotional reactions of those receiving the resources) when judging the fairness of unequal resource distributions. Participants (N = 130, 3- to 8-year-olds) were introduced to a Fair Coin (different color on each side) and an Unfair Coin (same color on both sides). In two between-subjects conditions, they watched a researcher flip either the Fair or Unfair Coin in order to distribute resources unequally between two child recipients. Participants then rated the fairness of this event, provided verbal justifications for their ratings (coded for references to procedures and/or outcomes), and rated the emotional state of each recipient (from which an Emotion Difference Score was computed). Results revealed that participants rated the event as more fair in the Fair Coin than the Unfair Coin condition. References to the outcome in children’s justifications predicted lower fairness ratings, while references to the procedure only predicted lower ratings in the Unfair Coin condition. Greater Emotion Difference Scores predicted lower fairness ratings, and this effect increased with age. Together, these results show that children consider procedures, outcomes, and emotions when judging the fairness of inequality. Moreover, results suggest age-related increases in consideration of recipients’ emotions makes inequality seem less fair, even when fair procedures are used. Implications for the development of fairness are discussed.

On the complexity of fair coin flipping

A two-party coin-flipping protocol is ε-fair if no efficient adversary can bias the output of the honest party (who always outputs a bit, even if the other party aborts) by more than ε. Cleve [STOC '86] showed that r-round o(1/r)-fair coin-flipping protocols do not exist. Awerbuch, Blum, Chor, Goldwasser, and Micali [Manuscript '85] constructed a Θ(1/r)-fair coin-flipping protocol, assuming the existence of one-way functions. Moran, Naor, and Segev [Journal of Cryptology '16] constructed an r-round coin-flipping protocol that is Θ(1/r)-fair (thus matching the aforementioned lower bound of Cleve [STOC '86]), assuming the existence of oblivious transfer.The above gives rise to the intriguing question of whether oblivious transfer, or more generally “public-key primitives,” is required for an o(1/r)-fair coin-flipping protocol. Towards answering this intriguing question, Maji and Wang [Crypto '18] have recently showed that in the random oracle model (ROM), any coin-flipping protocol can be biased by Ω(1/r). This implies that o(1/r)-fair coin-flipping protocol cannot be constructed from one-way function, or from a family of collision-resistant hash functions, in a black-box way. This result does not rule out, however, non black-box constructions, and black-box constructions based on primitives that cannot be realized in the ROM.We make a different progress towards answering above question by showing that, for any constant r∈N, the existence of an 1/(c⋅r)-fair, r-round coin-flipping protocol implies the existence of an infinitely-often key-agreement protocol, where c denotes some universal constant (independent of r). Our reduction is non black-box and makes a novel use of the recent dichotomy for two-party protocols of Haitner, Nissim, Omri, Shaltiel, and Silbak [SICOMP '20] to facilitate a two-party variant of the recent attack of Beimel, Haitner, Makriyannis, and Omri [FOCS '18] on multi-party coin-flipping protocols.

SMALL-SCALE EQUIDISTRIBUTION OF RANDOM WAVES GENERATED BY AN UNFAIR COIN FLIP

AbstractIn this paper we study the small-scale equidistribution property of random waves whose coefficients are determined by an unfair coin. That is, the coefficients take value $+1$ with probability p and $-1$ with probability $1-p$ . Random waves whose coefficients are associated with a fair coin are known to equidistribute down to the wavelength scale. We obtain explicit requirements on the deviation from the fair ( $p=0.5$ ) coin to retain equidistribution.

Synthesizing optimal bias in randomized self-stabilization

Randomization is a key concept in distributed computing to tackle impossibility results. This also holds for self-stabilization in anonymous networks where coin flips are often used to break symmetry. Although the use of randomization in self-stabilizing algorithms is rather common, it is unclear what the optimal coin bias is so as to minimize the expected convergence time. This paper proposes a technique to automatically synthesize this optimal coin bias. Our algorithm is based on a parameter synthesis approach from the field of probabilistic model checking. It over- and under-approximates a given parameter region and iteratively refines the regions with minimal convergence time up to the desired accuracy. We describe the technique in detail and present a simple parallelization that gives an almost linear speed-up. We show the applicability of our technique to determine the optimal bias for the well-known Herman’s self-stabilizing token ring algorithm. Our synthesis obtains that for small rings, a fair coin is optimal, whereas for larger rings a biased coin is optimal where the bias grows with the ring size. We also analyze a variant of Herman’s algorithm that coincides with the original algorithm but deviates for biased coins. Finally, we show how using speed reducers in Herman’s protocol improve the expected convergence time.

No fact of the middle

AbstractAmiddle factis a true proposition about what would have happened hadAbeen true (whereAis in fact false), whose truth isn't entailed by any non‐counterfactual facts. I argue that there are no middle facts; if there were, we wouldn't know them, and our ignorance of them would result in ignorance about whether regret is fitting in cases where we clearly know it is. But there's a problem. Consider an unflipped fair coin which is such that no non‐counterfactual fact determines that it would have landed heads had it been flipped (or tails had it been flipped). If there are no middle facts, it's not true that it would have landed heads had it been flipped nor that it would have landed tails had it been flipped. Yet each counterfactual is still possibly true for all we know. I argue that we can resolve this tension in the anti‐middle fact position, further strengthening the case against middle facts.

Re-exploring the Penney-Ante Game

We propose a single loop diagram and use it to devise a single loop matrix method to computationally solve the Penney-Ante game. This method avoids the nuances of repeated use of conditional probability and Markov chain representations. We remove the limitations of Conway’s trick as applied to a fair coin and generalize the method where the coin is allowed to be biased. A uniform random number generator is used to simulate the game and formulate implicit mathematical relations to explore nontransitivity.

Biases and Variability from Costly Bayesian Inference.

When humans infer underlying probabilities from stochastic observations, they exhibit biases and variability that cannot be explained on the basis of sound, Bayesian manipulations of probability. This is especially salient when beliefs are updated as a function of sequential observations. We introduce a theoretical framework in which biases and variability emerge from a trade-off between Bayesian inference and the cognitive cost of carrying out probabilistic computations. We consider two forms of the cost: a precision cost and an unpredictability cost; these penalize beliefs that are less entropic and less deterministic, respectively. We apply our framework to the case of a Bernoulli variable: the bias of a coin is inferred from a sequence of coin flips. Theoretical predictions are qualitatively different depending on the form of the cost. A precision cost induces overestimation of small probabilities, on average, and a limited memory of past observations, and, consequently, a fluctuating bias. An unpredictability cost induces underestimation of small probabilities and a fixed bias that remains appreciable even for nearly unbiased observations. The case of a fair (equiprobable) coin, however, is singular, with non-trivial and slow fluctuations in the inferred bias. The proposed framework of costly Bayesian inference illustrates the richness of a ‘resource-rational’ (or ‘bounded-rational’) picture of seemingly irrational human cognition.

Experimental test of fair three-sided coins

A simple model for a fair ‘three-sided coin’ is proposed and tested. Describing the coin as a cylinder with a given height and basis radius, this model efficiently characterizes the problem, constraining the size of the coin. A statistical analysis of the data collected from actual realizations of such coins has been performed, supporting the proposed model. Besides studying the case of a fair three-sided coin, this work represents a model for an explicit application of the scientific method, in which all parts (problem characterization, statement of a hypothesis, experiment, analysis, description, conclusions) have clearly directed its development. Thus, it represents an useful illustration of such method for undergraduate students.