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

In 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.

How strong can the Parrondo effect be? II

Parrondo’s coin-tossing games comprise two games, A A and B B . The result of game A A is determined by the toss of a fair coin. The result of game B B is determined by the toss of a p 0 p_0 -coin if capital is a multiple of r r , and by the toss of a p 1 p_1 -coin otherwise. In either game, the player wins one unit with heads and loses one unit with tails. Game B B is fair if ( 1 − p 0 ) ( 1 − p 1 ) r − 1 = p 0 p 1 r − 1 (1-p_0)(1-p_1)^{r-1}=p_0\,p_1^{r-1} . In a previous paper we showed that, if the parameters of game B B , namely r r , p 0 p_0 , and p 1 p_1 , are allowed to be arbitrary, subject to the fairness constraint, and if the two (fair) games A A and B B are played in an arbitrary periodic sequence, then the rate of profit can not only be positive (the so-called Parrondo effect), but also be arbitrarily close to 1 (i.e., 100%). Here we prove the same conclusion for a random sequence of the two games instead of a periodic one, that is, at each turn game A A is played with probability γ \gamma and game B B is played otherwise, where γ ∈ ( 0 , 1 ) \gamma \in (0,1) is arbitrary.

Savage vs. Anscombe-Aumann: an experimental investigation of ambiguity frameworks

The Savage and the Anscombe–Aumann frameworks are the two most popular approaches used when modeling ambiguity. The former is more flexible, but the latter is often preferred for its simplicity. We conduct an experiment where subjects place bets on the joint outcome of an ambiguous urn and a fair coin. We document that more than a third of our subjects make choices that are incompatible with Anscombe–Aumann for any preferences, while the Savage framework is flexible enough to account for subjects’ behaviors.

Memories of the Future. Predictable and Unpredictable Information in Fractional Flipping a Biased Coin

Some uncertainty about flipping a biased coin can be resolved from the sequence of coin sides shown already. We report the exact amounts of predictable and unpredictable information in flipping a biased coin. Fractional coin flipping does not reflect any physical process, being defined as a binomial power series of the transition matrix for “integer” flipping. Due to strong coupling between the tossing outcomes at different times, the side repeating probabilities assumed to be independent for “integer” flipping get entangled with one another for fractional flipping. The predictable and unpredictable information components vary smoothly with the fractional order parameter. The destructive interference between two incompatible hypotheses about the flipping outcome culminates in a fair coin, which stays fair also for fractional flipping.

Penney's Game Odds From No-Arbitrage

Penney's game is a two player zero-sum game in which each player chooses a three-flip pattern of heads and tails and the winner is the player whose pattern occurs first in repeated tosses of a fair coin. Because the players choose sequentially, the second mover has the advantage. In fact, for any three-flip pattern, there is another three-flip pattern that is strictly more likely to occur first. This paper provides a novel no-arbitrage argument that generates the winning odds corresponding to any pair of distinct patterns. The resulting odds formula is equivalent to that generated by Conway's ``leading number'' algorithm. The accompanying betting odds intuition adds insight into why Conway's algorithm works. The proof is simple and easy to generalize to games involving more than two outcomes, unequal probabilities, and competing patterns of various length. Additional results on the expected duration of Penney's game are presented. Code implementing and cross-validating the algorithms is included.

Exploring the randomness of mentally generated head–tail sequences

It is well known that people deviate from randomness as they attempt to mentally generate head–tail sequences as randomly as possible. This deviation from randomness is quantified by an excess of repetitions or alternations between successive responses more than would be expected by chance. We conducted an experiment in which a sample of students was asked to mentally simulate a sequence as if it is produced by a fair coin. We propose several models based on Markov chains for analysing the dynamic of head–tail outcomes in these sequences. First, we explore observed Markov chains and suggest some practical solutions to reduce the number of parameters. However, there is a need for more sophisticated models, and in this case, we propose latent Markov models and mixture of Markov chains to analyse these head–tail sequences. A generalization of the so-called mixture transition distribution (MTD) model is also considered.

Infinity, Causation, and Paradox

Infinity is paradoxical in many ways. A particular large family of paradoxes is examined that on its face iswidely varied. Some involve deterministic supertasks, such as Thomson’s Lamp where a switch is toggled an infinite number of times over a finite period of time, or the Grim Reaper, where it seems that infinitely many reapers can produce a result without doing anything. Others involve infinite lotteries. Yet others involve paradoxical results in decision theory, such as the surprising observation that if you perform a sequence of fair coin-flips that goes infinitely far back into the past but only finitely into the future, you can leverage information about past coin-flips to predict future ones with only finitely many mistakes. It turns out that these, and a number of other paradoxes have a common structure: their most natural embodiment involves an infinite number of items causally impinging on a single output. These paradoxes can all be solved with a single move: embrace causal finitism, the view that it is impossible for a single output to have an infinite causal history. The book exposits such paradoxes, defends causal finitism at length, and ends up considering connections with the philosophy of physics, where causal finitism favors, but does not require, discretist theories of space and time, and the philosophy of religion, where we get a cosmological argument reminiscent of the Kalām argument for the existence of God.

Patterns in Randomness

I do an extra-sensory perception (ESP) experiment on the first day of my statistics classes. I show the students an ordinary coin— sometimes borrowed from a student—and flip the coin ten times. After each flip, I think about the outcome intently while the students try to read my mind. They write their guesses down, and I record the actual flips by circling H or T on a piece of paper that has been designed so that the students cannot tell from the location of my pencil which letter I am circling. Anyone who guesses all ten flips correctly wins a one-pound box of chocolates from a local gourmet chocolate store. If you want to try this at home, guess my ten coin flips in the stats class I taught in the spring of 2017. My brain waves may still be out there somewhere. Write your guesses down, and we’ll see how well you do. After ten flips, I ask the students to raise their hands and I begin revealing my flips. If a student misses, the hand goes down, Anyone with a hand up at the end wins the chocolates. I had a winner once, which is to be expected since more than a thousand students have played this game. I don’t believe in ESP, so the box of chocolates is not the point of this experiment. I offer the chocolates in order to persuade students to take the test seriously. My real intent is to demonstrate that most people, even bright college students, have a misperception about what coin flips and other random events look like. This misperception fuels our mistaken belief that data patterns uncovered by computers must be meaningful. Back in the 1930s, the Zenith Radio Corporation broadcast a series of weekly ESP experiments. A “sender” in the radio studio randomly chose a circle or square, analogous to flipping a fair coin, and visualized the shape, hoping that the image would reach listeners hundreds of miles away. After five random draws, listeners were encouraged to mail in their guesses. These experiments did not support the idea of ESP, but they did provide compelling evidence that people underestimate how frequently patterns appear in random data.

Coin trials

According to the JUSTIFIED FAIR COINS principle, if I know that a coin is fair, and I lack justification for believing that it won’t be flipped, then I lack justification for believing that it won’t land tails. What this principle says, in effect, is that the only way to have justification for believing that a fair coin won’t land tails, is by having justification for believing that it won’t be flipped at all. Although this seems a plausible and innocuous principle, in a recent paper Dorr, Goodman and Hawthorne use it in devising an intriguing puzzle which places all justified beliefs about the future in jeopardy. They point out, further, that one very widespread theory of justification predicts that JUSTIFIED FAIR COINS is false, giving us additional reason to reject it. In this paper, I will attempt to turn this dialectic around. I will argue that JUSTIFIED FAIR COINS does not inevitably lead to scepticism about the future, and the fact that it is incompatible with a widespread theory of justification may give us reason to doubt the theory. I will outline an alternative theory of justification that predicts both that JUSTIFIED FAIR COINS is true and that we have justification for believing much about the future.