Related papers: Probabilistic vs deterministic gamblers
In the theory of algorithmic randomness, one of the central notions is that of computable randomness. An infinite binary sequence X is computably random if no recursive martingale (strategy) can win an infinite amount of money by betting on…
Consider a universal Turing machine that produces a partial or total function (or a binary stream), based on the answers to the binary queries that it makes during the computation. We study the probability that the machine will produce a…
We introduce a notion of computable randomness for infinite sequences that generalises the classical version in two important ways. First, our definition of computable randomness is associated with imprecise probability models, in the sense…
While it is well known that a Turing machine equipped with the ability to flip a fair coin cannot compute more that a standard Turing machine, we show that this is not true for a biased coin. Indeed, any oracle set $X$ may be coded as a…
A decision maker observes the evolving state of the world while constantly trying to predict the next state given the history of past states. The ability to benefit from such predictions depends not only on the ability to recognize patters…
A sequence of spin-1/2 particles polarised in one of two possible directions is presented to an experimenter, who can wager in a double-or-nothing game on the outcomes of measurements in freely chosen polarisation directions. Wealth is…
Betting strategies are often expressed formally as martingales. A martingale is called integer-valued if each bet must be an integer value. Integer-valued strategies correspond to the fact that in most betting situations, there is a minimum…
The classic model of computable randomness considers martingales that take real or rational values. Recent work by Bienvenu et al. (2012) and Teutsch (2014) shows that fundamental features of the classic model change when the martingales…
We look at the Florida Lottery records of winners of prizes worth $600 or more. Some individuals claimed large numbers of prizes. Were they lucky, or up to something? We distinguish the "plausibly lucky" from the "implausibly lucky" by…
Chances of a gambler are always lower than chances of a casino in the case of an ideal, mathematically perfect roulette, if the capital of the gambler is limited and the minimum and maximum allowed bets are limited by the casino. However, a…
As inductive inference and machine learning methods in computer science see continued success, researchers are aiming to describe ever more complex probabilistic models and inference algorithms. It is natural to ask whether there is a…
We develop a systematic approach to quantum probability as a theory of rational betting in quantum gambles. In these games of chance the agent is betting in advance on the outcomes of several (finitely many) incompatible measurements. One…
Many constructions in computability theory rely on "time tricks". In the higher setting, relativising to some oracles shows the necessity of these. We construct an oracle~$A$ and a set~$X$, higher Turing reducible to~$X$, but for which…
In a recent paper [1], it has been claimed that the outcomes of a quantum coin toss which is idealized as an infinite binary sequence is 1-random. We also defend the correctness of this claim and assert that the outcomes of quantum…
We derive an explicit formula for the probability of ruin of a gambler playing against an infinitely-rich adversary, when the games have payoff given by a general integer-valued probability distribution.
We introduce the sequence-set betting game, a generalization of An. A. Muchnik's non-monotonic betting game. Instead of successively partitioning the infinite binary strings by their value of a bit at a chosen position, as in the…
Do completely unpredictable events exist in nature? Classical theory, being fully deterministic, completely excludes fundamental randomness. On the contrary, quantum theory allows for randomness within its axiomatic structure. Yet, the fact…
A fruitful way of obtaining meaningful, possibly concrete, algorithmically random numbers is to consider a potential behaviour of a Turing machine and its probability with respect to a measure (or semi-measure) on the input space of binary…
Suppose that we are given an infinite binary sequence which is random for a Bernoulli measure of parameter $p$. By the law of large numbers, the frequency of zeros in the sequence tends to~$p$, and thus we can get better and better…
I defend an analog of probabilism that characterizes rationally coherent estimates for chances. Specifically, I demonstrate the following accuracy-dominance result for stochastic theories in the C*-algebraic framework: supposing an…