Related papers: The Three Doors Problem...-s
These are the notes of my lectures at the 1996 European Congress of Mathematicians. {} Polynomials appear in mathematics frequently, and we all know from experience that low degree polynomials are easier to deal with than high degree ones.…
We consider the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex. This problem is at the interface between convex analysis, convex optimization, variational…
We study computational aspects of three prominent voting rules that use approval ballots to elect multiple winners. These rules are satisfaction approval voting, proportional approval voting, and reweighted approval voting. We first show…
We study a simple example of a sequential game illustrating problems connected with making rational decisions that are universal for social sciences. The set of chooser's optimal decisions that manifest his preferences in case of a constant…
Any problem is concerned with the mind, but what do minds make a decision on? Here we show that there are three conditions for the mind to make a certain answer. We found that some difficulties in physics and mathematics are in fact…
The decision-maker (DM) sequentially evaluates up to N of different, rankable options. DM must select exactly the best one at the moment of its appearance. In the process of searching, DM finds out with each applicant whether she is the…
I explain the shutdown problem: the problem of designing artificial agents that (1) shut down when a shutdown button is pressed, (2) don't try to prevent or cause the pressing of the shutdown button, and (3) otherwise pursue goals…
In classical Monty Hall problem, one player can always win with probability 2/3. We generalize the problem to the quantum domain and show that a fair two-party zero-sum game can be carried out if the other player is permitted to adopt…
The evolution of mathematics is shaped importantly by interestingness: researchers choose which problems to pursue, and students choose which problems to engage with, based on expectations of interest and challenge. As AI systems,…
The Deviants' Dilemma is a two-person game with the individual gain conflicting with the choice for global good. Evolutionary considerations yield fixed point attractors, with the phenomena of exclusion potentially playing an important role…
Two active hypothesis testing problems are formulated. In these problems, the agent can perform a fixed number of experiments and then decide on one of the hypotheses. The agent is also allowed to declare its experiments inconclusive if…
One presents many Concatenated and Operation Sequences, P-Q Relationships, Digital Sequences, Magic Squares, Prime Conjectures, k-Divisibility and Strong Divisibility Sequences, Geometric Conjectures, Proposed problems.
Solving a decision theory problem usually involves finding the actions, among a set of possible ones, which optimize the expected reward, possibly accounting for the uncertainty of the environment. In this paper, we introduce the…
A conjecture of Hopkins (2018) posits that for certain high-dimensional hypothesis testing problems, no polynomial-time algorithm can outperform so-called "simple statistics", which are low-degree polynomials in the data. This conjecture…
New methods for solving the college admissions problem with indifference are presented and characterised with a Monte Carlo simulation in a variety of simple scenarios. Based on a qualifier defined as the average rank, it is found that…
In considering the reliability of numerical programs, it is normal to "limit our study to the semantics dealing with numerical precision" (Martel, 2005). On the other hand, there is a great deal of work on the reliability of programs that…
Recently, the educational initiative TED-Ed has published a popular brain teaser coined the 'frog riddle', which illustrates non-intuitive implications of conditional probabilities. In its intended form, the frog riddle is a reformulation…
Algebraic statistics is concerned with the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry. This article presents a list of open mathematical problems in this emerging field,…
We study the phase diagram and the algorithmic hardness of the random `locked' constraint satisfaction problems, and compare them to the commonly studied 'non-locked' problems like satisfiability of boolean formulas or graph coloring. The…
In a context where a decision has to be taken collectively by several agents, the social choice problem consists in deciding whether there exists a socially acceptable rule that aggregates the individual preferences of the agents into a…