Searcharxiv⌕ Search

arXiv subjects

Yaakov Malinovsky

Publications and source records attributed to Yaakov Malinovsky.

At least 19 recordsLinked to original sources
arXiv2026

The Game of Marginal Utilities

We study a noncooperative resource-allocation game in which $m$ players distribute fixed resources among $n$ projects and the payoff of player $j$ is given by \[ F^j(x) = \sum_{i=1}^n \frac{a_i x_i^j}{b_i+\sum_{\ell=1}^m x_i^\ell}, \] where $a_i$ and $b_i$ are project parameters, while $x_i^j$ is the amount of resources player $j$ allocates to project $i$. This specification combines diminishing returns with congestion generated by competitors. We prove that the game has a unique Nash equilibrium and characterize it by an equimarginal principle. We show that, after the projects are ordered by $a_i/b_i$, each player invests in an initial segment of projects, and these segments are nested across players, so the equilibrium decomposes into consecutive activity zones; players with larger resources invest weakly more in every project. In the fully active regime, where every player invests in every project, we reduce the equilibrium to a single scalar nonlinear equation for the aggregate marginal-utility rate; all individual marginal rates, investments, and payoffs then follow from explicit formulas. We also provide a projected marginal-utility algorithm with global linear convergence under an explicit step-size condition, together with a structure-exploiting \textit{Block Pandora} algorithm that reconstructs and certifies the equilibrium conditional on a proposed nested cutoff structure.

cs.GT↗
arXiv2026

Distinct Extreme Scores in Random Round-Robin Tournaments

We consider a general round-robin tournament model with equally strong players, where $X_{ij}$ denotes the score of player $i$ against player $j$. We assume that $X_{ij}$ takes values in a countable subset of $[0,1]$ and satisfies $X_{ij}+X_{ji}=1$. We prove that if $k(n)\to\infty$ as $n\to\infty$ and $\frac{k(n)^2\sqrt{\log(n/k(n))}}{\sqrt n}\to 0,$ then, with probability tending to one, the largest $k(n)$ scores are all distinct. In particular, this holds whenever $k(n)=o\!\Bigl(\bigl(n/\log n\bigr)^{1/4}\Bigr).$ By symmetry, the same conclusion also holds for the lowest $k(n)$ scores. The obtained scale coincides with the one arising in classical problems on distinct extreme degrees in Erd\H{o}s--R\'enyi random graphs, despite the fundamentally different dependence structure. This suggests that distinctness of extreme values may persist under broad classes of models exhibiting weak dependence.

math.PR↗
arXiv2026

Extreme Score Distributions in Countable-Outcome Round-Robin Tournaments of Equally Strong Players

We consider a general class of round-robin tournament models of equally strong players. In these models, each of the $n$ players competes against every other player exactly once. For each match between two players, the outcome is a value from a countable subset of the unit interval, and the scores of the two players in a match sum to one. The final score of each player is defined as the sum of the scores obtained in matches against all other players. We study the distribution of extreme scores, including the maximum, second maximum, and lower-order extremes. Since the exact distribution is computationally intractable even for small values of $n$, we derive asymptotic results as the number of players $n$ tends to infinity, including limiting distributions, and rates of convergence.

math.PR↗
arXiv2025

Maximal Counts in the Stopped Occupancy Problem

We revisit a version of the classic occupancy scheme, where balls are thrown until almost all boxes receive a given number of balls. Special cases are widely known as coupon-collectors and dixie cup problems. We show that as the number of boxes tends to infinity, the distribution of the maximal occupancy count does not converge, but can be approximated by a convolution of two Gumbel distributions, with the approximating distribution having oscillations close to periodic on a logarithmic scale. We pursue two approaches: one relies on lattice point processes obtained by poissonisation of the number of balls and boxes, and the other employs interpolation of the multiset of occupancy counts to a point process on reals. This way we gain considerable insight in known asymptotics obtained previously by mostly analytic tools. Further results concern the moments of maximal occupancy counts and ties for the maximum.

math.PR↗
arXiv2025

The Optimality of a Nested Generalized Pairwise Group Testing Procedure

We study the problem of identifying defective units in a finite population of \( n \) units, where each unit \( i \) is independently defective with known probability \( p_i \). This setting is referred to as the \emph{Generalized Group Testing Problem}. A testing procedure is called optimal if it minimizes the expected number of tests. It has been conjectured that, when all probabilities \( p_i \) lie within the interval \( \left[1 - \frac{1}{\sqrt{2}},\, \frac{3 - \sqrt{5}}{2} \right] \), the \emph{generalized pairwise testing {algorithm}}, applied to the \( p_i \) arranged in nondecreasing order, constitutes the optimal nested testing strategy among all such order-preserving nested strategies. In this work, we confirm this conjecture and establish the optimality of the procedure within the specified regime. Additionally, we provide a complete structural characterization of the procedure and derive a closed-form expression for its expected number of tests. These results offer new insights into the theory of optimal nested strategies in generalized group testing.

math.ST↗
arXiv2025

Hitting k primes by dice rolls

Let $S=(d_1,d_2,d_3, \ldots )$ be an infinite sequence of rolls of independent fair dice. For an integer $k \geq 1$, let $L_k=L_k(S)$ be the smallest $i$ so that there are $k$ integers $j \leq i$ for which $\sum_{t=1}^j d_t$ is a prime. Therefore, $L_k$ is the random variable whose value is the number of dice rolls required until the accumulated sum equals a prime $k$ times. It is known that the expected value of $L_1$ is close to $2.43$. Here we show that for large $k$, the expected value of $L_k$ is $(1+o(1)) k\log_e k$, where the $o(1)$-term tends to zero as $k$ tends to infinity. We also include some computational results about the distribution of $L_k$ for $k \leq 100$.

math.PR↗
arXiv2024

Optimal single threshold stopping rules and sharp prophet inequalities

This paper considers a finite horizon optimal stopping problem for a sequence of independent and identically distributed random variables, where the objective is to design stopping rules that attempt to select the random variable with the highest value in the sequence. The performance of any stopping rule may be benchmarked relative to the selection of a ``prophet" that has perfect foreknowledge of the largest value. Such comparisons are typically stated in the form of ``prophet inequalities." In this paper we develop a game-theoretic characterization that supports a principled approach for deriving sharp non-asymptotic prophet inequalities for single threshold stopping rules. We demonstrate that sharp constants in the ratio- and difference-type prophet inequalities are determined by the optimal values of infinite two-person zero-sum game on the unit square with particular payoff kernels, while the the solutions to the game provide optimal stopping rules and least favorable distributions. Among other things, this formulation also allows a systematic way to tackle restricted classes of distributions. The proposed framework leads to a numerically efficient algorithmic paradigm that allows computing sharp constants in prophet inequalities with any prescribed level of accuracy.

math.PR↗
arXiv2022

Hitting a prime in 2.43 dice rolls (on average)

What is the number of rolls of fair 6-sided dice until the first time the total sum of all rolls is a prime? We compute the expectation and the variance of this random variable up to an additive error of less than 10^{-4}. This is a solution to a puzzle suggested by DasGupta (2017) in the Bulletin of the Institute of Mathematical Statistics, where the published solution is incomplete. The proof is simple, combining a basic dynamic programming algorithm with a quick Matlab computation and basic facts about the distribution of primes.

math.PR↗
arXiv2022

On Round-Robin Tournaments with a Unique Maximum Score

Richard Arnold Epstein (1927-2016) published the first edition of "The Theory of Gambling and Statistical Logic" in 1967. He introduced some material on round-robin tournaments (complete oriented graphs) with n labeled vertices in Chapter 9; in particular, he stated, without proof, that the probability that there is a unique vertex with the maximum score tends to 1 as n tends to infinity. Our object here is to give a proof of this result along with some historical remarks and comments.

math.CO↗
arXiv2022

On Tournaments and Negative Dependence

Negative dependence of sequences of random variables is often an interesting characteristic of their distribution, as well as a useful tool for studying various asymptotic results, including central limit theorems, Poisson approximations, the rate of increase of the maximum, and more. In the study of probability models of tournaments, negative dependence of participants' outcomes arises naturally with application to various asymptotic results. In particular, the property of negative orthant dependence was proved in several articles for different tournament models, with a special proof for each model. In this note we unify these results by proving a stronger property, negative association, a generalization leading to a very simple proof. We also present a natural example of a knockout tournament where the scores are negatively orthant dependent but not negatively associated. The proof requires a new result on a preservation property of negative orthant dependence that is of independent interest.

math.PR↗
arXiv2022

A note on the distribution of the extreme degrees of a random graph via the Stein-Chen method

We offer an alternative proof, using the Stein-Chen method, of Bollob\'{a}s' theorem concerning the distribution of the extreme degrees of a random graph. Our proof also provides a rate of convergence of the extreme degree to its asymptotic distribution. The same method also applies in a more general setting where the probability of every pair of vertices being connected by edges depends on the number of vertices.

math.CO↗
arXiv2022

On the distribution of winners' scores in a round-robin tournament

In a classical chess round-robin tournament, each of $n$ players wins, draws, or loses a game against each of the other $n-1$ players. A win rewards a player with 1 points, a draw with 1/2 point, and a loss with 0 points. We are interested in the distribution of the scores associated with ranks of $n$ players after ${\displaystyle {n \choose 2}}$ games, i.e. the distribution of the maximal score, second maximum, and so on. The exact distribution for a general $n$ seems impossible to obtain; we obtain a limit distribution.

math.PR↗
arXiv2021

Is Group Testing Ready for Prime-time in Disease Identification?

Large scale disease screening is a complicated process in which high costs must be balanced against pressing public health needs. When the goal is screening for infectious disease, one approach is group testing in which samples are initially tested in pools and individual samples are retested only if the initial pooled test was positive. Intuitively, if the prevalence of infection is small, this could result in a large reduction of the total number of tests required. Despite this, the use of group testing in medical studies has been limited, largely due to skepticism about the impact of pooling on the accuracy of a given assay. While there is a large body of research addressing the issue of testing errors in group testing studies, it is customary to assume that the misclassification parameters are known from an external population and/or that the values do not change with the group size. Both of these assumptions are highly questionable for many medical practitioners considering group testing in their study design. In this article, we explore how the failure of these assumptions might impact the efficacy of a group testing design and, consequently, whether group testing is currently feasible for medical screening. Specifically, we look at how incorrect assumptions about the sensitivity function at the design stage can lead to poor estimation of a procedure's overall sensitivity and expected number of tests. Furthermore, if a validation study is used to estimate the pooled misclassification parameters of a given assay, we show that the sample sizes required are so large as to be prohibitive in all but the largest screening programs

stat.AP↗