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Ilia Krasikov

Publications and source records attributed to Ilia Krasikov.

At least 19 recordsLinked to original sources
arXiv2026

Reduced Forms: Feasibility, Extremality, Optimality

We study independent private values auction environments in which the auctioneer's revenue depends nonlinearly on bidders' interim winning probabilities. Our framework accommodates heterogeneity among bidders and places no ad hoc constraints on the mechanisms available to the auctioneer. Within this general setting, we show that feasibility of interim winning probabilities can be tested along a unidimensional curve -- the principal curve -- and use this insight to explicitly characterize the extreme points of the feasible set. We then combine our results on feasibility and extremality to solve for the optimal auction under a natural regularity condition. We show that the optimal mechanism allocates the good based on principal virtual values, which extend Myerson's virtual values to nonlinear settings and are constructed to equalize bidders' marginal revenue along the principal curve. We apply our approach to the classical linear model, settings with endogenous valuations due to ex ante investments, and settings with non-expected utility preferences, where previous results were largely limited either to symmetric environments with symmetric allocations or to two-bidder environments.

econ.TH↗
arXiv2024

Scoring and Favoritism in Optimal Procurement Design

We study buyer-optimal procurement mechanisms when quality is contractible. When some costs are borne by every participant of a procurement auction regardless of winning, the classic analysis should be amended. We show that an optimal symmetric mechanism is a scoring auction with a score function that may be either flatter or steeper than classically. This depends on the relative degrees of information asymmetry over the all-pay and winner-pay costs. However, the symmetry of the optimal mechanism is not granted due to the presence of all-pay costs. When ex-post efficiency is less important than the duplication of costs, favoritism becomes optimal. We show that, depending on the degree of convexity of costs, the solution takes one of two novel formats with a partially asymmetric treatment of firms, which we call a score floor and a score ceiling auction. Interestingly, these auctions feature side payments from or to the buyer, which has nothing to do with corruption.

econ.TH↗
arXiv2024

On the class reconstruction number of trees

Harary and Lauri conjectured that the class reconstruction number of trees is 2, that is, each tree has two unlabelled vertex-deleted subtrees that are not both in the deck of any other tree. We show that each tree $T$ can be reconstructed up to isomorphism given two of its unlabelled subgraphs $T-u$ and $T-v$ under the assumption that $u$ and $v$ are chosen in a particular way. Our result does not completely resolve the conjecture of Harary and Lauri since the special property defining $u$ and $v$ cannot be recognised from the given subtrees $T-u$ and $T-v$.

math.CO↗
arXiv2023

On Turán inequality for ultraspherical polynomials

We show that the normalised ultraspherical polynomials, $G_n^{(λ)}(x)=C_n^{(λ)}(x)/C_n^{(λ)}(1)$, satisfy the following stronger version of Turán inequality, $$|x|^θ\left(G_n^{(λ)}(x)\right)^2 -G_{n-1}^{(λ)}(x)G_{n+1}^{(λ)}(x) \ge 0 ,\;\;\;|x| \le 1, $$ where $θ=4/(2-λ)$ if $-1/2 <λ\le 0$, and $θ=2/(1+2λ)$ if $λ\ge 0$. We also provide a similar generalisation of Turán inequalities for some symmetric orthogonal polynomials with a finite or infinite support defined by a three term recurrence.

math.CA↗
arXiv2022

Behavioral epidemiology: An economic model to evaluate optimal policy in the midst of a pandemic

This paper combines a canonical epidemiology model of disease dynamics with government policy of lockdown and testing, and agents' decision to social distance in order to avoid getting infected. The model is calibrated with data on deaths and testing outcomes in the Unites States. It is shown that an intermediate but prolonged lockdown is socially optimal when both mortality and GDP are taken into account. This is because the government wants the economy to keep producing some output and the slack in reducing infection is picked up by social distancing agents. Social distancing best responds to the optimal government policy to keep the effective reproductive number at one and avoid multiple waves through the pandemic. Calibration shows testing to have been effective, but it could have been even more instrumental if it had been aggressively pursued from the beginning of the pandemic. Not having any lockdown or shutting down social distancing would have had extreme consequences. Greater centralized control on social activities would have mitigated further the spread of the pandemic.

econ.GN↗
arXiv2020

Absorption Paths and Equilibria in Quitting Games

We study quitting games and define the concept of absorption paths, which is an alternative definition to strategy profiles that accomodates both discrete time aspects and continuous time aspects, and is parameterized by the total probability of absorption in past play rather than by time. We then define the concept of sequentially 0perfect absorption paths, which are shown to be limits of $ε$-equilibrium strategy profiles as $ε$ goes to 0. We finally identify a class of quitting games that possess sequentially 0-perfect absorption paths.

math.OC↗
arXiv2016

On approximation of ultraspherical polynomials in the oscillatory region

For $k \ge 2$ even, and $ α\ge -(2k+1)/4 $, we provide a uniform approximation of the ultraspherical polynomials $ P_k^{(α,\, α)}(x) $ in the oscillatory region with a very explicit error term. In fact, our result covers all $α$ for which the expression "oscillatory region" makes sense. We show that there the function $g(x)={c \sqrt{b(x)} \, (1-x^2)^{(α+1)/2} P_k^{(α, α)}(x)=\cos \mathcal{B}(x)+ r(x)}$, where $c=c(k, α)$ is defined by the normalization, $\mathcal{B}(x)=\int_{0}^ x b(x) dx$, and the functions $c,\, b(x), \, \mathcal{B}(x)$, as well as bounds on the error term $r(x)$ are given by some rather simple elementary functions.

math.CA↗
arXiv2011

Some asymptotics for the Bessel functions with an explicit error term

We show how one can obtain an asymptotic expression for some special functions satisfying a second order differential equation with a very explicit error term starting from appropriate upper bounds. We will work out the details for the Bessel function $J_ν(x)$ and the Airy function $Ai(x)$ and find a sharp approximation for their zeros. We also answer the question raised by Olenko by showing that $$c_1 | ν^2-1/4\,| < \sup_{x \ge 0} x^{3/2}|J_ν(x)-\sqrt{\frac{2}{πx}} \, \cos (x-\frac{πν}{2}-\fracπ{4}\,)| <c_2 |ν^2-1/4\,|, $$ $ ν\ge -1/2 \, ,$ for some explicit numerical constants $c_1$ and $c_2.$

math.CA↗
arXiv2011

Turán Inequalities for Three Term Recurrences with Monotonic Coefficients

We establish some new Turán's type inequalities for orthogonal polynomials defined by a three-term recurrence with monotonic coefficients. As a corollary we deduce asymptotic bounds on the extreme zeros of orthogonal polynomials with polynomially growing coefficients of the three-term recurrence.

math.CA↗
arXiv2006

On Erdélyi-Magnus-Nevai conjecture for Jacobi polynomials

T. Erdélyi, A.P. Magnus and P. Nevai conjectured that for $α, β\ge - {1/2} ,$ the orthonormal Jacobi polynomials ${\bf P}_k^{(α, β)} (x)$ satisfy the inequality \begin{equation*} \max_{x \in [-1,1]}(1-x)^{α+{1/2}}(1+x)^{β+{1/2}}({\bf P}_k^{(α, β)} (x) )^2 =O (\max \left\{1,(α^2+β^2)^{1/4} \right\}), \end{equation*} [Erdélyi et al.,Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994), 602-614]. Here we will confirm this conjecture in the ultraspherical case $α= β\ge \frac{1+ \sqrt{2}}{4},$ even in a stronger form by giving very explicit upper bounds. We also show that \begin{equation*} \sqrt{δ^2-x^2} (1-x^2)^α({\bf P}_{2k}^{(α, α)} (x))^2 < \frac{2}π (1+ \frac{1}{8(2k+ α)^2} ) \end{equation*} for a certain choice of $δ,$ such that the interval $(- δ, δ)$ contains all the zeros of ${\bf P}_{2k}^{(α, α)} (x).$ Slightly weaker bounds are given for polynomials of odd degree.

math.CA↗
arXiv2006

An upper bound on Jacobi polynomials

Let ${\bf P}_k^{(α, β)} (x)$ be an orthonormal Jacobi polynomial of degree $k.$ We will establish the following inequality \begin{equation*} \max_{x \in [δ_{-1},δ_1]}\sqrt{(x- δ_{-1})(δ_1-x)} (1-x)^α(1+x)^β ({\bf P}_{k}^{(α, β)} (x))^2 < \frac{3 \sqrt{5}}{5}, \end{equation*} where $δ_{-1}<δ_1$ are appropriate approximations to the extreme zeros of ${\bf P}_k^{(α, β)} (x) .$ As a corollary we confirm, even in a stronger form, T. Erdélyi, A.P. Magnus and P. Nevai conjecture [Erdélyi et al., Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials, SIAM J. Math. Anal. 25 (1994), 602-614], by proving that \begin{equation*} \max_{x \in [-1,1]}(1-x)^{α+{1/2}}(1+x)^{β+{1/2}}({\bf P}_k^{(α, β)} (x))^2 < 3 α^{1/3} (1+ \fracα{k})^{1/6}, \end{equation*} in the region $k \ge 6, α, β\ge \frac{1+ \sqrt{2}}{4}.$

math.CA↗
arXiv2004

New bounds on the Hermite polynomials

We shall establish two-side explicit inequalities, which are asymptotically sharp up to a constant factor, on the maximum value of $|H_k(x)| e^{-x^2/2},$ on the real axis, where $H_k$ are the Hermite polynomials.

math.CA↗
arXiv2004

Turan inequalities and zeros of orthogonal polynomials

We use Turan type inequalities to give new non-asymptotic bounds on the extreme zeros of orthogonal polynomials in terms of the coefficients of their three term recurrence. Most of our results deal with symmetric polynomials satisfying the three term recurrence $p_{k+1}=x p_k-c_k p_{k-1},$ with a nondecreasing sequence $\{c_k\}$. As a special case they include a non-asymptotic version of Mate, Nevai and Totik result on the largest zeros of orthogonal polynomials with $c_k=k^δ (1+ o(k^{-2/3})).$

math.CA↗
arXiv2003

On extreme zeros of classical orthogonal polynomials

Let $x_1$ and $x_k$ be the least and the largest zeros of the Laguerre or Jacobi polynomial of degree $k.$ We shall establish sharp inequalities of the form $x_1 B,$ which are uniform in all the parameters involved. Together with inequalities in the opposite direction, recently obtained by the author, this locates the extreme zeros of classical orthogonal polynomials with the relative precision, roughly speaking, $O(k^{-2/3}).$

math.CA↗
arXiv2002

Bounds for the 3x+1 Problem using Difference Inequalities

We study difference inequality systems for the 3x+1 problem introduced by the first author in 1989. These systemes can be used to give lower bounds for the number of integers below x that contain 1 in their forward orbit under the 3x+1 map. Previous methods gave away some information in these inequalities. We give an improvement which apparantly extracts full information from the inequalities. By computer aided proof we show that at least x^{0.84} of the integers below x contain 1 in their forward orbit under the 3x+1 map.

math.NT↗
arXiv2002

Multiplicity of zeros and discrete orthogonal polynomials

We consider a problem of bounding the maximal possible multiplicity of a zero at of some expansions $\sum a_i F_i(x)$, at a certain point $c,$ depending on the chosen family $\{F_i \}$. The most important example is a polynomial with $c=1.$ It is shown that this question naturally leads to discrete orthogonal polynomials. Using this connection we derive some new bounds, in particular on the multiplicity of the zero at one of a polynomial with a prescribed norm.

math.CA↗