arXiv subjects

Kota Saito

Publications and source records attributed to Kota Saito.

At least 19 recordsLinked to original sources

arXiv2026

Approximating Choice Data by Discrete Choice Models

We obtain a necessary and sufficient condition under which random-coefficient discrete choice models, such as mixed-logit models, are rich enough to approximate any nonparametric random utility models arbitrarily well across choice sets. The condition turns out to be the affine-independence of the set of characteristic vectors. When the condition fails, resulting in some random utility models that cannot be closely approximated, we identify preferences and substitution patterns that are challenging to approximate accurately. We also propose algorithms to quantify the magnitude of approximation errors.

econ.TH↗

arXiv2026

Random Utility with Aggregation

We study random utility (RU) rationality with aggregation when the underlying alternatives in each aggregate vary across consumers and are unobserved, as is typical for an outside option. RUM over the underlying alternatives is the natural assumption on the data generating process, while an aggregated random utility model (ARUM) is the standard empirical tool. We characterize RU rationality in three frameworks and show its testable implications are substantially weaker than those of an ARUM. We provide two independent conditions for their equivalence: non-overlapping preferences within aggregates and menu-independent aggregation. Simulations show that violating either condition produces meaningful estimation bias when imposing an ARUM.

econ.TH↗

arXiv2026

I Choose For You: an Experimental Study

We investigate whether risk and time preferences differ when individuals make decisions for others compared to making decisions for themselves. We introduce a novel ``skin in the game'' experimental design, where choices for others incur a direct cost to the decision-maker, ensuring a genuine trade-off between self-interest and surrogate allocation. The modal outcome is that participants are more risk-averse and impatient when choosing for others than for themselves. Our methodology reveals significant heterogeneity, successfully identifying selfish types often missed by the more standard ``no skin in the game'' approaches. The message is nuanced, as even non-selfish participants behave differently when they have skin in the game. Furthermore, our framework yields more consistent behavior and superior out-of-sample predictive power.

econ.GN↗

arXiv2025

Transcendency of variants of Mills' constant

Let $\lfloor x\rfloor$ denote the integer part of $x$. For every sequence $(C_k)_{k\ge 1}$ of positive integers, we define $ξ(C_k)$ as the smallest real number $ξ>1$ such that $\lfloor ξ^{C_k} \rfloor$ is a prime number for every positive integer $k$. The number $ξ(3^k)$ is called Mills' constant. Recently, the author showed that $ξ(3^k)$ is irrational; however, the transcendency remains open. In this paper, we show that Mills' constant is transcendental under the Density Hypothesis of the Riemann zeta function. Furthermore, we obtain four classes of sequences $(C_k)_{k\ge 1}$ for which we can verify the arithmetic properties of $ξ(C_k)$. For simplicity, we give four representative examples belonging to each class: (A) $ξ(\lfloor b^k\rfloor)$ is irrational for every real number $b\ge 1+\sqrt{2}$; (B) $ξ((1+\sqrt{2})^k+(1-\sqrt{2})^k)$ is transcendental; (C) $ξ(r3^k-1)$ is transcendental for every integer $r\ge 4.003\times 10^{14}$; (D) $ξ(3^{k-\lfloor (\log k)^{1/2} \rfloor}2^{\lfloor (\log k)^{1/2}\rfloor})$ is transcendental.

math.NT↗

arXiv2025

Random Utility with Unobservable Alternatives

The random utility model, a cornerstone in economics, is axiomatized by Falmagne (1978) and McFadden and Richter (1990) with the assumption that if a menu is observable, the choice frequencies of all alternatives are also observable. However, in practice, it is common for choice frequencies of some alternatives to remain unobserved. To address this discrepancy, we obtain the testable implications of the random utility model when the choice frequencies of some alternatives are unobservable, which consist of nonredundant inequality constraints on observed choice frequencies. Our findings indicate that the widespread empirical practice of aggregating unobserved alternatives into a single "outside option" fails to capture significant implications of random utility models.

econ.TH↗

arXiv2025

Finiteness of solutions to linear Diophantine equations on Piatetski-Shapiro sequences

A sequence of integers of the form $\lfloor n^α\rfloor$ $(n=1,2,\ldots)$ for some fixed non-integral $α>1$ is called a Piatetski-Shapiro sequence, where $\lfloor x\rfloor$ denotes the integer part of $x$. Let $\mathrm{PS}(α)$ denote the set of all those terms. In this article, we show that $x+y=z$ has only finitely many solutions $(x,y,z)\in \mathrm{PS}(α)^3$ for almost every $α>3$. Furthermore, we show that $\mathrm{PS}(α)$ has only finitely many arithmetic progressions of length $3$ for almost every $α>10$. In addition, we estimate upper bounds for the Hausdorff dimension of the set of $α\in [s,t]$ such that $y=a_1x_1+\cdots +a_nx_n$ has infinitely many solutions on $\mathrm{PS}(α)$.

math.NT↗

arXiv2025

Mills' constant is irrational

Let $ \lfloor x \rfloor $ denote the integer part of $ x $. In 1947, Mills constructed a real number $ ξ> 1 $ such that $\lfloor ξ^{3^k} \rfloor$ is always a prime number for every positive integer $k$. We define Mills' constant as the smallest real number $ξ$ satisfying this property. Determining whether this number is irrational has been a long-standing problem. In this paper, we show that Mills' constant is irrational. Furthermore, we obtain partial results on the transcendency of this number.

math.NT↗

arXiv2025

Intervals without primes near an iterated linear recurrence sequence

Let $M$ be a fixed positive integer. Let $(R_{j}(n))_{n\ge 1}$ be a linear recurrence sequence for every $j=0,1,\ldots, M$, and we set $f(n)=(R_0\circ \cdots \circ R_M)(n)$, where $(S\circ T)(n)= S(T(n))$. In this paper, we obtain sufficient conditions on $(R_{0}(n))_{n\ge 1},\ldots, (R_{M}(n))_{n\ge 1}$ so that the intervals $(|f(n)|-c\log n, |f(n)|+c\log n)$ do not contain any prime numbers for infinitely many integers $n\ge 1$, where $c$ is an explicit positive constant depending only on the orders of $R_0,\ldots, R_M$. As a corollary, we show that if for each $j=1,2,\ldots, M$, the sequence $(R_j(n))_{n\ge 1}$ is positive, strictly increasing, and the constant term of its characteristic polynomial is $\pm 1$, then for every Pisot or Salem number $α$, the numbers $\lfloor α^{(R_1\circ \cdots \circ R_M)(n)} \rfloor $ are composite for infinitely many integers $n\ge 1$.

math.NT↗

arXiv2024

Normality of algebraic numbers and the Riemann zeta function

A real number is called simply normal to base $b$ if every digit $0,1,\ldots ,b-1$ should appear in its $b$-adic expansion with the same frequency $1/b$. A real number is called normal to base $b$ if it is simply normal to every base $b, b^2, \ldots$. In this article, we discover a relation between the normality of algebraic numbers and a mean of the Riemann zeta function on vertical arithmetic progressions. Consequently, we reveal that a positive algebraic irrational number $α$ is normal to base $b$ if and only if we have \[ \lim_{N\to \infty}\frac{1}{\log N} \sum_{1\leq |n|\leq N} ζ\left(-k+\frac{2πi n}{\log b} \right) \frac{e^{2πi n \log α/\log b}}{n^{k+1}} =0 \] for every integer $k\geq 0$.

math.NT↗

arXiv2024

Did Harold Zuercher Have Time-Separable Preferences?

This paper proposes an empirical model of dynamic discrete choice to allow for non-separable time preferences, generalizing the well-known Rust (1987) model. Under weak conditions, we show the existence of value functions and hence well-defined optimal choices. We construct a contraction mapping of the value function and propose an estimation method similar to Rust's nested fixed point algorithm. Finally, we apply the framework to the bus engine replacement data. We improve the fit of the data with our general model and reject the null hypothesis that Harold Zuercher has separable time preferences. Misspecifying an agent's preference as time-separable when it is not leads to biased inferences about structure parameters (such as the agent's risk attitudes) and misleading policy recommendations.

econ.EM↗

arXiv2023

The simple normality of the fractional powers of two and the Riemann zeta function

A real number is called simply normal to base $b$ if its base-$b$ expansion has each digit appearing with average frequency tending to $1/b$. In this article, we discover a relation between the frequency that the digit $1$ appears in the binary expansion of $2^{p/q}$ and a mean value of the Riemann zeta function on arithmetic progressions. As a consequence, we show that \[ \lim_{l\to \infty} \frac{1}{l}\sum_{0<|n|\leq 2^l } ζ\left(\frac{2 nπi}{\log 2}\right) \frac{e^{2nπi p/q} }{n} =0 \] if and only if $2^{p/q}$ is simply normal to base $2$.

math.NT↗

arXiv2023

Some remarks on the $[x/n]$-sequence

After the work of Bordellès, Dai, Heyman, Pan and Shparlinki (2018) and Heyman (2019), several authors studied the averages of arithmetic functions over the sequence $[x/n]$ and the integers of the form $[x/n]$. In this paper, we give three remarks on this topic. Firstly, we improve the result of Wu and Yu (2022) on the distribution of the integers of the form $[x/n]$ in arithmetic progressions by using a variant of Dirichlet's hyperbola method. Secondly, we prove an asymptotic formula for the number of primitive lattice points with coordinates of the form $[x/n]$, for which we introduce a certain averaging trick. Thirdly, we study a certain "multiplicative" analog of the Titchmarsh divisor problem. We derive asymptotic formulas for such "multiplicative" Titchmarsh divisor problems for "small" arithmetic functions and the Euler totient function with the von Mangoldt function. However, it turns out that the average of the Euler totient function over the $[x/p]$-sequence seems rather difficult and we propose a hypothetical asymptotic formula for this average.

math.NT↗

arXiv2023

A system of certain linear Diophantine equations on analogs of squares

This study investigates the existence of tuples $(k, \ell, m)$ of integers such that all of $k$, $\ell$, $m$, $k+\ell$, $\ell+m$, $m+k$, $k+\ell+m$ belong to $S(α)$, where $S(α)$ is the set of all integers of the form $\lfloor αn^2 \rfloor$ for $n\geq α^{-1/2}$ and $\lfloor x\rfloor$ denotes the integer part of $x$. We show that $T(α)$, the set of all such tuples, is infinite for all $α\in (0,1)\cap \mathbb{Q}$ and for almost all $α\in (0,1)$ in the sense of the Lebesgue measure. Furthermore, we show that if there exists $α>0$ such that $T(α)$ is finite, then there is no perfect Euler brick. We also examine the set of all integers of the form $\lceil αn^2 \rceil$ for $n\in \mathbb{N}$.

math.NT↗

arXiv2022

Adjacencies on random ordering polytopes and flow polytopes

The Multiple Choice Polytope (MCP) is the prediction range of a random utility model due to Block and Marschak (1960). Fishburn (1998) offers a nice survey of the findings on random utility models at the time. A complete characterization of the MCP is a remarkable achievement of Falmagne (1978). Apart for a recognition of the facets by Suck (2002), the geometric structure of the MCP was apparently not much investigated. Recently, Chang, Narita and Saito (2022) refer to the adjacency of vertices while Turansick (2022) uses a condition which we show to be equivalent to the non-adjacency of two vertices. We characterize the adjacency of vertices and the adjacency of facets. To derive a more enlightening proof of Falmagne Theorem and of Suck result, Fiorini (2004) assimilates the MCP with the flow polytope of some acyclic network. Our results on adjacencies also hold for the flow polytope of any acyclic network. In particular, they apply not only to the MCP, but also to three polytopes which Davis-Stober, Doignon, Fiorini, Glineur and Regenwetter (2018) introduced as extended formulations of the weak order polytope, interval order polytope and semiorder polytope (the prediction ranges of other models, see for instance Fishburn and Falmagne, 1989, and Marley and Regenwetter, 2017).

math.CO↗

arXiv2022

Topological properties and algebraic independence of sets of prime-representing constants

Let $(c_k)_{k\in \mathbb{N}}$ be a sequence of positive integers. We investigate the set of $A>1$ such that the integer part of $A^{c_1\cdots c_k}$ is always a prime number for every positive integer $k$. Let $\mathcal{W}(c_k)$ be this set. The first goal of this article is to determine the topological structure of $\mathcal{W}(c_k)$. Under some conditions on $(c_k)_{k\in \mathbb{N}}$, we reveal that $\mathcal{W}(c_k)\cap [0,a]$ is homeomorphic to the Cantor middle third set for some $a$. The second goal is to propose an algebraically independent subset of $\mathcal{W}(c_k)$ if $c_k$ is rapidly increasing. As a corollary, we disclose that the minimum of $\mathcal{W}(k)$ is transcendental. In addition, we apply the main result to the set of $A>1$ such that the integer part of $A^{3^{k!}}$ is always a prime number. As a consequence, we give a certain infinite subset of this set which is algebraically independent. Furthermore, we also get results on the rational approximation, $\mathbb{Q}$-linear independence, and numerical calculations of elements in $\mathcal{W}(c_k)$.

math.NT↗

arXiv2021

Decision Making under Uncertainty: An Experimental Study in Market Settings

We implement nonparametric revealed-preference tests of subjective expected utility theory and its generalizations. We find that a majority of subjects' choices are consistent with the maximization of some utility function. They respond to price changes in the direction subjective expected utility theory predicts, but not to a degree that makes them consistent with the theory. Maxmin expected utility a dds no explanatory power. The degree of deviations from the theory is uncorrelated with demographic characteristics. Our findings are essentially the same in laboratory data with a student population and in a panel survey with a general sample of the U.S. population.

econ.GN↗

arXiv2021

Measurement of ion displacement via RF power variation for excess micromotion compensation

We demonstrate a method of micromotion minimization of a trapped ion in a linear Paul trap based on the precision measurement of the ion trapping position displacement due to a stray electric field in the radial plane by ion fluorescence imaging. The amount of displacement in the radial plane is proportional to the strength of a stray electric field. Therefore, we evaluated the micromotion compensation condition by measuring the ion displacements from the ion equilibrium position using two different radial trap frequencies with various combinations of the compensation voltage. The residual electric field uncertainty of this technique reached a few volts per meter. This compensation technique does not depend on the orientation of the incident cooling laser or the detuning and imaging direction. Therefore, this method is suitable for a planar ion trap, a stylus ion trap, which limits the propagation angle of lasers, or miniaturized ion trap systems for sensing and metrological applications.

physics.atom-ph↗

arXiv2021

Approximate Expected Utility Rationalization

We propose a new measure of deviations from expected utility theory. For any positive number~$e$, we give a characterization of the datasets with a rationalization that is within~$e$ (in beliefs, utility, or perceived prices) of expected utility theory. The number~$e$ can then be used as a measure of how far the data is to expected utility theory. We apply our methodology to data from three large-scale experiments. Many subjects in those experiments are consistent with utility maximization, but not with expected utility maximization. Our measure of distance to expected utility is correlated with subjects' demographic characteristics.

econ.GN↗