How should recommender systems be designed when recommendations shape access to scarce, short-lived opportunities? We study this question in a production setting: Timee, Japan's largest platform for spot work, where workers favorite job templates and receive notifications when firms post shifts from those templates. Maximizing predicted favoriting can generate misdirected concentration: recommendations accumulate on popular templates that create few viable job openings, while templates with unmet labor demand receive too little exposure. We design exposure-control mechanisms for favorite-list management, reallocating template exposure based on posting activity and unfilled capacity. The proposed recommender, thresholded eligibility control (TEC), is fully parallelizable and suitable for large-scale digital platforms. In simulations calibrated to Timee data, TEC raises the per-round job-finding rate from 57.6% to 70.0%. A prefecture-level randomized field experiment increases realized matches and exposure per active template, reduces the share of low-exposure templates, and improves impression-level favoriting and downstream matching.
In this paper, we study favorite sites of one-dimensional asymmetric simple random walks. We show that almost surely, for any fixed integer $r\geq 1$, ``$r$ favorite sites" occurs infinitely often. We also give the asymptotic growth rate of the number of favorite sites.
On the trace of a discrete-time simple random walk on $\mathbb{Z}^d$ for $d\geq 2$, we consider the evolution of favorite sites, i.e., sites that achieve the maximal local time at a certain time. For $d=2$, we show that almost surely three favorite sites occur simultaneously infinitely often and eventually there is no simultaneous occurrence of four favorite sites. For $d\geq 3$, we derive sharp asymptotics of the number of favorite sites. This answers an open question of Erdős and Révész (1987), which was brought up again in Dembo (2005).
This article represents a personal tribute to Richard Askey together with a new look at some of his favorite integrals, including the Cauchy beta integral. The article also provides some new multidimensional extensions of Cauchy's beta integral in which the domain of integration is the space of real symmetric matrices, and these multidimensional integrals are used to obtain some special cases of the Cauchy--Selberg integrals.
For many years, I have been collecting math jokes and posting them on my website. I have more than 400 jokes there. In this paper, which is an extended version of my talk at the G4G15, I would like to present 66 of them.
Random walk is a very important Markov process and has important applications in many fields.For a one-dimensional simple symmetric random walk $(S_n)$, a site $x$ is called a favorite downcrossing site at time $n$ if its downcrossing local time at time $n$ achieves the maximum among all sites. In this paper, we study the cardinality of the favorite downcrossing site set, and will show that with probability 1 there are only finitely many times at which there are at least four favorite downcrossing sites and three favorite downcrossing sites occurs infinitely often. Some related open questions will be introduced.
This work introduces a natural variant of the online machine scheduling problem on unrelated machines, which we refer to as the favorite machine model. In this model, each job has a minimum processing time on a certain set of machines, called favorite machines, and some longer processing times on other machines. This type of costs (processing times) arise quite naturally in many practical problems. In the online version, jobs arrive one by one and must be allocated irrevocably upon each arrival without knowing the future jobs. We consider online algorithms for allocating jobs in order to minimize the makespan. We obtain tight bounds on the competitive ratio of the greedy algorithm and characterize the optimal competitive ratio for the favorite machine model. Our bounds generalize the previous results of the greedy algorithm and the optimal algorithm for the unrelated machines and the identical machines. We also study a further restriction of the model, called the symmetric favorite machine model, where the machines are partitioned equally into two groups and each job has one of the groups as favorite machines. We obtain a 2.675-competitive algorithm for this case, and the best poss
Consider a simple symmetric random walk on the integer lattice $\mathbb{Z}$. Let $E(n)$ denote a favorite edge of the random walk at time $n$. In this paper, we study the escape rate of $E(n)$, and show that almost surely $\liminf_{n\to\infty}\frac{|E(n)|}{\sqrt{n}\cdot(\log n)^{-γ}}$ equals 0 if $γ\le 1$, and is infinity otherwise. We also obtain a law of the iterated logarithm for $E(n)$.
Erdős and Révész initiated the study of favorite sites by considering the one-dimensional simple random walk. We investigate in this paper the same problem for a class of null-recurrent randomly biased walks on a supercritical Gaton-Watson tree. We prove that there is some parameter $κ\in (1, \infty]$ such that the set of the favorite sites of the biased walk is almost surely bounded in the case $κ\in (2, \infty]$, tight in the case $κ=2$, and oscillates between a neighborhood of the root and the boundary of the range in the case $κ\in (1, 2)$. Moreover, our results yield a complete answer to the cardinality of the set of favorite sites in the case $κ\in (2, \infty]$. The proof relies on the exploration of the Markov property of the local times process with respect to the space variable and on a precise tail estimate on the maximum of local times, using a change of measure for multi-type Galton-Watson trees.
Recursive noun phrases (NPs) have interesting semantic properties. For example, "my favorite new movie" is not necessarily my favorite movie, whereas "my new favorite movie" is. This is common sense to humans, yet it is unknown whether language models have such knowledge. We introduce the Recursive Noun Phrase Challenge (RNPC), a dataset of three textual inference tasks involving textual entailment and event plausibility comparison, precisely targeting the understanding of recursive NPs. When evaluated on RNPC, state-of-the-art Transformer models only perform around chance. Still, we show that such knowledge is learnable with appropriate data. We further probe the models for relevant linguistic features that can be learned from our tasks, including modifier semantic category and modifier scope. Finally, models trained on RNPC achieve strong zero-shot performance on an extrinsic Harm Detection evaluation task, showing the usefulness of the understanding of recursive NPs in downstream applications.
This paper is concerned with asymptotic behavior (at zero and at infinity) of the favorite points of Lévy processes. By exploring Molchan's idea for deriving lower tail probabilities of Gaussian processes with stationary increments, we extend the result of Marcus (2001) on the favorite points to a larger class of symmetric Lévy processes.
We study single-candidate voting embedded in a metric space, where both voters and candidates are points in the space, and the distances between voters and candidates specify the voters' preferences over candidates. In the voting, each voter is asked to submit her favorite candidate. Given the collection of favorite candidates, a mechanism for eliminating the least popular candidate finds a committee containing all candidates but the one to be eliminated. Each committee is associated with a social value that is the sum of the costs (utilities) it imposes (provides) to the voters. We design mechanisms for finding a committee to optimize the social value. We measure the quality of a mechanism by its distortion, defined as the worst-case ratio between the social value of the committee found by the mechanism and the optimal one. We establish new upper and lower bounds on the distortion of mechanisms in this single-candidate voting, for both general metrics and well-motivated special cases.
For a one-dimensional simple symmetric random walk $(S_n)$, an edge $x$ (between points $x-1$ and $x$) is called a favorite edge at time $n$ if its local time at $n$ achieves the maximum among all edges. In this paper, we show that with probability 1 three favorite edges occurs infinitely often. Our work is inspired by Tóth and Werner [Combin. Probab. Comput. {\bf 6} (1997) 359-369], and Ding and Shen [Ann. Probab. {\bf 46} (2018) 2545-2561], disproves a conjecture mentioned in Remark 1 on page 368 of Tóth and Werner [Combin. Probab. Comput. {\bf 6} (1997) 359-369].
For $s \in (1/4,1)$ and any degree the only $W^{s,\frac{1}{s}}$-minimizers for $\mathbb{S}^1 \to \mathbb{S}^1$ maps are Blaschke products. This gives a resolution of Open Problems 23 and 24 in Brezis-Mironescu's mappings to the circle book, as well as Brezis' Favorite Open Problem 5.4 in this $s$-range. Previous results of this type were partial and restricted only to a small neighborhood of $s=\frac{1}{2}$. In particular, Brezis' Favorite Open Problem 5.1 is completely settled. Moreover, as a consequence of the argument, one also obtains linearized stability results.
In 2023, H.\,Brezis published a list of his ``favorite open problems", which he described as challenges he had ``raised throughout his career and has resisted so far". We provide a complete resolution to the first one--Open Problem 1.1--in Brezis's favorite open problems list: the existence of solutions to the long-standing Brezis-Nirenberg problem on a three-dimensional ball. Furthermore, using the building blocks of Del Pino-Musso-Pacard-Pistoia sign-changing solutions to the Yamabe problem, we establish the existence of infinitely many sign-changing, nonradial solutions for the full range of the parameter.
In-group favoritism refers to the phenomena of favoring members of one's in-group over out-group members and is widely observed in numerous social cooperative behaviors. Recently, in-group favoritism biases have also been identified in generative language models. However, whether the in-group favoritism exists when persona agents are faced with contradicting information (e.g., misinformation), and how to mitigate the adverse effects of in-group favoritism biases in persona agents have been understudied. To address these problems, we propose a Truth or Tribe simulation framework to study the agent cooperation within the spread of contradicting information through a triadic interaction paradigm, and conduct controlled trials to evaluate the primary moderating factors. Extensive results showcase that persona agents display strong in-group favoritism, accepting incorrect answers from identity-similar peers at much higher rates than from dissimilar peers. In-group favoritism continues to emerge in defeasible reasoning contexts where no absolute truth exists, and it intensifies as cognitive complexity increases. Furthermore, three intervention strategies--Identity-Blind Instruction, Stru
We introduce OTTER, a unified open-set multi-label tagging framework that harmonizes the stability of a curated, predefined category set with the adaptability of user-driven open tags. OTTER is built upon a large-scale, hierarchically organized multi-modal dataset, collected from diverse online repositories and annotated through a hybrid pipeline combining automated vision-language labeling with human refinement. By leveraging a multi-head attention architecture, OTTER jointly aligns visual and textual representations with both fixed and open-set label embeddings, enabling dynamic and semantically consistent tagging. OTTER consistently outperforms competitive baselines on two benchmark datasets: it achieves an overall F1 score of 0.81 on Otter and 0.75 on Favorite, surpassing the next-best results by margins of 0.10 and 0.02, respectively. OTTER attains near-perfect performance on open-set labels, with F1 of 0.99 on Otter and 0.97 on Favorite, while maintaining competitive accuracy on predefined labels. These results demonstrate OTTER's effectiveness in bridging closed-set consistency with open-vocabulary flexibility for multi-modal tagging applications.
New fairness notions aligned with the merit principle are proposed for designing exchange rules. We show that for an obviously strategy-proof, efficient and individually rational rule, (i) an agent receives her favorite object when others unanimously perceive her object the best, if and only if preferences are single-peaked, and (ii) an upper bound on fairness attainable is that, if two agents' objects are considered the best by all agents partitioned evenly into two groups, it is guaranteed that one, not both, gets her favorite object. This on the one hand reveals the importance of single-peaked preferences in a private economy and provides a support of "Gul's Conjecture", and on the other hand also indicates an unambiguous trade-off between incentives and fairness in design of exchange rules.
Despite a large body of theoretical literature on voting mechanisms, there is no documented evidence from real-world panel evaluations about the effect of trimming the extreme votes on sincere voting. We provide the first such evidence by comparing subjective evaluations of experts from different countries in competitive settings with and without a trimming mechanism. In these evaluations, some of the evaluated subjects are experts' compatriots. Using data on 29,383 subjective evaluations, we find that experts assign significantly higher scores to their compatriots in panels without trimming. However, in panels with trimming, this favoritism is generally insignificant.
This paper aims to foster social interaction between parents and young adult children living apart via music. Our approach transforms their music-listening moment into an opportunity to listen to the other's favorite songs and enrich interaction in their daily lives. To this end, we explore the current practice and needs of parent-child communication and the experience and perception of music-mediated interaction. Based on the findings, we developed DJ-Fam, a mobile application that enables parents and children to listen to their favorite songs and use them as conversation starters to foster parent-child interaction. From our deployment study with seven families over four weeks in South Korea, we show the potential of DJ-Fam to influence parent-child interaction and their mutual understanding and relationship positively. Specifically, DJ-Fam considerably increases the frequency of communication and diversifies the communication channels and topics, all of which are satisfactory to the participants.