We study a learning-augmented variant of the secretary problem, recently introduced by Fujii and Yoshida (2023). In this variant, the decision-maker has access to machine-learned predictions of candidate values in advance. The key challenge is to balance consistency and robustness: when the predictions are accurate, the algorithm should hire a near-best secretary; however, if they are inaccurate, the algorithm should still achieve a bounded competitive ratio. We consider both the standard Random Order Secretary Problem (ROSP), where candidates arrive in a uniform random order, and a more natural model in the learning-augmented setting, where the decision-maker can choose the arrival order based on the predicted candidate values. This model, which we call the Chosen Order Secretary Problem (COSP), can capture scenarios such as an interview schedule that is set by the decision-maker. We propose a novel algorithm that applies to both ROSP and COSP. Building on the approach of Fujii and Yoshida, our method switches from fully trusting predictions to a threshold-based rule when a large deviation of a prediction is observed. Importantly, unlike the algorithm of Fujii and Yoshida, our algorithm uses randomization as part of its decision logic. We show that if ε ∈ [0,1] denotes the maximum multiplicative prediction error, then for ROSP our algorithm achieves competitive ratio max {0.221, (1-ε)/(1+ε)}, improving on a previous bound of max {0.215, (1-ε)/(1+ε)} due to Fujii and Yoshida [Fujii and Yoshida, 2023]. For COSP, our algorithm achieves max {0.262, (1-ε)/(1+ε)}. This surpasses a 0.25 upper bound on the worst-case competitive ratio that applies to the approach of Fujii and Yoshida, and gets closer to the classical secretary benchmark of 1/e ≈ 0.368, which is an upper bound for any algorithm. Our result for COSP highlights the benefit of integrating predictions with arrival-order control in online decision-making.

We quantify the value of the monopoly's bargaining power in terms of competition complexity—that is, the number of additional bidders the monopoly must attract in simple auctions to match the expected revenue of the optimal mechanisms —within the setting of multi-item auctions. We show that for simple auctions that sell items separately, the competition complexity is Θ(n/α) in an environment with n original bidders under the slightly stronger assumption of α-strong regularity, in contrast to the standard regularity assumption in the literature, which requires Ω (n · ln m/n) additional bidders. This significantly reduces the value of learning the distribution to design the optimal mechanisms, especially in large markets with many items for sale. For simple auctions that sell items as a grand bundle, we establish a constant competition complexity bound in a single-bidder environment when the number of items is small or when the value distribution has a monotone hazard rate. Some of our competition complexity results also hold when we compete against the first best benchmark (i.e., optimal social welfare).

Weitzman (1979) introduced Pandora’s box problem as a mathematical model of sequential search with inspection costs, in which a searcher is allowed to select a prize from one of n alternatives. Several decades later, Doval (2018) introduced a close version of the problem, where the searcher does not need to incur the inspection cost of an alternative, and can select it uninspected. Unlike the original problem, the optimal solution to the nonobligatory inspection variant is proved to need adaptivity by Doval (2018), and by recent work of Fu Li and Liu (2022), finding the optimal solution is NP-hard.

We study the outcome in a matching market where both sides have limited ability to consider options. For example, in the national residency matching program, doctors are limited to apply to a small set of hospitals, and hospitals are limited by the time required to interview candidates. Our main findings are the following: (1) In markets where jobs can only consider a limited number of candidates for interview, it increases the size of the resulting matching if the system has a limit on the number of applications a candidate can send. (2) The fair system of all applicants being allowed to apply to the exact same number of positions maximizes the expected size of the matching. More particularly, starting from an integer k as the number of applications, the matching size decreases as a few applicants are allowed to apply to one additional position (and then increases again as they are all allowed to apply to k+1). Although it seems natural to expect that the size of the matching would be a monotone increasing and concave function in the number of applications, our results show that neither is true. These results hold even in a market where a-priori all jobs and all candidates are equally likely to be good, and the judgments of different employers and candidates are independent. Our main technical contribution is computing the expected size of the matching found via the deferred acceptance algorithm as a function of the number of interviews and applications in a market where preferences are uniform and independent. Through simulations we confirm that these findings extend to markets where rankings become correlated after the interviews.

The classical Perceptron algorithm provides a simple and elegant procedure for learning a linear classifier. In each step, the algorithm observes the sample's position and label and updates the current predictor accordingly if it makes a mistake. However, in presence of strategic agents that desire to be classified as positive and that are able to modify their position by a limited amount, the classifier may not be able to observe the true position of agents but rather a position where the agent pretends to be. Unlike the original setting with perfect knowledge of positions, in this situation the Perceptron algorithm fails to achieve its guarantees, and we illustrate examples with the predictor oscillating between two solutions forever, making an unbounded number of mistakes even though a perfect large-margin linear classifier exists. Our main contribution is providing a modified Perceptron-style algorithm which makes a bounded number of mistakes in presence of strategic agents with both $\ell_2$ and weighted $\ell_1$ manipulation costs. In our baseline model, knowledge of the manipulation costs (i.e., the extent to which an agent may manipulate) is assumed. In our most general model, we relax this assumption and provide an algorithm which learns and refines both the classifier and its cost estimates to achieve good mistake bounds even when manipulation costs are unknown.

The Competition Complexity of an auction setting refers to the number of additional bidders necessary in order for the (deterministic, prior-independent, dominant strategy truthful) Vickrey-Clarke-Groves mechanism to achieve greater revenue than the (randomized, prior-dependent, Bayesian-truthful) optimal mechanism without the additional bidders.

Modeling is one of the major research areas in social network analysis whose goal is to study networks structure and its evolution. Motivated by the intuition that members in social networks behave selfishly, network creation games have been introduced for modeling social networks. In this paper, our aim is to measure how much the output graphs of a given network creation game are compatible with a social network. We first show that the precise measurement is not possible in polynomial time. Then we propose a method for its approximation; finally, we show the usability of our method by conducting experiments on real network data.