arXiv subjects

Aleck Johnsen

Publications and source records attributed to Aleck Johnsen.

11 recordsLinked to original sources

arXiv2025

Scale-robust Auctions

We study auctions that are robust at any scale, i.e., they can be applied to sell both expensive and cheap items and achieve the best multiplicative approximation of the optimal revenue in the worst case. We first show that it is without loss of optimality to restrict attention to scale-invariant mechanisms whenever the family of possible distributions is closed under every positive rescaling. This conclusion uses no regularity or other distributional shape restriction. We then solve the two-agent, single-item problem with values drawn i.i.d. from an unknown regular distribution when only a high value bidder can receive a positive allocation. The robustly optimal mechanism in this class randomizes between the second-price auction, with probability approximately 0.806, and a markup auction that offers the item to the highest-valued bidder at a price equal to 2.447 times the second-highest value. Its worst-case approximation ratio is approximately 1.907.

cs.GT↗

arXiv2024

Subgame Optimal and Prior-Independent Online Algorithms

This paper takes a game theoretic approach to the design and analysis of online algorithms and illustrates the approach on the finite-horizon ski-rental problem. This approach allows beyond worst-case analysis of online algorithms. First, we define "subgame optimality" which is stronger than worst case optimality in that it requires the algorithm to take advantage of an adversary not playing a worst case input. Algorithms only focusing on the worst case can be far from subgame optimal. Second, we consider prior-independent design and analysis of online algorithms, where rather than choosing a worst case input, the adversary chooses a worst case independent and identical distribution over inputs. Prior-independent online algorithms are generally analytically intractable; instead we give a fully polynomial time approximation scheme to compute them. Highlighting the potential improvement from these paradigms for the finite-horizon ski-rental problem, we empirically compare worst-case, subgame optimal, and prior-independent algorithms in the prior-independent framework.

cs.GT↗

arXiv2023

Extension of Minimax for Algorithmic Lower Bounds

This paper considers the use of Yao's Minimax Theorem in robust algorithm design, e.g., for online algorithms, where the algorithm designer aims to minimize the ratio of the algorithm's performance to the optimal performance. When applying Minimax, the objective becomes the expectation of these ratios. The main result of the paper is that it is equally valid for the objective (after applying Minimax) to be the ratio of the expectations.

cs.GT↗

arXiv2023

Screening with Disadvantaged Agents

Motivated by school admissions, this paper studies screening in a population with both advantaged and disadvantaged agents. A school is interested in admitting the most skilled students, but relies on imperfect test scores that reflect both skill and effort. Students are limited by a budget on effort, with disadvantaged students having tighter budgets. This raises a challenge for the principal: among agents with similar test scores, it is difficult to distinguish between students with high skills and students with large budgets. Our main result is an optimal stochastic mechanism that maximizes the gains achieved from admitting ``high-skill" students minus the costs incurred from admitting ``low-skill" students when considering two skill types and $n$ budget types. Our mechanism makes it possible to give higher probability of admission to a high-skill student than to a low-skill, even when the low-skill student can potentially get higher test-score due to a higher budget. Further, we extend our admission problem to a setting in which students uniformly receive an exogenous subsidy to increase their budget for effort. This extension can only help the school's admission objective and we show that the optimal mechanism with exogenous subsidies has the same characterization as optimal mechanisms for the original problem.

cs.GT↗

arXiv2022

Bias-Variance Games

Firms engaged in electronic commerce increasingly rely on predictive analytics via machine-learning algorithms to drive a wide array of managerial decisions. The tuning of many standard machine learning algorithms can be understood as trading off bias (i.e., accuracy) with variance (i.e., precision) in the algorithm's predictions. The goal of this paper is to understand how competition between firms affects their strategic choice of such algorithms. To this end, we model the interaction of two firms choosing learning algorithms as a game and analyze its equilibria. Absent competition, players care only about the magnitude of predictive error and not its source. In contrast, our main result is that with competition, players prefer to incur error due to variance rather than due to bias, even at the cost of higher total error. In addition, we show that competition can have counterintuitive implications -- for example, reducing the error incurred by a firm's algorithm can be harmful to that firm -- but we provide conditions under which such phenomena do not occur. In addition to our theoretical analysis, we also validate our insights by applying our metrics to several publicly available datasets.

cs.GT↗

arXiv2021

Lower Bounds for Prior Independent Algorithms

The prior independent framework for algorithm design considers how well an algorithm that does not know the distribution of its inputs approximates the expected performance of the optimal algorithm for this distribution. This paper gives a method that is agnostic to problem setting for proving lower bounds on the prior independent approximation factor of any algorithm. The method constructs a correlated distribution over inputs that can be generated both as a distribution over i.i.d. good-for-algorithms distributions and as a distribution over i.i.d. bad-for-algorithms distributions. Prior independent algorithms are upper-bounded by the optimal algorithm for the latter distribution even when the true distribution is the former. Thus, the ratio of the expected performances of the Bayesian optimal algorithms for these two decompositions is a lower bound on the prior independent approximation ratio. The techniques of the paper connect prior independent algorithm design, Yao's Minimax Principle, and information design. We apply this framework to give new lower bounds on several canonical prior independent mechanism design problems.

cs.GT↗

arXiv2020

Mechanisms for a No-Regret Agent: Beyond the Common Prior

A rich class of mechanism design problems can be understood as incomplete-information games between a principal who commits to a policy and an agent who responds, with payoffs determined by an unknown state of the world. Traditionally, these models require strong and often-impractical assumptions about beliefs (a common prior over the state). In this paper, we dispense with the common prior. Instead, we consider a repeated interaction where both the principal and the agent may learn over time from the state history. We reformulate mechanism design as a reinforcement learning problem and develop mechanisms that attain natural benchmarks without any assumptions on the state-generating process. Our results make use of novel behavioral assumptions for the agent -- centered around counterfactual internal regret -- that capture the spirit of rationality without relying on beliefs.

econ.TH↗

arXiv2020

Benchmark Design and Prior-independent Optimization

This paper compares two leading approaches for robust optimization in the models of online algorithms and mechanism design. Competitive analysis compares the performance of an online algorithm to an offline benchmark in worst-case over inputs, and prior-independent mechanism design compares the expected performance of a mechanism on an unknown distribution (of inputs, i.e., agent values) to the optimal mechanism for the distribution in worst case over distributions. For competitive analysis, a critical concern is the choice of benchmark. This paper gives a method for selecting a good benchmark. We show that optimal algorithm/mechanism for the optimal benchmark are equal to the prior-independent optimal algorithm/mechanism. We solve a central open question in prior-independent mechanism design, namely we identify the prior-independent revenue-optimal mechanism for selling a single item to two agents with i.i.d. and regularly distributed values. Via this solution and the above equivalence of prior-independent mechanism design and competitive analysis (a.k.a. prior-free mechanism design) we show that the standard method for lower bounds of prior-free mechanisms is not generally tight for the benchmark design program.

cs.GT↗

arXiv2020

Inference from Auction Prices

Econometric inference allows an analyst to back out the values of agents in a mechanism from the rules of the mechanism and bids of the agents. This paper gives an algorithm to solve the problem of inferring the values of agents in a dominant-strategy mechanism from the social choice function implemented by the mechanism and the per-unit prices paid by the agents (the agent bids are not observed). For single-dimensional agents, this inference problem is a multi-dimensional inversion of the payment identity and is feasible only if the payment identity is uniquely invertible. The inversion is unique for single-unit proportional weights social choice functions (common, for example, in bandwidth allocation); and its inverse can be found efficiently. This inversion is not unique for social choice functions that exhibit complementarities. Of independent interest, we extend a result of Rosen (1965), that the Nash equilbria of "concave games" are unique and pure, to an alternative notion of concavity based on Gale and Nikaido (1965).

cs.GT↗

arXiv2019

Dashboard Mechanisms for Online Marketplaces

This paper gives a theoretical model for design and analysis of mechanisms for online marketplaces where a bidding dashboard enables the bid-optimization of long-lived agents. We assume that a good allocation algorithm exists when given the true values of the agents and we develop online winner-pays-bid and all-pay mechanisms that implement the same outcome of the algorithm with the aid of a bidding dashboard. The bidding dashboards that we develop work in conjunction with the mechanism to guarantee that bidding according to the dashboard is strategically equivalent (with vanishing utility difference) to bidding truthfully in the sequential truthful implementation of the allocation algorithm. Our dashboard mechanism makes only a single call to the allocation algorithm in each stage.

cs.GT↗

arXiv2014

A manually-checkable proof for the NP-hardness of 11-color pattern self-assembly tile set synthesis

Patterned self-assembly tile set synthesis (PATS) aims at finding a minimum tile set to uniquely self-assemble a given rectangular (color) pattern. For k >= 1, k-PATS is a variant of PATS that restricts input patterns to those with at most $k$ colors. A computer-assisted proof has been recently proposed for 2-PATS by Kari et al. [arXiv:1404.0967 (2014)]. In contrast, the best known manually-checkable proof is for the NP-hardness of 29-PATS by Johnsen, Kao, and Seki [ISAAC 2013, LNCS 8283, pp.~699-710]. We propose a manually-checkable proof for the NP-hardness of 11-PATS.

cs.DM↗