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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Gameplay under various forms of uncertainty has been widely studied. Feldman et al. [Michal Feldman et al., 2010] studied a particularly low-information setting in which one observes the opponent’s actions but no payoffs, not even one’s own, and introduced an algorithm which guarantees one’s payoff nonetheless approaches the minimax optimal value (i.e., zero) in a symmetric zero-sum game. Against an opponent playing a minimax-optimal strategy, approaching the value of the game is the best one can hope to guarantee. However, a wealth of research in behavioral economics shows that people often do not make perfectly rational, optimal decisions. Here we consider whether it is possible to actually win in this setting if the opponent is behaviorally biased. We model several deterministic, biased opponents and show that even without knowing the game matrix in advance or observing any payoffs, it is possible to take advantage of each bias in order to win nearly every round (so long as the game has the property that each action beats and is beaten by at least one other action). We also provide a partial characterization of the kinds of biased strategies that can be exploited to win nearly every round, and provide algorithms for beating some kinds of biased strategies even when we don't know which strategy the opponent uses.

Avrim Blum and Melissa Dutz. Winning Without Observing Payoffs: Exploiting Behavioral Biases to Win Nearly Every Round. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{blum_et_al:LIPIcs.ITCS.2024.18, author = {Blum, Avrim and Dutz, Melissa}, title = {{Winning Without Observing Payoffs: Exploiting Behavioral Biases to Win Nearly Every Round}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {18:1--18:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.18}, URN = {urn:nbn:de:0030-drops-195463}, doi = {10.4230/LIPIcs.ITCS.2024.18}, annote = {Keywords: Game theory, Behavioral bias} }

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**Published in:** LIPIcs, Volume 256, 4th Symposium on Foundations of Responsible Computing (FORC 2023)

We consider the problem of helping agents improve by setting goals. Given a set of target skill levels, we assume each agent will try to improve from their initial skill level to the closest target level within reach (or do nothing if no target level is within reach). We consider two models: the common improvement capacity model, where agents have the same limit on how much they can improve, and the individualized improvement capacity model, where agents have individualized limits. Our goal is to optimize the target levels for social welfare and fairness objectives, where social welfare is defined as the total amount of improvement, and we consider fairness objectives when the agents belong to different underlying populations. We prove algorithmic, learning, and structural results for each model.
A key technical challenge of this problem is the non-monotonicity of social welfare in the set of target levels, i.e., adding a new target level may decrease the total amount of improvement; agents who previously tried hard to reach a distant target now have a closer target to reach and hence improve less. This especially presents a challenge when considering multiple groups because optimizing target levels in isolation for each group and outputting the union may result in arbitrarily low improvement for a group, failing the fairness objective. Considering these properties, we provide algorithms for optimal and near-optimal improvement for both social welfare and fairness objectives. These algorithmic results work for both the common and individualized improvement capacity models. Furthermore, despite the non-monotonicity property and interference of the target levels, we show a placement of target levels exists that is approximately optimal for the social welfare of each group. Unlike the algorithmic results, this structural statement only holds in the common improvement capacity model, and we illustrate counterexamples of this result in the individualized improvement capacity model. Finally, we extend our algorithms to learning settings where we have only sample access to the initial skill levels of agents.

Saba Ahmadi, Hedyeh Beyhaghi, Avrim Blum, and Keziah Naggita. Setting Fair Incentives to Maximize Improvement. In 4th Symposium on Foundations of Responsible Computing (FORC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 256, pp. 5:1-5:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ahmadi_et_al:LIPIcs.FORC.2023.5, author = {Ahmadi, Saba and Beyhaghi, Hedyeh and Blum, Avrim and Naggita, Keziah}, title = {{Setting Fair Incentives to Maximize Improvement}}, booktitle = {4th Symposium on Foundations of Responsible Computing (FORC 2023)}, pages = {5:1--5:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-272-3}, ISSN = {1868-8969}, year = {2023}, volume = {256}, editor = {Talwar, Kunal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FORC.2023.5}, URN = {urn:nbn:de:0030-drops-179261}, doi = {10.4230/LIPIcs.FORC.2023.5}, annote = {Keywords: Algorithmic Fairness, Learning for Strategic Behavior, Incentivizing Improvement} }

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**Published in:** LIPIcs, Volume 218, 3rd Symposium on Foundations of Responsible Computing (FORC 2022)

In this work, we consider classification of agents who can both game and improve. For example, people wishing to get a loan may be able to take some actions that increase their perceived credit-worthiness and others that also increase their true credit-worthiness. A decision-maker would like to define a classification rule with few false-positives (does not give out many bad loans) while yielding many true positives (giving out many good loans), which includes encouraging agents to improve to become true positives if possible. We consider two models for this problem, a general discrete model and a linear model, and prove algorithmic, learning, and hardness results for each.
For the general discrete model, we give an efficient algorithm for the problem of maximizing the number of true positives subject to no false positives, and show how to extend this to a partial-information learning setting. We also show hardness for the problem of maximizing the number of true positives subject to a nonzero bound on the number of false positives, and that this hardness holds even for a finite-point version of our linear model. We also show that maximizing the number of true positives subject to no false positive is NP-hard in our full linear model. We additionally provide an algorithm that determines whether there exists a linear classifier that classifies all agents accurately and causes all improvable agents to become qualified, and give additional results for low-dimensional data.

Saba Ahmadi, Hedyeh Beyhaghi, Avrim Blum, and Keziah Naggita. On Classification of Strategic Agents Who Can Both Game and Improve. In 3rd Symposium on Foundations of Responsible Computing (FORC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 218, pp. 3:1-3:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ahmadi_et_al:LIPIcs.FORC.2022.3, author = {Ahmadi, Saba and Beyhaghi, Hedyeh and Blum, Avrim and Naggita, Keziah}, title = {{On Classification of Strategic Agents Who Can Both Game and Improve}}, booktitle = {3rd Symposium on Foundations of Responsible Computing (FORC 2022)}, pages = {3:1--3:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-226-6}, ISSN = {1868-8969}, year = {2022}, volume = {218}, editor = {Celis, L. Elisa}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FORC.2022.3}, URN = {urn:nbn:de:0030-drops-165269}, doi = {10.4230/LIPIcs.FORC.2022.3}, annote = {Keywords: Strategic Classification, Social Welfare, Learning} }

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**Published in:** LIPIcs, Volume 156, 1st Symposium on Foundations of Responsible Computing (FORC 2020)

Multiple fairness constraints have been proposed in the literature, motivated by a range of concerns about how demographic groups might be treated unfairly by machine learning classifiers. In this work we consider a different motivation; learning from biased training data. We posit several ways in which training data may be biased, including having a more noisy or negatively biased labeling process on members of a disadvantaged group, or a decreased prevalence of positive or negative examples from the disadvantaged group, or both. Given such biased training data, Empirical Risk Minimization (ERM) may produce a classifier that not only is biased but also has suboptimal accuracy on the true data distribution. We examine the ability of fairness-constrained ERM to correct this problem. In particular, we find that the Equal Opportunity fairness constraint [Hardt et al., 2016] combined with ERM will provably recover the Bayes optimal classifier under a range of bias models. We also consider other recovery methods including re-weighting the training data, Equalized Odds, and Demographic Parity, and Calibration. These theoretical results provide additional motivation for considering fairness interventions even if an actor cares primarily about accuracy.

Avrim Blum and Kevin Stangl. Recovering from Biased Data: Can Fairness Constraints Improve Accuracy?. In 1st Symposium on Foundations of Responsible Computing (FORC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 156, pp. 3:1-3:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{blum_et_al:LIPIcs.FORC.2020.3, author = {Blum, Avrim and Stangl, Kevin}, title = {{Recovering from Biased Data: Can Fairness Constraints Improve Accuracy?}}, booktitle = {1st Symposium on Foundations of Responsible Computing (FORC 2020)}, pages = {3:1--3:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-142-9}, ISSN = {1868-8969}, year = {2020}, volume = {156}, editor = {Roth, Aaron}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FORC.2020.3}, URN = {urn:nbn:de:0030-drops-120192}, doi = {10.4230/LIPIcs.FORC.2020.3}, annote = {Keywords: fairness in machine learning, equal opportunity, bias, machine learning} }

Document

**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

We study methods for improving fairness to subgroups in settings with overlapping populations and sequential predictions. Classical notions of fairness focus on the balance of some property across different populations. However, in many applications the goal of the different groups is not to be predicted equally but rather to be predicted well. We demonstrate that the task of satisfying this guarantee for multiple overlapping groups is not straightforward and show that for the simple objective of unweighted average of false negative and false positive rate, satisfying this for overlapping populations can be statistically impossible even when we are provided predictors that perform well separately on each subgroup. On the positive side, we show that when individuals are equally important to the different groups they belong to, this goal is achievable; to do so, we draw a connection to the sleeping experts literature in online learning. Motivated by the one-sided feedback in natural settings of interest, we extend our results to such a feedback model. We also provide a game-theoretic interpretation of our results, examining the incentives of participants to join the system and to provide the system full information about predictors they may possess. We end with several interesting open problems concerning the strength of guarantees that can be achieved in a computationally efficient manner.

Avrim Blum and Thodoris Lykouris. Advancing Subgroup Fairness via Sleeping Experts. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 55:1-55:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{blum_et_al:LIPIcs.ITCS.2020.55, author = {Blum, Avrim and Lykouris, Thodoris}, title = {{Advancing Subgroup Fairness via Sleeping Experts}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {55:1--55:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.55}, URN = {urn:nbn:de:0030-drops-117402}, doi = {10.4230/LIPIcs.ITCS.2020.55}, annote = {Keywords: Online learning, Fairness, Game theory} }

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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

We study the classic Maximum Independent Set problem under the notion of stability introduced by Bilu and Linial (2010): a weighted instance of Independent Set is gamma-stable if it has a unique optimal solution that remains the unique optimal solution under multiplicative perturbations of the weights by a factor of at most gamma >= 1. The goal then is to efficiently recover this "pronounced" optimal solution exactly. In this work, we solve stable instances of Independent Set on several classes of graphs: we improve upon previous results by solving O~(Delta/sqrt(log Delta))-stable instances on graphs of maximum degree Delta, (k - 1)-stable instances on k-colorable graphs and (1 + epsilon)-stable instances on planar graphs (for any fixed epsilon > 0), using both combinatorial techniques as well as LPs and the Sherali-Adams hierarchy.
For general graphs, we present a strong lower bound showing that there are no efficient algorithms for O(n^(1/2 - epsilon))-stable instances of Independent Set, assuming the planted clique conjecture. To complement our negative result, we give an algorithm for (epsilon n)-stable instances, for any fixed epsilon > 0. As a by-product of our techniques, we give algorithms as well as lower bounds for stable instances of Node Multiway Cut (a generalization of Edge Multiway Cut), by exploiting its connections to Vertex Cover. Furthermore, we prove a general structural result showing that the integrality gap of convex relaxations of several maximization problems reduces dramatically on stable instances.
Moreover, we initiate the study of certified algorithms for Independent Set. The notion of a gamma-certified algorithm was introduced very recently by Makarychev and Makarychev (2018) and it is a class of gamma-approximation algorithms that satisfy one crucial property: the solution returned is optimal for a perturbation of the original instance, where perturbations are again multiplicative up to a factor of gamma >= 1 (hence, such algorithms not only solve gamma-stable instances optimally, but also have guarantees even on unstable instances). Here, we obtain Delta-certified algorithms for Independent Set on graphs of maximum degree Delta, and (1+epsilon)-certified algorithms on planar graphs. Finally, we analyze the algorithm of Berman and Fürer (1994) and prove that it is a ((Delta + 1)/3 + epsilon)-certified algorithm for Independent Set on graphs of maximum degree Delta where all weights are equal to 1.

Haris Angelidakis, Pranjal Awasthi, Avrim Blum, Vaggos Chatziafratis, and Chen Dan. Bilu-Linial Stability, Certified Algorithms and the Independent Set Problem. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 7:1-7:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{angelidakis_et_al:LIPIcs.ESA.2019.7, author = {Angelidakis, Haris and Awasthi, Pranjal and Blum, Avrim and Chatziafratis, Vaggos and Dan, Chen}, title = {{Bilu-Linial Stability, Certified Algorithms and the Independent Set Problem}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {7:1--7:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.7}, URN = {urn:nbn:de:0030-drops-111288}, doi = {10.4230/LIPIcs.ESA.2019.7}, annote = {Keywords: Bilu-Linial stability, perturbation resilience, beyond worst-case analysis, Independent Set, Vertex Cover, Multiway Cut} }

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Complete Volume

**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

LIPIcs, Volume 124, ITCS'19, Complete Volume

10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@Proceedings{blum:LIPIcs.ITCS.2019, title = {{LIPIcs, Volume 124, ITCS'19, Complete Volume}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019}, URN = {urn:nbn:de:0030-drops-101704}, doi = {10.4230/LIPIcs.ITCS.2019}, annote = {Keywords: Theory of computation, Mathematics of computing} }

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Front Matter

**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

Front Matter, Table of Contents, Preface, Conference Organization

10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 0:i-0:xii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{blum:LIPIcs.ITCS.2019.0, author = {Blum, Avrim}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {0:i--0:xii}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.0}, URN = {urn:nbn:de:0030-drops-100937}, doi = {10.4230/LIPIcs.ITCS.2019.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

Document

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Given a finite set of points P subseteq R^d, we would like to find a small subset S subseteq P such that the convex hull of S approximately contains P. More formally, every point in P is within distance epsilon from the convex hull of S. Such a subset S is called an epsilon-hull. Computing an epsilon-hull is an important problem in computational geometry, machine learning, and approximation algorithms.
In many applications, the set P is too large to fit in memory. We consider the streaming model where the algorithm receives the points of P sequentially and strives to use a minimal amount of memory. Existing streaming algorithms for computing an epsilon-hull require O(epsilon^{(1-d)/2}) space, which is optimal for a worst-case input. However, this ignores the structure of the data. The minimal size of an epsilon-hull of P, which we denote by OPT, can be much smaller. A natural question is whether a streaming algorithm can compute an epsilon-hull using only O(OPT) space.
We begin with lower bounds that show, under a reasonable streaming model, that it is not possible to have a single-pass streaming algorithm that computes an epsilon-hull with O(OPT) space. We instead propose three relaxations of the problem for which we can compute epsilon-hulls using space near-linear to the optimal size. Our first algorithm for points in R^2 that arrive in random-order uses O(log n * OPT) space. Our second algorithm for points in R^2 makes O(log(epsilon^{-1})) passes before outputting the epsilon-hull and requires O(OPT) space. Our third algorithm, for points in R^d for any fixed dimension d, outputs, with high probability, an epsilon-hull for all but delta-fraction of directions and requires O(OPT * log OPT) space.

Avrim Blum, Vladimir Braverman, Ananya Kumar, Harry Lang, and Lin F. Yang. Approximate Convex Hull of Data Streams. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 21:1-21:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{blum_et_al:LIPIcs.ICALP.2018.21, author = {Blum, Avrim and Braverman, Vladimir and Kumar, Ananya and Lang, Harry and Yang, Lin F.}, title = {{Approximate Convex Hull of Data Streams}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {21:1--21:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.21}, URN = {urn:nbn:de:0030-drops-90254}, doi = {10.4230/LIPIcs.ICALP.2018.21}, annote = {Keywords: Convex Hulls, Streaming Algorithms, Epsilon Kernels, Sparse Coding} }

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**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

In this work we propose a model where the value of a buyer for some
product (like a slice of pizza) is a combination of their personal
desire for the product (how hungry they are for pizza) and the
quality of the product (how good the pizza is). Sellers in this
setting have a two-dimensional optimization problem of determining
both the quality level at which to make their product (how expensive
ingredients to use) and the price at which to sell it. We analyze
optimal seller strategies as well as analogs of Walrasian equilibria
in this setting. A key question we are interested in is: to what
extent will the price of a good be a reliable indicator of the
good's quality?
One result we show is that indeed in this model, price will be a
surprisingly robust signal for quality under optimal seller
behavior. In particular, while the specific quality and price that
a seller should choose will depend highly on the specific
distribution of buyers, for optimal sellers, price and quality will
be linearly related, independent of that distribution. We also show
that for the case of multiple buyers and sellers, an analog of
Walrasian equilibrium exists in this setting, and can be found via a
natural tatonnement process. Finally, we analyze markets with a combination of "locals" (who know the quality of each good) and "tourists" (who do not) and analyze under what conditions the market will become a tourist trap, setting quality to zero while keeping prices high.

Avrim Blum and Yishay Mansour. On Price versus Quality. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 16:1-16:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{blum_et_al:LIPIcs.ITCS.2018.16, author = {Blum, Avrim and Mansour, Yishay}, title = {{On Price versus Quality}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {16:1--16:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.16}, URN = {urn:nbn:de:0030-drops-83264}, doi = {10.4230/LIPIcs.ITCS.2018.16}, annote = {Keywords: Algorithmic game theory, pricing, Cobb-Douglas valuations} }

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