arXiv subjects

Gabriel Ziegler

Publications and source records attributed to Gabriel Ziegler.

11 recordsLinked to original sources

arXiv2026

Efficiency Adjustments Break the Logarithmic Rank Barrier

We study the expected average rank achieved by the Efficiency-Adjusted Deferred Acceptance (EADA) mechanism in i.i.d.\ matching markets. While student-proposing Deferred Acceptance gives students an expected average rank of logarithmic order, we prove that EADA's expected average rank is at most $4\log\log n+O(1)$. Therefore, EADA improves the asymptotic order of students' assignments. At the cost of a weaker bound, $O((\log\log n)^2)$, we extend this conclusion to a much larger class of mechanisms. Namely, every Pareto-efficient mechanism that weakly Pareto-dominates DA breaks DA's logarithmic barrier. These are the first asymptotic guarantees for the expected average rank of EADA and of the broader class of Pareto-efficient improvements of DA. The conclusions extend to many-to-one markets with bounded quotas and random markets with correlated preferences.

cs.GT↗

arXiv2026

The Distribution of Envy in Matching Markets

We study the distribution of envy in random matching markets under the Deferred Acceptance (DA) algorithm. Using tools from applied probability, we compute the expected number of proposing agents whom nobody envies and those who envy nobody. We obtain an exact finite-market expression for the former, based on a connection with the coupon collector problem, and asymptotic bounds for the latter. To put these quantities into perspective, we compare them to their counterparts under Random Serial Dictatorship (RSD): while RSD assigns a constant fraction of agents to their top choice, both DA and RSD leave exactly $H_n$ proposing agents unenvied in expectation. Our results show that these clearly unimprovable proposing agents constitute a vanishing fraction of the market.

econ.TH↗

arXiv2025

Reasoning about Bounded Reasoning

In experimental applications of bounded-reasoning models, behavior is often summarized by distributions of "levels". We argue that such summaries conflate two conceptually distinct dimensions: a player's type, capturing beliefs about what types their opponents might be, and the depth of higher-order reasoning about rationality. Distinguishing these dimensions matters for interpreting experimental evidence and for understanding when cross-environment variation should be read as changes in beliefs versus changes in cognitive depth, but existing frameworks provide no language to do so. We develop a unified framework by "lifting" static complete-information games into incomplete-information versions in which players are explicitly uncertain about opponents' types. Within this framework, bounded reasoning about opponents' types is represented by transparent first-order belief restrictions, while (higher-order) reasoning depth is captured by bounds on belief in rationality. We analyze three benchmark instances: downward rationalizability, a robust baseline, and two refinements, $\mathsf{L}$-rationalizability and $\mathsf{C}$-rationalizability, which provide epistemic foundations -- with an important nuance -- for classic level-$k$ and Cognitive Hierarchy, respectively, and clarify what "level-$k$" behavior can and cannot reveal about underlying reasoning processes.

econ.TH↗

arXiv2025

Capturing Misalignment

We introduce and formalize misalignment, a phenomenon of interactive environments perceived from an analyst's perspective where an agent holds beliefs about another agent's beliefs that do not correspond to the actual beliefs of the latter. We demonstrate that standard frameworks, such as type structures, fail to capture misalignment, necessitating new tools to analyze this phenomenon. To this end, we characterize misalignment through non-belief-closed state spaces and introduce agent-dependent type structures, which provide a flexible tool to understand the varying degrees of misalignment. Furthermore, we establish that appropriately adapted modal operators on agent-dependent type structures behave consistently with standard properties, enabling us to explore the implications of misalignment for interactive reasoning. Finally, we show how speculative trade can arise under misalignment, even when imposing the corresponding assumptions that rule out such trades in standard environments.

econ.TH↗

arXiv2025

What Pareto-Efficiency Adjustments Cannot Fix

The Deferred Acceptance (DA) algorithm is stable and strategy-proof, but can produce outcomes that are Pareto-inefficient for students, and thus several alternative mechanisms have been proposed to correct this inefficiency. However, we show that these mechanisms cannot correct DA's rank-inefficiency and inequality, because these shortcomings can arise even in cases where DA is Pareto-efficient. We also examine students' segregation in settings with advantaged and marginalized students. We prove that the demographic composition of every school is perfectly preserved under any Pareto-efficient mechanism that dominates DA, and consequently fully segregated schools under DA maintain their extreme homogeneity.

econ.TH↗

arXiv2024

The Large and Likely Inefficiency of Stable Matching Mechanisms

We prove that any stable matching mechanism suffers from systematic inefficiency of striking magnitude: in large random markets, any stable allocation is Pareto-inefficient with high probability, and almost all students can simultaneously improve their placements without harming anyone else. We establish this result by showing that the envy digraph generated by the student-proposing Deferred Acceptance mechanism contains a unique giant strongly connected component, implying that nearly all students are improvable via trading cycles. Finally, we show that every maximal cycle packing covers almost all students, revealing a surprising asymptotic equivalence among all efficient mechanisms that Pareto-dominate DA.

econ.TH↗

arXiv2022

Strategic Behavior under Context Misalignment

We study the behavioral implications of Rationality and Common Strong Belief in Rationality (RCSBR) with contextual assumptions allowing players to entertain misaligned beliefs, i.e., players can hold beliefs concerning their opponents' beliefs where there is no opponent holding those very beliefs. Taking the analysts' perspective, we distinguish the infinite hierarchies of beliefs actually held by players ("real types") from those that are a byproduct of players' hierarchies ("imaginary types") by introducing the notion of separating type structure. We characterize the behavioral implications of RCSBR for the real types across all separating type structures via a family of subsets of Full Strong Best-Reply Sets of Battigalli & Friedenberg (2012). By allowing misalignment, in dynamic games we can obtain behavioral predictions inconsistent with RCSBR (in the standard framework), contrary to the case of belief-based analyses for static games--a difference due to the dichotomy "non-monotonic vs. monotonic" reasoning.

econ.TH↗

arXiv2022

Informational Robustness of Common Belief in Rationality

In this note, I explore the implications of informational robustness under the assumption of common belief in rationality. That is, predictions for incomplete-information games which are valid across all possible information structures. First, I address this question from a global perspective and then generalize the analysis to allow for localized informational robustness.

econ.TH↗

arXiv2021

Optimism and Pessimism in Strategic Interactions under Ignorance

We study players interacting under the veil of ignorance, who have -- coarse -- beliefs represented as subsets of opponents' actions. We analyze when these players follow $\max \min$ or $\max\max$ decision criteria, which we identify with pessimistic or optimistic attitudes, respectively. Explicitly formalizing these attitudes and how players reason interactively under ignorance, we characterize the behavioral implications related to common belief in these events: while optimism is related to Point Rationalizability, a new algorithm -- Wald Rationalizability -- captures pessimism. Our characterizations allow us to uncover novel results: ($i$) regarding optimism, we relate it to wishful thinking \'a la Yildiz (2007) and we prove that dropping the (implicit) "belief-implies-truth" assumption reverses an existence failure described therein; ($ii$) we shed light on the notion of rationality in ordinal games; ($iii$) we clarify the conceptual underpinnings behind a discontinuity in Rationalizability hinted in the analysis of Weinstein (2016).

econ.TH↗

arXiv2021

Binary Classification Tests, Imperfect Standards, and Ambiguous Information

New binary classification tests are often evaluated relative to a pre-established test. For example, rapid Antigen tests for the detection of SARS-CoV-2 are assessed relative to more established PCR tests. In this paper, I argue that the new test can be described as producing ambiguous information when the pre-established is imperfect. This allows for a phenomenon called dilation -- an extreme form of non-informativeness. As an example, I present hypothetical test data satisfying the WHO's minimum quality requirement for rapid Antigen tests which leads to dilation. The ambiguity in the information arises from a missing data problem due to imperfection of the established test: the joint distribution of true infection and test results is not observed. Using results from Copula theory, I construct the (usually non-singleton) set of all these possible joint distributions, which allows me to assess the new test's informativeness. This analysis leads to a simple sufficient condition to make sure that a new test is not a dilation. I illustrate my approach with applications to data from three COVID-19 related tests. Two rapid Antigen tests satisfy my sufficient condition easily and are therefore informative. However, less accurate procedures, like chest CT scans, may exhibit dilation.

econ.EM↗

arXiv2020

How many people are infected? A case study on SARS-CoV-2 prevalence in Austria

Using recent data from voluntary mass testing, I provide credible bounds on prevalence of SARS-CoV-2 for Austrian counties in early December 2020. When estimating prevalence, a natural missing data problem arises: no test results are generated for non-tested people. In addition, tests are not perfectly predictive for the underlying infection. This is particularly relevant for mass SARS-CoV-2 testing as these are conducted with rapid Antigen tests, which are known to be somewhat imprecise. Using insights from the literature on partial identification, I propose a framework addressing both issues at once. I use the framework to study differing selection assumptions for the Austrian data. Whereas weak monotone selection assumptions provide limited identification power, reasonably stronger assumptions reduce the uncertainty on prevalence significantly.

econ.GN↗