This thesis comprises four independent parts and an appendix.
\r\n\r\n1. We define and study expansion problems on countable structures in the setting of descriptive combinatorics. We consider both expansions on countable Borel equivalence relations and on countable groups, in the Borel, measure, and category settings, and establish some basic correspondences between the two notions. We then explore in detail many examples, including finding spanning trees in graphs, finding monochromatic sets in Ramsey's Theorem, and linearizing partial orders.
\r\n\r\n2. Standard results in descriptive set theory provide sufficient conditions for a set P \u2286 \u2115\u2115 \u00d7 \u2115\u2115 to admit a Borel uniformization, namely, when P has small or large sections. We consider an invariant analogue of these results with respect to a Borel equivalence relation E. Given E, we show that every such P admits an E-invariant Borel uniformization if and only if E is smooth. We also compute the definable complexity of counterexamples in the case where E is not smooth, using category, measure, and Ramsey-theoretic methods. We also show that the set of pairs (E, P) such that P has large sections and admits an E-invariant Borel uniformization is \u03a312-complete.
\r\n\r\n3. Let E, F be Borel equivalence relations on X, Y, and P be an E-invariant Borel set whose sections contain countably many F-classes. We explore obstructions to the existence of Borel E-invariant uniformizing sets for P, i.e., sets choosing one F-class from every section. We survey known results, and prove new dichotomies for the case where P has \u03c3-bounded finite sections. On the way, we prove a dichotomy characterizing the essential values of Borel cocycles into residually finite Polish groups.
\r\n\r\n4. We show that the Kechris\u2013Solecki\u2013Todor\u010devi\u0107 dichotomy implies the Harrington\u2013Kechris\u2013Louveau dichotomy. We also give a simple proof of a graph-theoretic dichotomy of Miller for doubly-indexed sequences of analytic graphs, and show that this dichotomy generalizes to finite-dimensional hypergraphs but not to \u21350-dimensional hypergraphs.
\r\n\r\n5. An effective version of Nadkarni\u2019s Theorem was proved in Ditzen\u2019s unpublished Ph.D. thesis. The appendix contains a streamlined exposition of the proof and provides an alternative proof of the Effective Ergodic Decomposition Theorem for invariant measures (also originally proved by Ditzen). In addition, we show that the existence of an invariant Borel probability measure is not effective.
", "doi": "10.7907/w4nq-ag86", "publication_date": "2025", "thesis_type": "phd", "thesis_year": "2025" }, { "id": "thesis:16380", "collection": "thesis", "collection_id": "16380", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05132024-181232920", "type": "thesis", "title": "Essays on Rational Social Learning", "author": [ { "family_name": "Huang", "given_name": "Wanying", "orcid": "0009-0000-0413-9171", "clpid": "Huang-Wanying" } ], "thesis_advisor": [ { "family_name": "Tamuz", "given_name": "Omer", "orcid": "0000-0002-0111-0418", "clpid": "Tamuz-O" } ], "thesis_committee": [ { "family_name": "Cvitani\u0107", "given_name": "Jak\u0161a", "orcid": "0000-0001-6651-3552", "clpid": "Cvitani\u0107-J" }, { "family_name": "Echenique", "given_name": "Federico", "orcid": "0000-0002-1567-6770", "clpid": "Echenique-F" }, { "family_name": "Nielsen", "given_name": "Kirby", "orcid": "0000-0003-4536-1021", "clpid": "Nielsen-Kirby" }, { "family_name": "Tamuz", "given_name": "Omer", "orcid": "0000-0002-0111-0418", "clpid": "Tamuz-O" } ], "local_group": [ { "literal": "div_hss" } ], "abstract": "This dissertation contains three essays, each contributing to the study of social learning among rational agents in various contexts.
\r\n\r\nIn Chapter 1, I study whether individuals can learn the informativeness of their information technology through social learning. Building on the classic sequential social learning model, I introduce the possibility that a common source is completely uninformative. I then define asymptotic learning as the situation in which an outsider, who observes the actions of all agents, eventually distinguishes between uninformative and informative sources. I show that asymptotic learning in this setting is not guaranteed; it depends crucially on the relative tail distributions of private beliefs induced by uninformative and informative signals. Furthermore, I identify the phenomenon of perpetual disagreement as the cause of learning and provide a characterization of learning in the canonical Gaussian environment.
\r\n\r\nIn Chapter 2, co-authored with Omer Tamuz and Philipp Strack, we study the asymptotic rate at which the probability of a group of long-lived, rational agents in a social network taking the correct action converges to one. In every period, after observing the past actions of his neighbors, each agent receives a private signal, and chooses an action whose payoff depends only on the state. Since equilibrium actions depend on higher-order beliefs, characterizing agents' behavior becomes difficult. Nevertheless, we show that the rate of learning in any equilibrium is bounded from above by a constant, regardless of the size and shape of the network, the utility function, and the patience of the agents. This bound only depends on the private signal distribution.
\r\n \r\n\r\nIn Chapter 3, I study how fads emerge from social learning in a changing environment. I consider a simple sequential learning model in which rational agents arrive in order, each acting only once, and the underlying unknown state is constantly evolving. Each agent receives a private signal, observes all past actions of others, and chooses an action to match the current state. Because the state changes over time, cascades cannot last forever, and actions also fluctuate. I show that despite the rise of temporary information cascades, in the long run, actions change more often than the state. This provides a theoretical foundation for faddish behavior in which people often change their actions more frequently than necessary.
", "doi": "10.7907/p1xt-ys43", "publication_date": "2024", "thesis_type": "phd", "thesis_year": "2024" }, { "id": "thesis:16103", "collection": "thesis", "collection_id": "16103", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06112023-083707212", "primary_object_url": { "basename": "Senior_Thesis_Gorokhovsky.pdf", "content": "final", "filesize": 417535, "license": "other", "mime_type": "application/pdf", "url": "/16103/1/Senior_Thesis_Gorokhovsky.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Convergence of Time-Inhomogeneous Random Walks on Finite Groups with Applications to Universality for Random Groups", "author": [ { "family_name": "Gorokhovsky", "given_name": "Elia Peter", "orcid": "0000-0001-5901-9783", "clpid": "Gorokhovsky-Elia-Peter" } ], "thesis_advisor": [ { "family_name": "Tamuz", "given_name": "Omer", "orcid": "0000-0002-0111-0418", "clpid": "Tamuz-O" } ], "thesis_committee": [ { "family_name": "Tamuz", "given_name": "Omer", "orcid": "0000-0002-0111-0418", "clpid": "Tamuz-O" }, { "family_name": "Mazel-Gee", "given_name": "Aaron", "orcid": "0000-0002-7618-741X", "clpid": "MazelGee-Aaron" }, { "family_name": "Song", "given_name": "Antoine Y.", "clpid": "Song-Antoine" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We study time-inhomogeneous random walks on finite groups in the case where each random walk step need not be supported on a generating set of the group. When the supports of the random walk steps satisfy a natural condition involving normal subgroups of quotients of the group, we show that the random walk converges to the uniform distribution on the group, and give bounds for the convergence rate using spectral properties of the random walk steps. As applications, we prove a general universality theorem for quotients of the free group on n generators as n \u2192 \u221e, and another universality theorem for cokernels of random integer matrices with dependent entries.", "doi": "10.7907/s4c4-hk39", "publication_date": "2023", "thesis_type": "senior_major", "thesis_year": "2023" }, { "id": "thesis:15250", "collection": "thesis", "collection_id": "15250", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06012023-000259864", "primary_object_url": { "basename": "hann-caruthers_wade_2023.pdf", "content": "final", "filesize": 545003, "license": "other", "mime_type": "application/pdf", "url": "/15250/1/hann-caruthers_wade_2023.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Essays on Social Learning and Social Choice", "author": [ { "family_name": "Hann-Caruthers", "given_name": "Wade Daniel", "orcid": "0000-0002-4273-6249", "clpid": "Hann-Caruthers-Wade-Daniel" } ], "thesis_advisor": [ { "family_name": "Tamuz", "given_name": "Omer", "orcid": "0000-0002-0111-0418", "clpid": "Tamuz-O" } ], "thesis_committee": [ { "family_name": "Agranov", "given_name": "Marina", "orcid": "0000-0002-0642-7975", "clpid": "Agranov-M" }, { "family_name": "Tamuz", "given_name": "Omer", "orcid": "0000-0002-0111-0418", "clpid": "Tamuz-O" }, { "family_name": "Pomatto", "given_name": "Luciano", "orcid": "0000-0002-4331-8436", "clpid": "Pomatto-L" }, { "family_name": "Rosenthal", "given_name": "Jean-Laurent", "clpid": "Rosenthal-J-L" } ], "local_group": [ { "literal": "div_hss" } ], "abstract": "This dissertation contains three essays, two which contribute to the study of social learning (Chapters 1 and 2) and one which contributes to the study of social choice (Chapter 3).
\r\n\r\nIn Chapter 1, I introduce a fully rational model of social learning on networks with endogenous action timing. I show that the structure of the network can play an important role in the aggregation of information. When the social network contains high-degree vertices, agents can be arbitrarily likely to make good choices. In contrast, when the social network is linear, there is a bound on how likely agents are to make good choices which holds regardless of how patient they are. The main contribution of this chapter is the identification of a novel mechanism through which strategic behavior can substantially impede the flow of information through a social network.
\r\n\r\nIn Chapter 2, co-authored with Vadim Martynov and Omer Tamuz, we study the asymptotic rate at which the probability of taking the correct action converges to 1 in the classical sequential learning model with unbounded signals. We provide a characterization of the asymptotic law of motion of the public belief, and we use this characterization to show that convergence occurs more slowly than when agents directly observe private signals, and that the expected time until the last incorrect action can be finite or infinite.
\r\n\r\nIn Chapter 3, co-authored with Laurent Bartholdi, Maya Josyula, Omer Tamuz, and Leeat Yariv, we introduce equitability as a less stringent alternative to symmetry for modeling egalitarianism in voting rules. We then use techniques from group theory to show that equitable voting rules can have minimal winning coalitions comprising a vanishing fraction of the population, but they cannot be smaller than the square root of the population size.
", "doi": "10.7907/4v8h-z740", "publication_date": "2023", "thesis_type": "phd", "thesis_year": "2023" }, { "id": "thesis:14206", "collection": "thesis", "collection_id": "14206", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05302021-185858708", "type": "thesis", "title": "Analogues of Amenability", "author": [ { "family_name": "Frisch", "given_name": "Joshua Robert", "clpid": "Frisch-Joshua-Ropbert" } ], "thesis_advisor": [ { "family_name": "Tamuz", "given_name": "Omer", "orcid": "0000-0002-0111-0418", "clpid": "Tamuz-O" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "thesis_committee": [ { "family_name": "Katz", "given_name": "Nets H.", "orcid": "0000-0002-6239-5429", "clpid": "Katz-N-H" }, { "family_name": "Vidny\u00e1nszky", "given_name": "Zolt\u00e1n", "orcid": "0000-0001-8168-9353", "clpid": "Vidny\u00e1nszky-Zolt\u00e1n" }, { "family_name": "Tamuz", "given_name": "Omer", "orcid": "0000-0002-0111-0418", "clpid": "Tamuz-O" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In this thesis I study multiple notions related to and inspired by amenability coming from the points of the view of random walks on groups, dynamical systems, Borel equivalence relations, and descriptive linear algebra. In particular I study notions related to harmonic functions, invariant measures, hyperfiniteness, and dichotomy theorems.", "doi": "10.7907/yymw-xd55", "publication_date": "2021", "thesis_type": "phd", "thesis_year": "2021" }, { "id": "thesis:13701", "collection": "thesis", "collection_id": "13701", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05122020-180653707", "primary_object_url": { "basename": "Vadim_Martynov_Thesis_Caltech_Thesis_LaTeX_Template__with_logo___1_ (2).pdf", "content": "final", "filesize": 923929, "license": "other", "mime_type": "application/pdf", "url": "/13701/1/Vadim_Martynov_Thesis_Caltech_Thesis_LaTeX_Template__with_logo___1_ (2).pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Essays on Social Learning and Networks", "author": [ { "family_name": "Martynov", "given_name": "Vadim Vadimovich", "orcid": "0000-0001-5357-4161", "clpid": "Martynov-Vadim-Vadimovich" } ], "thesis_advisor": [ { "family_name": "Tamuz", "given_name": "Omer", "orcid": "0000-0002-0111-0418", "clpid": "Tamuz-O" } ], "thesis_committee": [ { "family_name": "Pomatto", "given_name": "Luciano", "orcid": "0000-0002-4331-8436", "clpid": "Pomatto-L" }, { "family_name": "Tamuz", "given_name": "Omer", "orcid": "0000-0002-0111-0418", "clpid": "Tamuz-O" }, { "family_name": "Echenique", "given_name": "Federico", "orcid": "0000-0002-1567-6770", "clpid": "Echenique-F" }, { "family_name": "Cvitani\u0107", "given_name": "Jak\u0161a", "orcid": "0000-0001-6651-3552", "clpid": "Cvitani\u0107-J" } ], "local_group": [ { "literal": "div_hss" } ], "abstract": "This thesis offers a contribution to the study of Social Learning and Networks. It studies information aggregation and its effect on individual's actions (Chapter 2, 3) and social network (Chapter 4).
\r\n\r\nChapter 2, co-authored with Omer Tamuz and Wade Hann-Caruthers, studies how quickly does the public belief converge to its true value when agents are able to observe actions of their predecessors. In the classical herding literature, agents receive a private signal regarding a binary state of nature, and sequentially choose an action, after observing the actions of their predecessors. When the informativeness of private signals is unbounded, it is known that agents converge to the correct action and correct belief. We study how quickly convergence occurs, and show that it happens more slowly than it does when agents observe signals. However, we also show that the speed of learning from actions can be arbitrarily close to the speed of learning from signals. In particular, the expected time until the agents stop taking the wrong action can be either finite or infinite, depending on the private signal distribution. In the canonical case of Gaussian private signals, we calculate the speed of convergence precisely, and show explicitly that, in this case, learning from actions is significantly slower than learning from signals.
\r\n\r\nIn Chapter 3, I investigate how social planning can reduce the inefficiencies of social learning, stemming from herding and informational cascades. A social planner is introduced to the classical sequential social learning model. She can tax or subsidize players' actions in order to maximize social welfare, a discounted sum of agents' utilities. We solve or accurately approximate the expected utility of the social planner and the optimal pricing strategy for various signal distributions. In equilibrium, it is optimal to increase the price for the better action, causing a reduction in current agent's utility, but also a net gain, due to the information this action reveals. The addition of the social planner significantly improves social welfare and the asymptotic speed of learning.
\r\n\r\nChapter 4 analyzes how different types of social connections between people shape their social networks. There are two possible types of ties between individuals, strong and weak, that differ in maintenance costs and reliability. A network formation game is played in which agents choose the number of ties of each type to maximize their chances of hearing about a new job opportunity. We find that in equilibrium, people maintain both types of connections, which was not explained in previous theoretical models. Furthermore, in the socially optimal symmetric network, there are more strong ties than in the equilibrium one.
", "doi": "10.7907/0m03-b330", "publication_date": "2020", "thesis_type": "phd", "thesis_year": "2020" }, { "id": "thesis:13707", "collection": "thesis", "collection_id": "13707", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05162020-233411502", "primary_object_url": { "basename": "Pooya_Vahidi_Ferdowsi_2020_thesis.pdf", "content": "final", "filesize": 500636, "license": "other", "mime_type": "application/pdf", "url": "/13707/1/Pooya_Vahidi_Ferdowsi_2020_thesis.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Strongly Amenable Groups, Choquet-Deny Groups, and the Infinite Conjugacy Class Property", "author": [ { "family_name": "Vahidi Ferdowsi", "given_name": "Pooya", "orcid": "0000-0002-3477-9492", "clpid": "Pooya-Vahidi-Ferdowsi" } ], "thesis_advisor": [ { "family_name": "Tamuz", "given_name": "Omer", "orcid": "0000-0002-0111-0418", "clpid": "Tamuz-O" } ], "thesis_committee": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Tamuz", "given_name": "Omer", "orcid": "0000-0002-0111-0418", "clpid": "Tamuz-O" }, { "family_name": "Frank", "given_name": "Rupert L.", "orcid": "0000-0001-7973-4688", "clpid": "Frank-R-L" }, { "family_name": "Panagiotopoulos", "given_name": "Aristotelis", "orcid": "0000-0002-7695-4842", "clpid": "Panagiotopoulos-Aristotelis" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis consists of two main parts. In the first part, we study a space of symbolic dynamical systems for countable discrete ICC groups and show that minimal proximal actions in that space are generic. This study leads to a characterization of countable discrete strongly amenable groups; a countable discrete group is strongly amenable if and only if it has no ICC quotients.
\r\n\r\nIn the second part, we show that a countable discrete group is Choquet-Deny if and only if it has no ICC quotients, where a group is called Choquet-Deny if the Poisson boundary of every non-degenerate measure on the group is trivial. Combining the aforementioned results, we get that a countable discrete group is Choquet-Deny if and only if it is strongly amenable.
\r\n\r\nIn the case of finitely generated groups, by an old result due to McLain (1956) and Duguid and McLain (1956) and our classifications, we see that strongly amenable groups and Choquet-Deny groups are the same as virtually nilpotent groups.
", "doi": "10.7907/58t8-h497", "publication_date": "2020", "thesis_type": "phd", "thesis_year": "2020" } ]