Exchangeable equilibria

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 183-188).The main contribution of this thesis is a new solution concept for symmetric games (of complete information in strategic form), the exchangeable equilibrium. This is an intermediate notion between symmetric Nash and symmetric correlated equilibrium. While a variety of weaker solution concepts than correlated equilibrium and a variety of refinements of Nash equilibrium are known, there is little previous work on "interpolating" between Nash and correlated equilibrium. Several game-theoretic interpretations suggest that exchangeable equilibria are natural objects to study. Moreover, these show that the notion of symmetric correlated equilibrium is too weak and exchangeable equilibrium is a more natural analog of correlated equilibrium for symmetric games. The geometric properties of exchangeable equilibria are a mix of those of Nash and correlated equilibria. The set of exchangeable equilibria is convex, compact, and semi-algebraic, but not necessarily a polytope. A variety of examples illustrate how it relates to the Nash and correlated equilibria. The same ideas which lead to the notion of exchangeable equilibria can be used to construct tighter convex relaxations of the symmetric Nash equilibria as well as convex relaxations of the set of all Nash equilibria in asymmetric games. These have similar mathematical properties to the exchangeable equilibria. An example game reveals an algebraic obstruction to computing exact exchangeable equilibria, but these can be approximated to any degree of accuracy in polynomial time. On the other hand, optimizing a linear function over the exchangeable equilibria is NP-hard. There are practical linear and semidefinite programming heuristics for both problems. A secondary contribution of this thesis is the computation of extreme points of the set of correlated equilibria in a simple family of games. These examples illustrate that in finite games there can be factorially many more extreme correlated equilibria than extreme Nash equilibria, so enumerating extreme correlated equilibria is not an effective method for enumerating extreme Nash equilibria. In the case of games with a continuum of strategies and polynomial utilities, the examples illustrate that while the set of Nash equilibria has a known finite-dimensional description in terms of moments, the set of correlated equilibria admits no such finite-dimensional characterization.by Noah D. Stein.Ph.D

Characterization and computation of equilibria in infinite games

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2007.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 79-82).Broadly, we study continuous games (those with continuous strategy spaces and utility functions) with a view towards computation of equilibria. We cover all of the game-theoretic background needed to understand these results in detail. Then we present new work, which can be divided into three parts. First, it is known that arbitrary continuous games may have arbitrarily complicated equilibria, so we investigate some properties of games with polynomial utility functions and a class of games with polynomial-like utility functions called separable games. We prove new bounds on the complexity of equilibria of separable games in terms of the complexity of the utility functions. In order to measure this complexity we propose a new definition for the rank of a continuous game; when applied to the case of finite games this improves on the results known in that setting. Furthermore, we prove a characterization theorem showing that several conditions which are necessary for a game to possess a finite-dimensional representation each define the class of separable games precisely, providing evidence that separable games are the natural class of continuous games in which to study computation. The characterization theorem also provides a natural connection between separability and the notion of the rank of a game. Second, we apply this theory to give an algorithm for computing e-Nash equilibria of two-player separable games with continuous strategy spaces. While a direct comparison to corresponding algorithms for finite games is not possible, the asymptotic running time in the complexity of the game grows slower for our algorithm than for any known algorithm for finite games.(cont.) Nonetheless, as in finite games, computing e-Nash equilibria still appears to be difficult for infinite games. Third, we consider computing approximate correlated equilibria in polynomial games. To do so, we first prove several new characterizations of correlated equilibria in continuous games which may be of independent interest. Then we introduce three algorithms for approximating correlated equilibria of polynomial games arbitrarily accurately. These include two discretization algorithms for computing a sample correlated equilibrium: a naive linear programming approach called static discretization which operates without regard to the structure of the game, and a semidefinite programming approach called adaptive discretization which exploits the structure of the game to achieve far better performance in practice. The third algorithm consists of a nested sequence of semidefinite programs converging to a description of the entire set of correlated equilibria.by Noah D. Stein.S.M

Large expert-curated database for benchmarking document similarity detection in biomedical literature search

Document recommendation systems for locating relevant literature have mostly relied on methods developed a decade ago. This is largely due to the lack of a large offline gold-standard benchmark of relevant documents that cover a variety of research fields such that newly developed literature search techniques can be compared, improved and translated into practice. To overcome this bottleneck, we have established the RElevant LIterature SearcH consortium consisting of more than 1500 scientists from 84 countries, who have collectively annotated the relevance of over 180 000 PubMed-listed articles with regard to their respective seed (input) article/s. The majority of annotations were contributed by highly experienced, original authors of the seed articles. The collected data cover 76% of all unique PubMed Medical Subject Headings descriptors. No systematic biases were observed across different experience levels, research fields or time spent on annotations. More importantly, annotations of the same document pairs contributed by different scientists were highly concordant. We further show that the three representative baseline methods used to generate recommended articles for evaluation (Okapi Best Matching 25, Term Frequency-Inverse Document Frequency and PubMed Related Articles) had similar overall performances. Additionally, we found that these methods each tend to produce distinct collections of recommended articles, suggesting that a hybrid method may be required to completely capture all relevant articles. The established database server located at https://relishdb.ict.griffith.edu.au is freely available for the downloading of annotation data and the blind testing of new methods. We expect that this benchmark will be useful for stimulating the development of new powerful techniques for title and title/abstract-based search engines for relevant articles in biomedical research.Peer reviewe

ERGODIC PROPERTIES OF A CLASS OF DISCRETE ABELIAN GROUP EXTENSIONS OF RANK-ONE TRANSFORMATIONS

Abstract. We define a class of discrete Abelian group extensions of rankone transformations and establish necessary and sufficient conditions for these extensions to be power weakly mixing. We show that all members of this class are multiply recurrent. We then study conditions sufficient for showing that Cartesian products of transformations are conservative for a class of invertible infinite measure-preserving transformations and provide examples of these transformations. 1

Macromolecular modeling and design in Rosetta: recent methods and frameworks

The Rosetta software for macromolecular modeling, docking and design is extensively used in laboratories worldwide. During two decades of development by a community of laboratories at more than 60 institutions, Rosetta has been continuously refactored and extended. Its advantages are its performance and interoperability between broad modeling capabilities. Here we review tools developed in the last 5 years, including over 80 methods. We discuss improvements to the score function, user interfaces and usability. Rosetta is available at .This Perspective reviews tools developed over the past five years in the macromolecular modeling, docking and design software Rosetta

Beyond the imitation game: Quantifying and extrapolating the capabilities of language models

Language models demonstrate both quantitative improvement and new qualitative capabilities with increasing scale. Despite their potentially transformative impact, these new capabilities are as yet poorly characterized. In order to inform future research, prepare for disruptive new model capabilities, and ameliorate socially harmful effects, it is vital that we understand the present and near-future capabilities and limitations of language models. To address this challenge, we introduce the Beyond the Imitation Game benchmark (BIG-bench). BIG-bench currently consists of 204 tasks, contributed by 442 authors across 132 institutions. Task topics are diverse, drawing problems from linguistics, childhood development, math, common-sense reasoning, biology, physics, social bias, software development, and beyond. BIG-bench focuses on tasks that are believed to be beyond the capabilities of current language models. We evaluate the behavior of OpenAI's GPT models, Google-internal dense transformer architectures, and Switch-style sparse transformers on BIG-bench, across model sizes spanning millions to hundreds of billions of parameters. In addition, a team of human expert raters performed all tasks in order to provide a strong baseline. Findings include: model performance and calibration both improve with scale, but are poor in absolute terms (and when compared with rater performance); performance is remarkably similar across model classes, though with benefits from sparsity; tasks that improve gradually and predictably commonly involve a large knowledge or memorization component, whereas tasks that exhibit "breakthrough" behavior at a critical scale often involve multiple steps or components, or brittle metrics; social bias typically increases with scale in settings with ambiguous context, but this can be improved with prompting