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

Structure of Extreme Correlated Equilibria: a Zero-Sum Example and its Implications

We exhibit the rich structure of the set of correlated equilibria by analyzing the simplest of polynomial games: the mixed extension of matching pennies. We show that while the correlated equilibrium set is convex and compact, the structure of its extreme points can be quite complicated. In finite games the ratio of extreme correlated to extreme Nash equilibria can be greater than exponential in the size of the strategy spaces. In polynomial games there can exist extreme correlated equilibria which are not finitely supported; we construct a large family of examples using techniques from ergodic theory. We show that in general the set of correlated equilibrium distributions of a polynomial game cannot be described by conditions on finitely many moments (means, covariances, etc.), in marked contrast to the set of Nash equilibria which is always expressible in terms of finitely many moments

Separable and Low-Rank Continuous Games

In this paper, we study nonzero-sum separable games, which are continuous games whose payoffs take a sum-of-products form. Included in this subclass are all finite games and polynomial games. We investigate the structure of equilibria in separable games. We show that these games admit finitely supported Nash equilibria. Motivated by the bounds on the supports of mixed equilibria in two-player finite games in terms of the ranks of the payoff matrices, we define the notion of the rank of an n-player continuous game and use this to provide bounds on the cardinality of the support of equilibrium strategies. We present a general characterization theorem that states that a continuous game has finite rank if and only if it is separable. Using our rank results, we present an efficient algorithm for computing approximate equilibria of two-player separable games with fixed strategy spaces in time polynomial in the rank of the game

Correlated Equilibria in Continuous Games: Characterization and Computation

We present several new characterizations of correlated equilibria in games with continuous utility functions. These have the advantage of being more computationally and analytically tractable than the standard definition in terms of departure functions. We use these characterizations to construct effective algorithms for approximating a single correlated equilibrium or the entire set of correlated equilibria of a game with polynomial utility functions.Comment: Games and Economic Behavior, In Press, Accepted Manuscript, Available online 16 April 201

Potential therapeutic implications of new insights into respiratory syncytial virus disease

Viral bronchiolitis is the most common cause of hospitalization in infants under 6 months of age, and 70% of all cases of bronchiolitis are caused by respiratory syncytial virus (RSV). Early RSV infection is associated with respiratory problems such as asthma and wheezing later in life. RSV infection is usually spread by contaminated secretions and infects the upper then lower respiratory tracts. Infected cells release proinflammatory cytokines and chemokines, including IL-1, tumor necrosis factor-α, IL-6, and IL-8. These activate other cells and recruit inflammatory cells, including macrophages, neutrophils, eosinophils, and T lymphocytes, into the airway wall and surrounding tissues. The pattern of cytokine production by T lymphocytes can be biased toward 'T-helper-1' or 'T-helper-2' cytokines, depending on the local immunologic environment, infection history, and host genetics. T-helper-1 responses are generally efficient in antiviral defense, but young infants have an inherent bias toward T-helper-2 responses. The ideal intervention for RSV infection would be preventive, but the options are currently limited. Vaccines based on protein subunits, live attenuated strains of RSV, DNA vaccines, and synthetic peptides are being developed; passive antibody therapy is at present impractical in otherwise healthy children. Effective vaccines for use in neonates continue to be elusive but simply delaying infection beyond the first 6 months of life might reduce the delayed morbidity associated with infantile disease

The Plant Structure Ontology, a Unified Vocabulary of Anatomy and Morphology of a Flowering Plant

Formal description of plant phenotypes and standardized annotation of gene expression and protein localization data require uniform terminology that accurately describes plant anatomy and morphology. This facilitates cross species comparative studies and quantitative comparison of phenotypes and expression patterns. A major drawback is variable terminology that is used to describe plant anatomy and morphology in publications and genomic databases for different species. The same terms are sometimes applied to different plant structures in different taxonomic groups. Conversely, similar structures are named by their species-specific terms. To address this problem, we created the Plant Structure Ontology (PSO), the first generic ontological representation of anatomy and morphology of a flowering plant. The PSO is intended for a broad plant research community, including bench scientists, curators in genomic databases, and bioinformaticians. The initial releases of the PSO integrated existing ontologies for Arabidopsis (Arabidopsis thaliana), maize (Zea mays), and rice (Oryza sativa); more recent versions of the ontology encompass terms relevant to Fabaceae, Solanaceae, additional cereal crops, and poplar (Populus spp.). Databases such as The Arabidopsis Information Resource, Nottingham Arabidopsis Stock Centre, Gramene, MaizeGDB, and SOL Genomics Network are using the PSO to describe expression patterns of genes and phenotypes of mutants and natural variants and are regularly contributing new annotations to the Plant Ontology database. The PSO is also used in specialized public databases, such as BRENDA, GENEVESTIGATOR, NASCArrays, and others. Over 10,000 gene annotations and phenotype descriptions from participating databases can be queried and retrieved using the Plant Ontology browser. The PSO, as well as contributed gene associations, can be obtained at www.plantontology.org.This is the publisher’s final pdf. The published article is copyrighted by the American Society of Plant Biologists and can be found at: http://www.plantphysiol.org/

Crystal Structure of HIV-1 gp41 Including Both Fusion Peptide and Membrane Proximal External Regions

The HIV-1 envelope glycoprotein (Env) composed of the receptor binding domain gp120 and the fusion protein subunit gp41 catalyzes virus entry and is a major target for therapeutic intervention and for neutralizing antibodies. Env interactions with cellular receptors trigger refolding of gp41, which induces close apposition of viral and cellular membranes leading to membrane fusion. The energy released during refolding is used to overcome the kinetic barrier and drives the fusion reaction. Here, we report the crystal structure at 2 Å resolution of the complete extracellular domain of gp41 lacking the fusion peptide and the cystein-linked loop. Both the fusion peptide proximal region (FPPR) and the membrane proximal external region (MPER) form helical extensions from the gp41 six-helical bundle core structure. The lack of regular coiled-coil interactions within FPPR and MPER splay this end of the structure apart while positioning the fusion peptide towards the outside of the six-helical bundle and exposing conserved hydrophobic MPER residues. Unexpectedly, the section of the MPER, which is juxtaposed to the transmembrane region (TMR), bends in a 90°-angle sideward positioning three aromatic side chains per monomer for membrane insertion. We calculate that this structural motif might facilitate the generation of membrane curvature on the viral membrane. The presence of FPPR and MPER increases the melting temperature of gp41 significantly in comparison to the core structure of gp41. Thus, our data indicate that the ordered assembly of FPPR and MPER beyond the core contributes energy to the membrane fusion reaction. Furthermore, we provide the first structural evidence that part of MPER will be membrane inserted within trimeric gp41. We propose that this framework has important implications for membrane bending on the viral membrane, which is required for fusion and could provide a platform for epitope and lipid bilayer recognition for broadly neutralizing gp41 antibodies

The ERI-6/7 Helicase Acts at the First Stage of an siRNA Amplification Pathway That Targets Recent Gene Duplications

Endogenous small interfering RNAs (siRNAs) are a class of naturally occuring regulatory RNAs found in fungi, plants, and animals. Some endogenous siRNAs are required to silence transposons or function in chromosome segregation; however, the specific roles of most endogenous siRNAs are unclear. The helicase gene eri-6/7 was identified in the nematode Caenorhabditis elegans by the enhanced response to exogenous double-stranded RNAs (dsRNAs) of the null mutant. eri-6/7 encodes a helicase homologous to small RNA factors Armitage in Drosophila, SDE3 in Arabidopsis, and Mov10 in humans. Here we show that eri-6/7 mutations cause the loss of 26-nucleotide (nt) endogenous siRNAs derived from genes and pseudogenes in oocytes and embryos, as well as deficiencies in somatic 22-nucleotide secondary siRNAs corresponding to the same loci. About 80 genes are eri-6/7 targets that generate the embryonic endogenous siRNAs that silence the corresponding mRNAs. These 80 genes share extensive nucleotide sequence homology and are poorly conserved, suggesting a role for these endogenous siRNAs in silencing of and thereby directing the fate of recently acquired, duplicated genes. Unlike most endogenous siRNAs in C. elegans, eri-6/7–dependent siRNAs require Dicer. We identify that the eri-6/7–dependent siRNAs have a passenger strand that is ∼19 nt and is inset by ∼3–4 nts from both ends of the 26 nt guide siRNA, suggesting non-canonical Dicer processing. Mutations in the Argonaute ERGO-1, which associates with eri-6/7–dependent 26 nt siRNAs, cause passenger strand stabilization, indicating that ERGO-1 is required to separate the siRNA duplex, presumably through endonucleolytic cleavage of the passenger strand. Thus, like several other siRNA–associated Argonautes with a conserved RNaseH motif, ERGO-1 appears to be required for siRNA maturation