66,148 research outputs found
Cautious Belief and Iterated Admissibility
We define notions of cautiousness and cautious belief to provide epistemic
conditions for iterated admissibility in finite games. We show that iterated
admissibility characterizes the behavioral implications of "cautious
rationality and common cautious belief in cautious rationality" in a terminal
lexicographic type structure. For arbitrary type structures, the behavioral
implications of these epistemic assumptions are characterized by the solution
concept of self-admissible set (Brandenburger, Friedenberg and Keisler 2008).
We also show that analogous conclusions hold under alternative epistemic
assumptions, in particular if cautiousness is "transparent" to the players.
KEYWORDS: Epistemic game theory, iterated admissibility, weak dominance,
lexicographic probability systems. JEL: C72
Nonlinear Adaptive Processes of Growth with General Increments: Attainable and Unattainable Components of Terminal Set
A local asymptotic theory of adaptive processes of growth with general increments is developed for the case when a terminal set consists of more than one connected component. The notions of an attainable and unattainable component are introduced. Sufficient conditions for attainability and unattainability are derived. The limit theorems are applied in the investigation of the rate of convergence to singleton stable components. The relation between the obtained results and the study of asymptotic properties of stochastic quasi-gradient algorithms in non-convex multiextremum problems is discussed. Specifically, the developed approach is used to explore the limit behavior of iterations in the Fabian modification of the Kiefer-Wolfowitz algorithm
Dynamics and Coalitions in Sequential Games
We consider N-player non-zero sum games played on finite trees (i.e.,
sequential games), in which the players have the right to repeatedly update
their respective strategies (for instance, to improve the outcome wrt to the
current strategy profile). This generates a dynamics in the game which may
eventually stabilise to a Nash Equilibrium (as with Kukushkin's lazy
improvement), and we argue that it is interesting to study the conditions that
guarantee such a dynamics to terminate.
We build on the works of Le Roux and Pauly who have studied extensively one
such dynamics, namely the Lazy Improvement Dynamics. We extend these works by
first defining a turn-based dynamics, proving that it terminates on subgame
perfect equilibria, and showing that several variants do not terminate. Second,
we define a variant of Kukushkin's lazy improvement where the players may now
form coalitions to change strategies. We show how properties of the players'
preferences on the outcomes affect the termination of this dynamics, and we
thereby characterise classes of games where it always terminates (in particular
two-player games).Comment: In Proceedings GandALF 2017, arXiv:1709.0176
Completion, closure, and density relative to a monad, with examples in functional analysis and sheaf theory
Given a monad T on a suitable enriched category B equipped with a proper
factorization system (E,M), we define notions of T-completion, T-closure, and
T-density. We show that not only the familiar notions of completion, closure,
and density in normed vector spaces, but also the notions of sheafification,
closure, and density with respect to a Lawvere-Tierney topology, are instances
of the given abstract notions. The process of T-completion is equally the
enriched idempotent monad associated to T (which we call the idempotent core of
T), and we show that it exists as soon as every morphism in B factors as a
T-dense morphism followed by a T-closed M-embedding. The latter hypothesis is
satisfied as soon as B has certain pullbacks as well as wide intersections of
M-embeddings. Hence the resulting theorem on the existence of the idempotent
core of an enriched monad entails Fakir's existence result in the non-enriched
case, as well as adjoint functor factorization results of Applegate-Tierney and
Day
Introduction to Categories and Categorical Logic
The aim of these notes is to provide a succinct, accessible introduction to
some of the basic ideas of category theory and categorical logic. The notes are
based on a lecture course given at Oxford over the past few years. They contain
numerous exercises, and hopefully will prove useful for self-study by those
seeking a first introduction to the subject, with fairly minimal prerequisites.
The coverage is by no means comprehensive, but should provide a good basis for
further study; a guide to further reading is included. The main prerequisite is
a basic familiarity with the elements of discrete mathematics: sets, relations
and functions. An Appendix contains a summary of what we will need, and it may
be useful to review this first. In addition, some prior exposure to abstract
algebra - vector spaces and linear maps, or groups and group homomorphisms -
would be helpful.Comment: 96 page
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