Forgetting complex propositions

This paper uses possible-world semantics to model the changes that may occur in an agent's knowledge as she loses information. This builds on previous work in which the agent may forget the truth-value of an atomic proposition, to a more general case where she may forget the truth-value of a propositional formula. The generalization poses some challenges, since in order to forget whether a complex proposition $\pi$ is the case, the agent must also lose information about the propositional atoms that appear in it, and there is no unambiguous way to go about this. We resolve this situation by considering expressions of the form $[\boldsymbol{\ddagger} \pi]\varphi$, which quantify over all possible (but minimal) ways of forgetting whether $\pi$. Propositional atoms are modified non-deterministically, although uniformly, in all possible worlds. We then represent this within action model logic in order to give a sound and complete axiomatization for a logic with knowledge and forgetting. Finally, some variants are discussed, such as when an agent forgets $\pi$ (rather than forgets whether $\pi$) and when the modification of atomic facts is done non-uniformly throughout the model

Decrement Operators in Belief Change

While research on iterated revision is predominant in the field of iterated belief change, the class of iterated contraction operators received more attention in recent years. In this article, we examine a non-prioritized generalisation of iterated contraction. In particular, the class of weak decrement operators is introduced, which are operators that by multiple steps achieve the same as a contraction. Inspired by Darwiche and Pearl's work on iterated revision the subclass of decrement operators is defined. For both, decrement and weak decrement operators, postulates are presented and for each of them a representation theorem in the framework of total preorders is given. Furthermore, we present two sub-types of decrement operators

Extending the Harper Identity to Iterated Belief Change

The field of iterated belief change has focused mainly on revision, with the other main operator of AGM belief change theory, i.e. contraction, receiving relatively little attention. In this paper we extend the Harper Identity from single-step change to define iterated contraction in terms of iterated revision. Specifically, just as the Harper Identity provides a recipe for defining the belief set resulting from contracting A in terms of (i) the initial belief set and (ii) the belief set resulting from revision by ¬A, we look at ways to define the plausibility ordering over worlds resulting from contracting A in terms of (iii) the initial plausibility ordering, and (iv) the plausibility ordering resulting from revision by ¬A. After noting that the most straightforward such extension leads to a trivialisation of the space of permissible orderings, we provide a family of operators for combining plausibility orderings that avoid such a result. These operators are characterised in our domain of interest by a pair of intuitively compelling properties, which turn out to enable the derivation of a number of iterated contraction postulates from postulates for iterated revision. We finish by observing that a salient member of this family allows for the derivation of counterparts for contraction of some well known iterated revision operators, as well as for defining new iterated contraction operators

Two-Dimensional Belief Change

The idea of two-dimensional belief change operators is that a belief state is transformed by an input sentence $A$ in such a way that $A$ gets accepted with at least the strength or certainty of a sentence $B$ (the reference sentence). The input of such a transformation may alternatively be conceived as `$B leq A$\u27 [`$B$ less-than-or-equal-to $A$\u27]. This notation makes explicit that the process induced is basically one of doxastic preference change. The principal case of two-dimensional belief change obtains when $B$ is a prior belief which is more strongly accepted than both $A$ and $eg A$, but the non-principal cases are interesting in their own right. Various two-dimensional revision operators were studied by Cantwell (1997, `raising\u27 and `lowering\u27), Fermé and Rott (2003, `revision by comparison\u27), and Rott (2007, `bounded revision\u27). Special choices of a fixed input sentence $A$ or a fixed reference sentence $B$ lead to some well-known unary oparators of belief change: `irrevocable\u27 (aka `radical\u27) revision, `severe withdrawal\u27 (aka `mild contraction\u27), `natural\u27 (aka `conservative\u27) and `lexicographic\u27 (aka `moderate\u27) revision. The talk gives a survey of several variants of two-dimensional belief change and their representations. I argue that two-dimensional belief change operators offer an interesting qualitative model with an expressive power between (all too poor) unary operators and (all too demanding) quantitative models of belief change

The consistency principle for set-valued solutions and a new direction for the theory of equilibrium refinements

game theory;equilibrium theory

A cognitive hierarchy theory of one-shot games: Some preliminary results

Strategic thinking, best-response, and mutual consistency (equilibrium) are three key modelling principles in noncooperative game theory. This paper relaxes mutual consistency to predict how players are likely to behave in in one-shot games before they can learn to equilibrate. We introduce a one-parameter cognitive hierarchy (CH) model to predict behavior in one-shot games, and initial conditions in repeated games. The CH approach assumes that players use k steps of reasoning with frequency f (k). Zero-step players randomize. Players using k (≥ 1) steps best respond given partially rational expectations about what players doing 0 through k - 1 steps actually choose. A simple axiom which expresses the intuition that steps of thinking are increasingly constrained by working memory, implies that f (k) has a Poisson distribution (characterized by a mean number of thinking steps τ ). The CH model converges to dominance-solvable equilibria when τ is large, predicts monotonic entry in binary entry games for τ < 1:25, and predicts effects of group size which are not predicted by Nash equilibrium. Best-fitting values of τ have an interquartile range of (.98,2.40) and a median of 1.65 across 80 experimental samples of matrix games, entry games, mixed-equilibrium games, and dominance-solvable p-beauty contests. The CH model also has economic value because subjects would have raised their earnings substantially if they had best-responded to model forecasts instead of making the choices they did

Essays on Economic Decision Making

My dissertation consists of three chapters and I take different approach in each chapter to investigate economic decision making behavior. The first chapter analyzes individuals strategic decisionmaking when players have replaceable identities and private information in a repeated prisoners dillmma game. The second chapter studies individuals non-strategic decision making when she has incomplete information about her underlying preference in a sequential choice situation. The third chapter experimentally examines a link between an individuals strategic thinking and nonstrategic decision making in a setting designed to elicit beliefs about independent random variables. In the first chapter, I focus on strategic decision making of economic agents when they are replaceable in a repeated prisoners dilemma. I assume that agents have different private information that restricts their set of actions, and that replacement of agent involves change of such private information. In this environment, some agents are required to signal their own private information to induce their opponents cooperative response, which may induce Pareto improvement of their expected continuation payoffs. Except for a trivial equilibrium, we can have non-trivial equilibria supporting cooperative action as a part of the equilibrium play; however, different from the environment with two long-run agents, replaceable agents environment puts a restriction on an existence of the equilibrium in which agents share the risk of type uncertainty equally regardless of the past history. Because of replacement, agents can avoid a full cost of signaling by shifting it to their successor upon their own replacement. As replacement incurs such a situation with a strictly positive probability, the equilibrium cannot avoid failure. In the second chapter I focus on an economic agents optimal decision making in a non-strategic environment. Especially, I study a sequential choice problem where an agents preferences evolve over time. I assume that an agent has an underlying preference, and she learns about her underlying preference depending on her choice histories. Given that an agent makes an optimal decision upon her current available menu, I characterize the sequential choice behavior that follows a Sequential Weak Axiom of Revealed Preference (WARP-S). Using this characterization, I provide criteria for sequential choice data that recovers agents underlying preference. In the third chapter I and my co-author, Duk-Gyoo Kim, focus on a link between an optimal decision making in a non-strategic environment and strategic environment. Our research investigates whether an individual decision maker follows own subjective optimization in a non-strategic decision making, and such a difference in subjective optimization is correlated with strategic decision making pattern. We conducted two separate sessions in the same subject. Each session is designed to identify subjects behavioral pattern in strategic and non-strategic decision making environment respectively. From the data, we observed that subjects behavioral pattern shows significant similarity in two sessions

Notes on contraposing conditionals

The contraposing conditional 'If A then C' is defined by the conjunction of A > C and ~C > ~A, where > is a conditional of the kind studied by Stalnaker, Lewis and others. This idea has recently been explored, under the name 'evidential conditional', in a sequence of papers by Crupi and Iacona and Raidl, and it has been found of independent interest by Booth and Chandler. I discuss various properties of these conditionals and compare them to the 'difference-making conditionals' studied by Rott, which are defined by the conjunction of A > C and not ~A > C. I raise some doubts about Crupi and Iacona's claim that contraposition captures the idea of evidence or support