Articles

A Simpler and More Realistic Subjective Decision Theory (with Haim Gaifman)
This article provides a simplification of Savage’s theory of subjective expected utilities. Savage motivated his system using a number of real-life examples including the well-known “omelet problem”, where the agents, guided by various rationality principles, are faced with finitary decision situations with finite sets of states, consequences, and acts. Yet in deriving the representation theorem, the system is supplemented with a number of structural assumptions, which leads to the construction of various infinitary structures. These rich mathematical constructs, we argue, may come in conflict with the finitary decision situations the theory initially intends to model. The present work provides a series of simplifications of Savage’s system within finitary framework. In particular, we show, by developing a novel technique of tripartition trees that which allows us to derive numerical probabilities in Savage style systems without appealing to a σ-algebra.
Epistemic Wild Card
In this paper, I provide a defense of the thesis that, while deliberating about what to do, one cannot rationally have subjective probabilities for what she is about to do (action credences as it is called). I argue that action credences cannot hold in classical subjectivist decision theory we owe to writers like Frank Ramsey and L. J. Savage.
On Countably Additive Subjective Probabilities
The present work concerns the issue of finitely versus countably additive probability in Savage’s theory of personal probability. The paper is divided into three main parts. First, we comment, by providing a brief historical review, on Savage’s reasons for no requiring subjective probability derived in his decision model to be countably additive. It is pointed out that Savage’s argument for avoiding countable additivity is inconclusive due to an oversight of set-theoretic details. In the second part, we discuss some defects of employing merely finitely additive probability measures in Savage’s system. A diagnosis is then attempted which links the insufficiency of finite additivity to the failure of continuity in a rich probability space. The analyses then lead, in the third part, to the introduction of countable additivity as a formal assumption of the theory, and we provide an analysis of the utility extension with countable additivity in sight.

e-Book

Elements of Bayesian Decision Theory
An introduction to von Neumann–Morgenstern utility theory, Anscombe-Aumann model, and Savage’s theory of subjective expected utility.

Notes

Context-dependent Utilities (with H. Gaifman)
A solution to the constant-act problem.
What Epistemic Models Cannot Be
A Puzzle in Hintikka’s system
On Completeness in the Begriffsschrift
A short proof of completeness of the first-order part of Begriffsschrift

Expositions

Incompleteness Results and Provability Logic
Gödel’s incompleteness results, Löb’s theorem, and Solovay’s arithmetic completeness theorems.
Uniform Distribution over the Natural Numbers
Set-theoretic construction of uniform distribution over the natural numbers.
Fixed-point Theorems in Game Theory
Sperner’s lemma, Brouwer and Kakutani fixed-point theorems used in showing Nash.
Visualizing Modal Systems
Just a nice diagram of various modal systems