Heart of DARCness (with Huw Price)
We propose a valid core for the much-disputed thesis that Deliberation Crowds Out Prediction, and identify terminological causes for some of the apparent disputes.
A Simpler and More Realistic Subjective Decision Theory (with Haim Gaifman)
The paper presents two mathematical results. The first, and the more difficult one, shows that the probability measures derived in Savage's theory of expected utility can be defined without the σ-algebra assumption. The second states that, as long as utilities are assigned to finite gambles only, the constant act assumption can be replaced by the more plausible and much weaker assumption that there are at least two non-equivalent constant acts. The second result also employs a novel way of deriving utilities in Savage-style systems---without appealing to von Neumann-Morgenstern lotteries. The paper discusses the notion of "idealized agent" that underlies Savage's approach, and argues that the simplified system, which is adequate for all the actual purposes for which the system is designed, involves a more realistic notion of an idealized agent.
The Sure-thing Principle and P2
This paper offers a fine analysis of different versions of the well known sure-thing principle. We show that Savage's formal formulation of the principle, i.e., his second postulate (P2), is strictly stronger than what is intended originally.
Frege's Begriffsschrift is Indeed First-order Complete
It is widely taken that the first-order part of Frege's Begriffsschrift is complete. However, there does not seem to have been a formal verification of this received claim. The general concern is that Frege's system is one axiom short in the first-order predicate calculus comparing to, by now, standard first-order theory. Yet Frege has one extra inference rule in his system. Then the question is whether Frege's first-order calculus is still deductively sufficient as far as first-order completeness is concerned. In this short note we confirm that the missing axiom is derivable from his stated axioms and inference rules, and hence the logic system in the Begriffsschrift is indeed first-order complete.
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Elements of Bayesian Decision Theory
An introduction to classical Bayesian Decision Theory including von Neumann–Morgenstern utility theory, Anscombe-Aumann model, and Savage's theory of subjective expected utility.
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
A nice diagram of various modal systems.