Decision Theory and the Future of Artificial Intelligence
(edited with Huw Price and Stephan Hartmann)
There is increasing interest in the challenges of ensuring that the long-term development of artificial intelligence (AI) is safe and beneficial. Moreover, despite different perspectives, there is much common ground between mathematical and philosophical decision theory, on the one hand, and AI, on the other. The aim of the special issue is to explore links and joint research at the nexus between decision theory and AI, broadly construed.
Special issue of Synthese
Countable Additivity, Idealization, and Conceptual Realism
This paper addresses the issue of finite versus countable additivity in Bayesian probability and decision Theory -- Savage's theory of subjective expected utility and personal probability in particular. I show that Savage's reason for not requiring countable additivity in his theory is inconclusive. The assessment leads to an analysis of various highly idealized assumptions commonly involved in Bayesian theory, where I argue that a healthy dose of, what I call, conceptual realism is often helpful in understanding the interpretational value of sophisticated mathematical structures employed in applied sciences like decision theory. In the last part, I introduce countable additivity into Savage's theory and explore some properties in relation to other axioms of the system.
'Click!' Bait for Causalists
(with Huw Price)
Causalists and Evidentialists can agree about the right course of action in an (apparent) Newcomb problem, if the causal facts are not as initially they seem. If declining $1,000 causes the Predictor to have placed $1m in the opaque box, CDT agrees with EDT that one-boxing is rational. This creates a difficulty for Causalists. We explain the problem with reference to Dummett's work on backward causation and Lewis's on chance and crystal balls. We show that the possibility that the causal facts might be properly judged to be non-standard in Newcomb problems leads to a dilemma for Causalism. One horn embraces a subjectivist understanding of causation, in a sense analogous to Lewis's own subjectivist conception of objective chance. In this case the analogy with chance reveals a terminological choice point, such that either (i) CDT is completely reconciled with EDT, or (ii) EDT takes precedence in the cases in which the two theories give different recommendations. The other horn of the dilemma rejects subjectivism, but now the analogy with chance suggests that it is simply mysterious why causation so construed should constrain rational action.
Heart of DARCness
(with Huw Price)
There is a long-standing disagreement in the philosophy of probability and Bayesian decision theory about whether an agent can hold a meaningful credence about an upcoming action, while she deliberates about what to do. Can she believe that it is, say, 70% probable that she will do A, while she chooses whether to do A? No, say some philosophers, for Deliberation Crowds Out Prediction (DCOP), but others disagree. In this paper, we propose a valid core for DCOP, 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.