A Subjective Foundation of Objective Probability

Proceedings of the National Academy of Sciences, 2013

We consider decision makers who know that payo¤ relevant observations are generated by a process that belongs to a given class M , as postulated in Wald [36]. We incorporate this Waldean piece of objective information within an otherwise subjective setting a la Savage [33] and show that this leads to a two-stage subjective expected utility model that accounts for both state and model uncertainty.

Journal of Mathematical Psychology, 2011

This paper proposes a theory of subjective expected utility based on primitives only involving the fact that an act can be judged either "attractive" or "unattractive". We give conditions implying that there are a utility function on the set of consequences and a probability distribution on the set of states such that attractive acts have a subjective expected utility above some threshold. The numerical representation that is obtained has strong uniqueness properties.

The Review of Economic Studies, 1991

arXiv, 2014

We provide foundations for decisions in face of unlikely events by extending the standard framework of Savage to include preferences indexed by a family of events. We derive a subjective lexicographic expected utility representation which allows for infinitely many lexicographically ordered levels of events and for event-dependent attitudes toward risk. Our model thus provides foundations for models in finance that rely on different attitudes toward risk (e.g. Skiadas [9]) and for off-equilibrium reasonings in infinite dynamic games, thus extending and generalizing the analysis in Blume, Brandenburger and Dekel [3].

Synthese, 2018

In his classic book Savage (1954, 1972) develops a formal system of rational decision making. It is based on (i) a set of possible states of the world, (ii) a set of consequences, (iii) a set of acts, which are functions from states to consequences, and (iv) a preference relation over the acts, which represents the preferences of an idealized rational agent. The goal and the culmination of the enterprise is a representation theorem: Any preference relation that satisfies certain arguably acceptable postulates determines a (finitely additive) probability distribution over the states and a utility assignment to the consequences, such that the preferences among acts are determined by their expected utilities. Additional problematic assumptions are however required in Savage's proofs. First, there is a Boolean algebra of events (sets of states) which determines the richness of the set of acts. The probabilities are assigned to members of this algebra. Savage's proof requires that this be a σ-algebra (i.e., closed under infinite countable unions and intersections), which makes for an extremely rich preference relation. On Savage's view we should not require subjective probabilities to be σ-additive. He therefore finds the insistence on a σ-algebra peculiar and is unhappy with it. But he sees no way of avoiding it. Second, the assignment of utilities requires the constant act assumption: for every consequence there is a constant act, which produces that consequence in every state. This assumption is known to be highly counterintuitive. The present work contains two mathematical results. The first, and the more difficult one, shows that the σ-algebra assumption can be dropped. 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.

2010

This paper shows that subjective expected utility can be obtained using primitives that are much poorer than a preference relation on the set of acts. Our primitives only involve the fact that an act can be judged either “attractive”, “neutral” or “unattractive”. These categories may be interpreted as denoting the position of an act vis-à-vis a status quo. We give conditions implying that there are a utility function on the set of consequences and a probability distribution on the set of states such that attractive (resp. unattractive) acts have a subjective expected utility that is above (resp. below) some threshold. The numerical representation that is obtained has strong uniqueness properties. Our derivation uses results in conjoint measurement with ordered categories and, hence, we adopt a framework involving a finite set of states.

AIP Conference Proceedings, 2004

Recent literature in the last Maximum Entropy workshop introduced an analogy between cumulative probability distributions and normalized utility functions [1]. Based on this analogy, a utility density function can de defined as the derivative of a normalized utility function. A utility density function is non-negative and integrates to unity. These two properties of a utility density function form the basis of a correspondence between utility and probability, which allows the application of many tools from one domain to the other. For example, La Place's principle of insufficient reason translates to a principle of insufficient preference. The notion of uninformative priors translates to uninformative utility functions about a decision maker's preferences. A natural application of this analogy is a maximum entropy principle to assign maximum entropy utility values. Maximum entropy utility interprets many of the common utility functions based on the preference information needed for their assignment, and helps assign utility values based on partial preference information. This paper reviews maximum entropy utility, provides axiomatic justification for its use, and introduces further results that stem from the duality between probability and utility, such as joint utility density functions, utility inference, and the notion of mutual preference.

Economica, 1997

We assume that a decision-maker has expected utility preferences over a large space which includes some variables not observable by the theorist. These will induce preferences over observable variables, which typically will not have the expected utility form. This paper focuses on implications for multi-period decisions. We show that such preferences are not vulnerable to 'Dutch books'. In particular, we consider preferences arising from non-additive subjective probabilities and show how they can arise as induced preferences.

Erkenntnis, 2014

It is well known that classical, aka ‘sharp’, Bayesian decision theory, which models belief states as single probability functions, faces a number of serious difficulties with respect to its handling of agnosticism. These difficulties have led to the increasing popularity of so-called ‘imprecise’ models of decision-making, which represent belief states as sets of probability functions. In a recent paper, however, Adam Elga has argued in favour of a putative normative principle of sequential choice that he claims to be borne out by the sharp model but not by any promising incarnation of its imprecise counterpart. After first pointing out that Elga has fallen short of establishing that his principle is indeed uniquely borne out by the sharp model, I cast aspersions on its plausibility. I show that a slight weakening of the principle is satisfied by at least one, but interestingly not all, varieties of the imprecise model and point out that Elga has failed to motivate his stronger commitment.

CalTech WP, 2005

This note shows that if the space of events is sufficiently rich and the subjective probability function of each individual is non-atomic, then there is a σ-algebra of events over which everyone will have the same probability function, and moreover, the range of these probabilities is the whole [0, 1] segment. 1 See Harsanyi [9], and more recently, Epstein and Segal [6], Broome [3], Kamm [11], or Karni and Safra [10].