In fact, only those propositions theagent considers to be possible (in the sense that she assigns them aprobability greater than zero) are, according to Jeffrey’stheory, included in her preference ordering. This disanalogy is due to the fact that there is nosense in which the \(p_i\)s that \(p\) is evaluated in terms of needto be ultimate outcomes; they can themselves be thought of asuncertain prospects that are evaluated in terms of their differentpossible realisations. The above result may seem remarkable; in particular, the fact that aperson’s preferences can determine a unique probability functionthat represents her beliefs. On a closer look, however, it is evidentthat some of our beliefs can be determined by examining ourpreferences. Suppose you are offered a choice between two lotteries,one that results in you winning a nice prize if a coin comes up headsbut getting nothing if the coin comes up tails, another that resultsin you winning the same prize if the coin comes up tails but gettingnothing if the coin comes up heads. Then assuming that thedesirability of the prize (and similarly the desirability of no prize)is independent of how the coin lands, your preference between the twolotteries should be entirely determined by your comparative beliefsfor the two ways in which the coin can land.
Normative and descriptive
It can actually be seen as a weak version ofIndependence and the Sure Thing Principle, and it plays a similar rolein Jeffrey’s theory. But it is not directly inconsistent withAllais’ preferences, and its plausibility does not depend on thetype of probabilistic independence that the STP implies. The postulaterequires that no proposition be strictly better or worse than all ofits possible realisations, which seems to be a reasonablerequirement.
- More generally, although people rarelythink of it this way, they constantly take gambles that have minusculechances of leading to immanent death, and correspondingly very highchances of some modest reward.
- On a closer look, however, it isevident that some of our beliefs can be determined by examining ourpreferences.
- This amounts to a minimal accountof rationality, one that sets aside more substantialquestions about appropriate values and preferences, and reasonablebeliefs, given the situation at hand.
For a domain of options we speak of anagent’s preference ordering, this being the ordering ofoptions that is generated by the agent’s preference between anytwo options in that domain. Ontologically bolder incarnations of the view have it that agents areso describable because they really do have degrees of beliefand desires, introspectively familiar psychological states, thatdetermine their preferences and choices in such a manner. There has been recent interest in yet a further challenge to expectedutility theory, namely, the challenge from unawareness. To keep things simple, we shall however focus onSavage’s expected utility theory to illustrate the challengeposed by unawareness. In effect, Non-Atomicityimplies that \(\bS\) contains events of arbitrarily small probability.It is not too difficult to imagine how that could be satisfied.
Savage showed that whenever these six axioms are satisfied, thecomparative belief relation can be represented by a uniqueprobability function. Lara Buchak (2013) has recently developed a decision theory that canaccommodate Allais’ preferences without re-describing theoutcomes. On Buchak’s interpretation, the explanation forAllais’ preferences is not the different value that theoutcome $0 has depending on what lottery it is part of. However, the contribution that $0makes towards the overall value of an option partly depends on whatother outcomes are possible, she suggests, which reflects the factthat the option-risk that the possibility of $0 generates depends onwhat other outcomes the option might result in.
This reasoning was madeprominent in a paper by Good (1967), where he proves that one shouldalways seek “free evidence” that may have a bearing on thedecision at hand. (Precursors of this theorem can be found in Ramsey(1990, published posthumously) and Savage (1954).) Strictly speaking,value-of-information reasoning should underpin all Bayesianexperimental design. It is certainly employed explicitly by Bayesianstatisticians in cases where delicate tradeoffs must be made betweenthe desiderata of well-informed inference and low-cost evidence (forone example, see Anscombe 1963). Instead of adding specific belief-postulates to Jeffrey’s theory,as Joyce suggests, one can get the same uniqueness result by enrichingthe set of prospects. It should moreover be evident, given the discussion of the SureThing Principle (STP) in Section 3.1, thatJeffrey’s theory does not have this axiom.
Dietrich and List (2013 & 2015) have proposed aneven more general framework for representing the reasons underlyingpreferences. In their framework, preferences satisfying some minimalconstraints are representable as dependent on the bundle of propertiesin terms of which each option is perceived by the agent in a givencontext. Properties can, in turn, be categorised as either optionproperties (which are intrinsic to the outcome), relationalproperties (which concern the outcome in a particular context),or context properties (which concern the context of choiceitself). Such a representation permits more detailed analysis of thereasons for an agent’s preferences and captures different kindsof context-dependence in an agent’s choices.
Descriptive vs Normative Decision Theory
Nevertheless, Savage’s theory has been much moreinfluential than Ramsey’s, perhaps because Ramsey neither gave afull proof of his result nor provided much detail of how it would go(Bradley 2004). However, the ingredients and structure of his theoremwill be laid out, highlighting its strengths and weaknesses. The last section provided an interval-valued utility representation ofa person’s preferences over lotteries, on the assumption thatlotteries are evaluated in terms of expected utility. The vNM theorem effectivelyshores up the gaps in reasoning by shifting attention back to thepreference relation.
Theorem 2 (von Neumann-Morgenstern)Let \(\bO\) be a finite set of outcomes, \(\bL\)a set of corresponding lotteries that is closed under probabilitymixture and \(\preceq\) a weak preference relationon \(\bL\). Then \(\preceq\) satisfies axioms 1–4 ifand only if there exists a function \(u\), from\(\bO\) into the set of real numbers, that is unique up topositive linear transformation, and relative to which \(\preceq\) canbe represented as maximising expected utility. That is, themain question of interest is what criteria an agent’s preferenceattitudes should satisfy in any genericcircumstances.
Types of decisions
Beyond this, there is room for argument aboutwhat preferences over options actually amount to, or in other words,what it is about an agent (perhaps oneself) that concerns us when wetalk about his/her preferences over options. This section considerssome elementary issues of interpretation that set the stage forintroducing (in the next section) the decision tables and expectedutility rule that for many is the familiar subject matter of decisiontheory. Further interpretive questions decision theory is concerned with regarding preferences andprospects will be addressed later, as they arise. The static model has familiar tabular or normalform, with each row representing an available act/option, and columnsrepresenting the different possible states of the world that yield agiven outcome for each act.
Probability theory
Whether or not it is true by definition, i.e., whether real agentscan fail to satisfy its demands, the EU characterisation ofrationality serves to structure and thus identify an agent’spreference attitudes. The substantial nature of these preferenceattitudes—the agent’s beliefs and desires—can thenbe examined, perhaps with an eye to transformation or reform. Then there is an ordinal utilityfunction that represents \(\preceq\) just in case \(\preceq\) iscomplete and transitive. A number of important formal results, known as “representationtheorems”, show that this claim about describability can bederived from a set of prima facie plausible generalprinciples, aka “postulates” or “axioms”,pertaining to the agents’ preferences over acts. Furthermore,not only are these axioms collectively sufficient to deriveSEU’s claim, but a significant proper subset of them also turnout to be individually necessary. Unsurprisingly then, muchof the work on assessing the empirical adequacy of SEU has focused onthe testing of the aforementioned axioms.
Hammond shows that only a fully Bayesian agent canplan to pursue any path in a sequential decision tree that is deemedoptimal at the initial choice node. This makes the Bayesian agentunique in that she will never make “self-defeatingchoices” on account of her preferences and norms for preferencechange. She will never choose a strategy that is worse by her ownlights than another strategy that she might otherwise have chosen, ifonly her preferences were such that she would choose differently atone or more future decision nodes.
