Furthermore,whether or not to seek more evidence is a pragmatic issue; it dependson the “value of information” one expects to gain withrespect to the decision problem at hand. The idea is that seeking moreevidence is an action that is choice-worthy just in case the expectedutility of seeking further evidence before making one’s decisionis greater than the expected utility of making the decision on thebasis of existing evidence. This reasoning was made prominent in apaper by Good (1967), where he proves that one should always seek“free evidence” that may have a bearing on the decision athand. In most ordinary choice situations, the objects of choice, over whichwe must have or form preferences, are not like this. Rather,decision-makers must consult their own probabilistic beliefsabout whether one outcome or another will result from a specifiedoption.

The standard representation of a decision problem requires that we specify the alternatives available to the decision-maker, the possible outcomes of the decision, the values of these outcomes, and… The above problems suggest there is a need for an alternative theoryof choice under uncertainty. Richard Jeffrey’s theory, whichwill be discuss next, avoids all of the problems that have beendiscussed so far.

1 Allais’ paradoxes

To accommodate this,they extend the Boolean algebra in Jeffrey’s decision theory tocounterfactual propositions, and show that Jeffrey’sextended theory can represent the value-dependencies one often findsbetween counterfactual and actual outcomes. In particular, theirtheory can capture the intuition that the (un)desirability of winningnothing partly depends on whether or not one was guaranteed to winsomething had one chosen differently. Therefore, their theory canrepresent Allais’ preferences as maximising the value of anextended Jeffrey-desirability function. These issues turn out to be rather controversial, raising a host ofinterpretive questions regarding sequential decision models and rational choice in this setting. Thesequestions will be addressed in turn, after the scene has been set withan old story of Ulysses.

For instance, anyevent \(F\) can be partitioned intotwo equiprobable sub-events according to whether some coin would comeup heads or tails if it were tossed. Each sub-event could be similarlypartitioned according to the outcome of the second toss of the samecoin, and so on, ad infinitum. Is there anyprobability \(p\) such that you wouldbe willing to accept a gamble that has that probability of you losingyour life and probability \((1-p)\)of you winning $10?

What are preferences over prospects?

In particular, economists decision theory is concerned with Karni and Vierø (2013,2015) have recently extended standard Bayesian conditionalisation tosuch learning events. Their theory, Reverse Bayesianism,informally says that awareness growth should not affect the ratios ofprobabilities of the states/outcomes that the agent was aware ofbefore the growth. Richard Bradley (2017) defends a similar principlein the context of the more general Jeffrey-style framework, and sodoes Roussos (2020); but the view is criticised by Steele andStefánsson (forthcoming-a, forthcoming-b) and by Mahtani(forthcoming). So under what conditions can a preference relation \(\preceq\) on theset \(\Omega\) be represented as maximising desirability? Some of therequired conditions on preference should be familiar by now and willnot be discussed further. In particular, \(\preceq\) has to betransitive, complete and continuous (recall our discussion in Section 2.3 of vNM’s Continuity preference axiom).

Those who are less inclined towards behaviourism might, however,not find this lack of uniqueness in Bolker’s theorem to be aproblem. However,if uniqueness is what we are after, then we can, as Joyce points out,supplement the Bolker-Jeffrey axioms with certain conditions on theagent’s comparative belief relation that guarantee that it givesrise to one and only one probability function (for instance, thecomparative belief conditions proposed in Villegas 1964). Having doneso, we can show that any function representing the agent’sdesires will be unique up to a positive linear transformation. Definition 1 is based on the simple observation that one wouldgenerally prefer to stake a good outcome on a more rather than lessprobable event. We could, for instance, imaginepeople who are instrumentally irrational, and as a result fail toprefer \(g\) to \(f\), even when the above conditions all hold andthey find \(F\) more likely than \(E\).

Broader implications of Expected Utility (EU) theory

A concave \(\Phi\) will overweigh low expected utilities, resulting inrelatively ambiguity-averse preferences. In decision theory, a decision problem is situation in which a decision maker, (a person, a company, or a society) chooses what to do from a set of alternative acts, where the outcomeof the… Is there anyprobability \(p\) such that you would be willing to accept a gamblethat has that probability of you losing your life and probability\((1-p)\) of you gaining $10?

1. And she shows that if an agentsatisfies a particular set of axioms, which is essentiallySavage’s except that the Sure Thing Principle is replaced with astrictly weaker one, then the agent’s preferences can berepresented as maximising risk weighted expected utility;which is essentially Savage-style expected utility weighted by a riskfunction.
2. If so, thiswould amount to a subtle shift in the question or problem of interest.In what follows, the standard interpretation of sequential decisionmodels will be assumed, and accordingly, it will be assumed thatrational agents pursue the sophisticated approach to choice (as perLevi 1991, Maher 1992, Seidenfeld 1994, amongst others).
3. Recallfrom Section 2.3 that people tend toprefer \(L_2\) over \(L_1\) and \(L_3\)over \(L_4\)—an attitude thathas been called Allais’ preferences—in violationof expected utility theory.
4. Perhaps no suchpeople exist (and Savage’s axiom P5indeed makes clear that his result does not pertain to suchpeople).
5. Under certain assumptions, the overallor aggregate preference ordering is compatible with EUtheory.

Then if it turns out thatyou are indifferent between \(p\) joined with \(r\) and \(q\) joinedwith \(r\), that must be because you find \(p\) and \(q\) equallyprobable. Otherwise, you would prefer the union that contains the oneof \(p\) and \(q\) that you find less probable, since that gives you ahigher chance of the more desirable proposition \(r\). It then followsthat for any other proposition \(s\) that satisfies the aforementionedconditions that \(r\) satisfies, you should also be indifferentbetween \(p\cup s\) and \(q\cup s\), since, again, the two unions areequally likely to result in \(s\).

The theorem is limited to evaluating options that come with aprobability distribution over outcomes—a situation decisiontheorists and economists often describe as “choice underrisk” (Knight 1921). It may yet be argued that EU theory does not go far enough instructuring an agent’s preference attitudes so that we mayunderstand and assess these attitudes. Various generalisations of thetheory have been suggested that offer more detailed analyses ofpreferences. For instance, the multiple criteria decisionframework (see, for instance, Keeney and Raiffa 1993) takes anagent’s overall preference ordering over options to be anaggregate of the set of preference orderings corresponding to all thepertinent dimensions of value. Under certain assumptions, the overallor aggregate preference ordering is compatible with EUtheory.

Expected utility theory has been criticised for not allowing forvalue interactions between outcomes in different, mutuallyincompatible states of the world. For instance, recall that whendeciding between two risky options you should, according toSavage’s version of the theory, ignore the states of the worldwhere the two options result in the same outcome. That seems veryreasonable if we can assume separability between outcomes indifferent states of the world; i.e., if the contribution that anoutcome in one state of the world makes towards the overall value ofan option is independent of what other outcomes the option mightresult in. For then identical outcomes (with equal probabilities)should cancel each other out in a comparison of two options, whichwould entail that if two options share an outcome in some state of theworld, then when comparing the options, it does not matter what thatshared outcome is. Even if we suspend doubts about the basic commitments of prominentversions of EU theory (which will be taken upin Section 5), there is a large question as towhat the theory really establishes about how agents should reason inthe real world.