And suppose now we find aproposition \(r\), that is pairwiseincompatible with both \(p\)and \(q\), and which you find moredesirable than both \(p\)and \(q\). Then if it turns out thatyou are indifferent between \(p\)joined with \(r\)and \(q\) joinedwith \(r\), that must be because youfind \(p\) and \(q\) equally probable. Otherwise, you would preferthe union that contains the one of \(p\) and \(q\) thatyou find less probable, since that gives you a higher chance of themore desirable proposition \(r\). Itthen follows that for any other proposition \(s\) that satisfies the aforementionedconditions \(r\) satisfies, youshould also be indifferent between \(p\cups\) and \(q\cup s\), since,again, the two unions are equally likely to resultin \(s\). 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.
Challenges to EU theory
- The idea is that we can reason about whether and how to goabout seeking further evidence only in relation to a decision problem;the subjective expected utility of making the decision on the basis ofexisting evidence is compared with the expected utility of seekingfurther evidence before making the decision.
- 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.
- Thevarious causal decision theories make causal hypotheses explicit, andrank acts according to their expected success in bringing about goodoutcomes.
- Early decision theorists, motivated by a misguided scientific methodology, thought of preferences as operationally defined in terms of overt choices, so that, by definition, an agent prefers A to B if and only if (iff) she will incur a cost to choose A over B.
- According to the standard model of decision-theoretic rationality, an action is rational just in case, relative to the agent’s beliefs and desires, it has the highest subjective expected utility of any available option.
For example,consider the predicament of a mountaineer deciding whether or not toattempt a dangerous summit ascent, where the key factor for her is theweather. Nevertheless, the weather statistics differfrom the lottery set-up in that they do not determine theprobabilities of the possible outcomes of attempting versus notattempting the summit on a particular day. Not least, the mountaineermust consider how confident she is in the data-collection procedure,whether the statistics are applicable to the day in question, and soon, when assessing her options in light of the weather. Let us conclude by summarising the main reasons why decision theory,as described above, is of philosophical interest. The aim is to characterise the attitudes of agents whoare practically rational, and various (static and sequential)arguments are typically made to show that certain practicalcatastrophes befall agents who do not satisfy standarddecision-theoretic constraints. For instance,Broome (1991c), Byrne and Hájek (1997) and Hájek andPettit (2004) suggest formulations of anti-Humeanism that are immuneto Lewis’ criticism, while Stefánsson (2014) and Bradleyand Stefánsson (2016) argue that Lewis’ proof relies on afalse assumption.
While Ulysses isrational at the first choice node by static decisionstandards, we might regard him irrational overall bysequential decision standards, understood in terms of the relativevalue of sequences of choices. It would have beenbetter were he able to sail unconstrained and continue on home toIthaca. This sequence could have been achieved if Ulysses werecontinuously rational over the extended time period; say, ifat all times he were to act as an EU maximiser, and change his beliefsand desires only in accordance with Bayesian norms (variants ofstandard conditionalisation). On this reading, sequentialdecision models introduce considerations of rationality-over-time.
Comparative Probability
AI decision-making can be completely automated or augmented with human intervention, depending on the complexity and time constraints of the decision to be made. AI can analyze large datasets without error, allowing business teams to focus on work relevant to their field. More complex models, like deep learning, can provide high accuracy but are less interpretable, making it difficult to understand the reasoning behind decisions. This can be problematic in fields like healthcare or finance, where understanding the rationale for decisions is crucial.
- As noted in Section 4, criticisms of the EU requirement of a complete preference orderingare motivated by both epistemic and desire/value considerations.
- Onthe value side, many contend that a rational agent may simply find twooptions incomparable due to their incommensurablequalities.
- The aim is to characterise theattitudes of agents who are practically rational, and various (staticand sequential) arguments are typically made to show that certainpractical catastrophes befall agents who do not satisfy standarddecision-theoretic constraints.
- Nevertheless, Lewis’ argument nodoubt provoked an interesting debate about the sorts of connectionsbetween belief and desire that EU theory permits.
- For instance, it is questionable whether an agent shouldbe able to compare the option whereby two additional people in theworld are made literate with the option whereby two additional peoplereach the age of sixty.
Intertemporal choice
In the end, preferences are best thought of as subjective judgments of the comparative merits of actions as promoters of desirable outcomes. While such judgments are closely tied to overt choice behavior, the relationship between the two is nowhere near as direct and unsophisticated as behaviorism suggests. The decision theories of Savage and Jeffrey, as well as those of theircritics, apparently concern a single or “one shot only”decision; at issue is an agent’s preference ordering, andultimately her choice of act, at a particular point in time. 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.
Only a few remarks will be made about the “causalchallenge” to EU theory, as the interested reader can consultthe SEP entry on Causal DecisionTheory and the references therein. The supposed problem for EUtheory (or at least Jeffrey’s version of it) is its commitmentto the desirability or choice-worthiness of acts being dependent onthe states of affairs that are probabilistically correlated with theacts. Dissenters point out thatcorrelation is not necessarily causation; in particular there may be acommon cause underlying the correlation between a particular act beingchosen and a good state of affairs or outcome. The positionof causal decision theorists is that we should choose actsbecause they actually bring about good outcomes, not because theyhappen to harbour news of good outcomes. Savage’s theory leavesopen this possibility, but offers no resources for determining whetheracts, states and outcomes are defined in a way that reflects causalrelationships rather than mere probabilistic relationships. Thevarious causal decision theories make causal hypotheses explicit, andrank acts according to their expected success in bringing about goodoutcomes.
Furthermore, it permits explicit restrictions on what countsas a legitimate reason for preference, or in other words, whatproperties legitimately feature in an outcome description; suchrestrictions may help to clarify the normative commitments of EUtheory. In most ordinary decision theory is concerned with 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. In our continuing investigation of rational preferences overprospects, the numerical representation (ormeasurement) of preference orderings will become important.The numerical measures in question are known as utilityfunctions.
2 On separability: Risk and regret attitudes
The agent is assumed to be an expected utilitymaximiser who takes a sophisticated (backwards reasoning) approach tosequential decision problems. Skyrms shows that any such agent whoplans to learn in a manner at odds with conditionalisation will makeself-defeating choices in some specially contrived sequential decisionsituations. A good conditionalising agent, by contrast, will nevermake choices that are self-defeating in this way. That is, the agent chooses a strategy that issurely worse, by her own lights, than another strategy that she mightotherwise have chosen, if only her learning rule was such that shewould choose differently at one or more future choice nodes.
Ulysses must make a choice about the manner in which hewill sail past an island inhabited by sweet-singing sirens. In the formercase, Ulysses will later have the choice, upon hearing the sirens, toeither continue sailing home to Ithaca or to stay on the islandindefinitely. In the latter case, he will not be free to make furtherchoices and the ship will sail onwards to Ithaca past thesweet-singing sirens. Ulysses’ decision problem is representedin tree (or extensive) form in Figure 1 (where thetwo boxes represent choice points for Ulysses). Nevertheless, it does seem that an argument canbe made that any reasonable person will satisfy this axiom. Supposeyou are indifferent between two propositions, \(p\) and \(q\), thatcannot be simultaneously true.
This section expands, in turn,on the epistemological and evaluative commitments of EU theory. Another important thing to notice about Jeffrey’s way ofcalculating desirability, is that it does not assume probabilisticindependence between the alternative that is being evaluated, \(p\),and the possible ways, the \(p_i\)s, that the alternative may berealised. Indeed, the probability of each \(p_i\) is explicitlyconditional on the \(p\) in question. When it comes to evaluatingacts, this is to say (in Savage’s terminology) that theprobabilities for the possible state-outcome pairs for the act areconditional on the act in question. A well-known sequential decision problem is the one facing Ulysseson his journey home to Ithaca in Homer’s great tale fromantiquity.
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. The decision theories of Savage and Jeffrey, as well as those oftheir critics, apparently concern a single “one shot only”decision; at issue is an agent’s preference ordering, andultimately her choice of act, at a particular point in time. The questionarises as to whether this framework is adequate for handling morecomplex scenarios, in particular those involving a series or sequenceof decisions; these are referred to as sequential decisionproblems.
Subjective Expected Utility
Specifically, for each consequence c there is a constant act c that produces c in every state of the world, and, for any acts A and B, and any disjunction of states E, there is a mixed act AE ∪ B~E that produces A ‘s consequence when E holds and B ‘s consequence when ~E holds. While real agents will typically be unable to realize such recherché prospects as these, imagining that decision makers have attitudes toward them often helps one determine which realistic acts should be performed. 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. As noted, a special case is when the content of\(p\) is such that it is recognisably something the agent can chooseto make true, i.e., an act.
The idea is that we can reason about whether and how to goabout seeking further evidence only in relation to a decision problem;the subjective expected utility of making the decision on the basis ofexisting evidence is compared with the expected utility of seekingfurther evidence before making the decision. 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).
