If, on the other hand, all options in the setare quite similar to each other, say, all options are investmentportfolios, then Completeness is more compelling. But even if we donot restrict the kinds of options under consideration, the question ofwhether or not Completeness should be satisfied turns on the meaningof preference. For instance, if preferences merely represent choicebehaviour or choice dispositions, as they do according to the“revealed preference theory” popular amongst economists(see Sen 1973), then Completeness is automatically satisfied, on theassumption that a choice must inevitably be made. By contrast, ifpreferences are understood rather as mental attitudes, i.e.,considered judgments about whether an option is better or moredesirable than another, then the doubts about Completeness alluded toabove are pertinent (for further discussion, see Mandler 2001).
Further interpretive questions regarding preferences andprospects will be addressed later, as they arise. Decision theory is concerned with the reasoning underlying anagent’s choices, whether this is a mundane choice between takingthe bus or getting a taxi, or a more far-reaching choice about whetherto pursue a demanding political career. (Note that “agent”here stands for an entity, usually an individual person, that iscapable of deliberation and action.) Standard thinking is that what anagent chooses to do on any given occasion is completely determined byher beliefs and desires or values, but this is not uncontroversial, aswill be noted below. In any case, decision theory is as much a theoryof beliefs, desires and other relevant attitudes as it is a theory ofchoice; what matters is how these various attitudes (call them“preference attitudes”) cohere together.
3 On completeness: Vague beliefs and desires
Nevertheless, it does seem that an argument can be made that anyreasonable person will satisfy this axiom. Suppose you are indifferentbetween two propositions, \(p\) and \(q\), that cannot besimultaneously true. And suppose now we find a proposition \(r\), thatis pairwise incompatible with both \(p\) and \(q\), and which you findmore desirable 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 you find \(p\) and \(q\) equallyprobable.
This approach allows decision-makers to refine their assessments and improve their choices over time. By integrating prior knowledge with observed data, Bayesian methods facilitate more informed decision-making, particularly in dynamic environments where conditions may change rapidly. The EU characterisation of rational desire is in many ways highlypermissive, but it may be more controversial than first meets theeye. Here the focus will be on the compatibility of EU theory withprominent ethical positions regarding the choice-worthiness of acts,as well as meta-ethical positions regarding the nature of value andits relationship to belief. In effect, Non-Atomicity implies that \(\bS\)contains events of arbitrarily small probability.
It’s time to build
- 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.
- And suppose now we find aproposition \(r\), that is pairwiseincompatible with both \(p\)and \(q\), and which you find moredesirable than both \(p\)and \(q\).
- Independence implies that when two alternatives have the sameprobability for some particular outcome, our evaluation of the twoalternatives should be independent of our opinion of that particularoutcome.
- Intuitively, Continuity guaranteesthat an agent’s evaluations of lotteries are appropriatelysensitive to the probabilities of the lotteries’ prizes.
- The only information contained in an ordinal utility representation ishow the agent whose preferences are being represented orders options,from least to most preferable.
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. The sequential decision model, on theother hand, has tree or extensive form (such asin Figure 1). It depicts a series of anticipatedchoice points, where the branches extending from a choice pointrepresent the options at that choice point. Some of these brancheslead to further choice points, often after the resolution of someuncertainty due to new evidence. David Lewis (1988, 1996) famously employed EU theory to argueagainst anti-Humeanism, the position that we are sometimesmoved entirely by our beliefs about what would be good, rather than byour desires as the Humean claims. Forinstance, Broome (1991c), Byrne and Hájek (1997) and Hájek and Pettit(2004) suggestformulations of anti-Humeanism that are immune to Lewis’criticism, while Stefánsson (2014) and Bradley and Stefánsson (2016) argue that Lewis’ proofrelies on a false assumption.
In such a case, some argue(e.g., Temkin 2012) that there is no reason why Transitivity should besatisfied with respect to the preferences concerning \(A\), \(B\)and \(C\). Others (e.g., Broome1991a) argue thatTransitivity is part of the very meaning of the betterness relation(or objective comparative desirability); if rational preference is ajudgment of betterness or desirability, then Transitivity isnon-negotiable. With respect to the car example, Broome would arguethat the desirability of a fully specified option should not vary,simply in virtue of what other options it is compared with.
As big data and machine learning continue to evolve, decision-makers will have access to more comprehensive datasets and sophisticated analytical tools. This evolution will enhance the ability to model complex decision problems and improve the accuracy of predictions. Furthermore, interdisciplinary approaches that integrate insights from behavioral economics and psychology will continue to enrich the field of Decision Theory.
The Representation of Decisions
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. This subjective expected utility (SEU) theory has its roots in the work of Blaise Pascal, Daniel Bernoulli, Vilfredo Pareto, and Frank decision theory is concerned with P. Ramsey, and finds its fullest expression in Leonard J. Savage’s Foundations of Statistics (1972). According to SEU a rational agent’s basic desires can be represented by a utility function u that assigns a real number u (c ) to each consequence c.
2 On separability: Risk and regret attitudes
In decisions under risk, the probabilities of outcomes are known, allowing for a more straightforward analysis. Conversely, decisions under uncertainty involve situations where the probabilities are unknown, requiring more complex modeling techniques. Understanding the type of decision problem is crucial for selecting the appropriate analytical methods. The above can be taken as a preliminary characterisation ofrational preference over options. Even this limited characterisationis contentious, however, and points to divergent interpretations of“preferences over prospects/options”. Less formally (and stated in terms of strict preference), the idea isthat if you prefer to stake the prize \(X\) on \(f\) rather than\(f’\), you must consider \(E\) more probable than \(F\).
Sequential decisions
- It would have beenbetter were he able to sail unconstrained and continue on home toIthaca.
- Interesting though these alternatives are, none has seriously challenged the normative status of SEU.
- 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.
- For one thing, in many real-world decision circumstances, it is hardto frame the decision model in such a way that states are intuitivelyprobabilistically independent of acts.
Decision-makers must weigh these trade-offs based on the specific context and goals of their AI initiatives. Decision theory is widely applied across various fields, including management, conservation planning, business statistics, and engineering design. It provides a structured approach to decision-making that can be particularly useful when dealing with complex problems or when there is significant uncertainty. The intuition behind the STP is that if \(g\) is weakly preferred to\(f\), then that must be because the consequence \(Y\) is consideredat least as desirable as \(X\), which by the same reasoning impliesthat \(g’\) is weakly preferred to \(f’\).
Grant and Quiggin (2013a, 2013b), for instance,suggest that these judgments are made based on induction from pastsituations where one experienced awareness growth. Then there is a desirability measure on \(\Omega\setminus \bot \) and a probability measure on \(\Omega\) relative towhich \(\preceq\) can be represented as maximising desirability. The proponents of fuzzy logic, possibility theory, Dempster–Shafer theory, and info-gap decision theory maintain that probability is only one of many alternatives and point to many examples where non-standard alternatives have been implemented with apparent success.
The above problems suggest there is a need for an alternativetheory of choice under uncertainty. Richard Jeffrey’s theory,which will be discuss next, avoids all of the problems that have beendiscussed so far. But as we will see, Jeffrey’s theory haswell-known problems of its own, albeit problems that are notinsurmountable. There are several tools and platforms available that can assist in automating decision-making processes. For instance, GiniMachine is an AI-powered decision management platform that can process terabytes of historical data, building, validating, and deploying predictive models in minutes. Other tools like Rationale AI assist in making tough decisions by providing pros and cons, SWOT analysis, multi-criteria analysis, or causal analysis.
Finally, decision theory should be of great interest tophilosophers of mind and psychology, and others who are interested inhow people can understand the behaviour and intentions of others; and,more generally, how we can interpret what goes on in otherpeople’s minds. Decision theorists typically assume that aperson’s behaviour can be fully explained in terms of herbeliefs and desires. But on anoptimistic reading of these results, they assure us that we canmeaningfully talk about what goes on in other people’s mindswithout much evidence beyond information about their dispositions tochoose. The vNM theorem is a very important result for measuring thestrength of a rational agent’s preferences over sure options(the lotteries effectively facilitate a cardinal measure over sureoptions). But this does not get us all the way to making rationaldecisions in the real world; we do not yet really have a decisiontheory. The theorem is limited to evaluating options that come with anobjective probability distribution over outcomes—a situationdecision theorists and economists often describe as “choiceunder risk” (Knight 1921).
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. Independence implies that when two alternatives have the sameprobability for some particular outcome, our evaluation of the twoalternatives should be independent of our opinion of that particularoutcome. Intuitively, this means that preferences between lotteriesshould be governed only by the features of the lotteries that differ;the commonalities between the lotteries should be effectivelyignored. Decision theory is an interdisciplinary field that deals with the logic and methodology of making choices, particularly under conditions of uncertainty.
The model does not seem able toaccommodate basic deontological notions like agent relativity,absolute prohibitions or permissible and yet suboptimal acts. There are also less general models that offer templates forunderstanding the reasons underlying preferences. One might otherwise seek tounderstand the role of time, or the temporal position of goods, onpreferences. To this end, outcomes are described in terms oftemporally-indexed bundles of goods, or consumption streams(for an early model of this kind see Ramsey 1928; a later influentialtreatment is Koopmans 1960). There may be systematic structure to anagent’s preferences over these consumption streams, over and above thestructure imposed by the EU axioms of preference.
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