An example of adecision-rule is the EU rule, requiring that we maximise expectedutility in each decision-problem. Stefánsson and Bradley (2019) suggest yet another way ofaccounting for Allais’ preferences in decision theory is concerned with an extension ofJeffrey’s decision theory; this time extended to chancepropositions, that is, propositions describing objectiveprobability distributions. The general idea is that the desirabilityof a particular increase or decrease in the chance of someoutcome—for instance, in the Allais case, a 0.01 increase in thechance of the $0-outcome—might depend on what the chances werebefore the increase or decrease. But they suggest thatwhat explains Allais’ preferences is that the value of winingnothing from a chosen lottery partly depends on what would havehappened had one chosen differently. To accommodate this, they extendthe 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. Others contendthat accounts of rational belief can and should be ultimatelyjustified on epistemic grounds; Joyce (1998), for instance, offers anon-pragmatic justification of probabilism that rests on the notion ofoverall accuracy of one’s beliefs.
Axiom 4
For instance, it may be that Bangkok isconsidered almost as desirable as Cardiff, but Amsterdam is a long waybehind Bangkok, relatively speaking. Or else perhaps Bangkok is onlymarginally better than Amsterdam, compared to the extent to whichCardiff is better than Bangkok. This kind of information about therelative distance between options, in terms of strength of preferenceor desirability, is precisely what is given by an interval-valuedutility function. Let \(S\) be afinite set of prospects, and \(\preceq\) a weak preference relation on\(S\). Then there is an ordinal utility function that represents\(\preceq\) just in case \(\preceq\) is complete and transitive. 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.
Normative and descriptive
The vNM theorem is a very important result for measuring the strengthof a rational agent’s preferences over sure options (thelotteries 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 aprobability distribution over outcomes—a situation decisiontheorists and economists often describe as “choice underrisk” (Knight 1921).
Minimax Criterion
The content is designed for B.Com/M.Com students and emphasizes the logical framework necessary for making informed decisions under certainty and uncertainty. The interest in descriptive decision theory arose in parallel with the development of normative theories. Given the enormous influence axiomatic theories had in the academic community in the latter half of the twentieth century, it became natural to test the axioms in empirical studies. Since many decision theorists advocate (some version of) the expected utility principle, it is hardly surprising that the axioms of expected utility theory are the most researched ones. Early studies cast substantial doubt on the expected utility principle as an accurate description of how people actually choose.
Alternatives to probability theory
The subject includes rational choice theory, which seeks to formulate and justify the normative principles that govern optimal decision making, and descriptive choice theory, which aims to explain how human beings actually make decisions. Within both these areas one may distinguish individual decision theory, which concerns the choices of a single agent with specific goals and knowledge, and game theory, which deals with interactions among individuals. This entry will focus on rational choice theory for the single agent, but some descriptive results will be mentioned in passing. Yes, decision theory can be applied to personal life decisions, ranging from financial planning and career choices to more mundane daily choices. While the formal models and calculations used in corporate or policy decisions might be overwhelming for individual use, the principles can be adapted.
A common way to rationalise Allais’ preferences, is that in thefirst choice situation, the risk of ending up with nothing when onecould have had $2400 for sure does not justify the increased chance ofa higher prize. In the second choice situation, however, the minimumone stands to gain is $0 no matter which choice one makes. Therefore,in that case many people do think that the slight extra risk of $0 isworth the chance of a better prize. Thenthere is an ordinal utility function that represents \(\preceq\) justin case \(\preceq\) is complete and transitive. Let us nonetheless proceed by first introducing basic candidateproperties of (rational) preference over options and only afterwardsturning to questions of interpretation. As noted above, preferenceconcerns the comparison of options; it is a relation between options.For a domain of options we speak of an agent’s preferenceordering, this being the ordering of options that is generated bythe agent’s preference between any two options in thatdomain.
For personal decisions, simple weighted lists of pros and cons, considering different outcomes and their likelihoods, can help individuals make more considered and rational choices. Let us conclude by summarising the main reasons why decision theory,as described above, is of philosophical interest. First, normativedecision theory is clearly a (minimal) theory of practicalrationality. 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 practical setbacksbefall agents who do not satisfy standard decision-theoreticconstraints. Under the assumption that an ethical choice must berational, the findings of decision theory have implications forethics, or more generally, for the ways in which we can valuestates of affairs.
Richard Jeffrey’s theory, whichwill be discussed next, avoids all of the problems that have beendiscussed so far. Another way to put this is that, when the above holds, the preferencerelation can be represented as maximising utility, asmeasured by u, since it always favours an option with higherutility. Illustrates decision trees as a visual decision-making tool, demonstrating an example involving insurance decision under uncertain fire risk.View Overview of Decision Theory focusing on decision making process amid uncertainty, Bayesian analysis, and the steps involved in decision making.View
These shapes are linked with arrows or arcs in specific ways to show the relationship among the elements. Figure 10.2 shows an example for making a decision regarding Vacation Activity (decision node). Weather Forecast and Weather Condition are random nodes namely uncertain events while Satisfaction is a value node which is a result of the decision. The elements of decision theory are a set of possible future conditions that can exist that will have affect the results of the decision, a list of possible alternatives to choose from and a calculated or known payoff for each of the possible alternatives under each of the possible future conditions. Decision analysis provides a “framework for analyzing a wide variety of management models”.
Addressing these challenges requires a combination of analytical rigor and awareness of human behavior, ensuring that decisions are both informed and aligned with the decision-maker’s objectives. Bayesian Decision Theory incorporates Bayes’ theorem to update the probabilities of outcomes as new information becomes available. 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.
Independence and the Sure-Thing Principle
- AI technologies such as machine learning, natural language processing, and computer vision are trusted aspects of business today, used to increase profits and reach set goals.
- This evolution will enhance the ability to model complex decision problems and improve the accuracy of predictions.
- While the principle is often defended as adecision-rule (see, e.g., Steel 2014), some think that anysuch interpretation of the principle is implausible (Peterson 2006,Stefánsson 2019).
- This approach has recently beenquestioned in the context of climate policy, where so-calledintegrated assessment models have been criticised for theirreliance on EU theory (see, for instance Pindyck 2022 and Stern et. al2022).
- But whether or notthe preference in question should be explained by the potential forregret, it would seem that the desirability of the $0-outcome dependson what could (or would) otherwise have been; in violation of theaforementioned assumption of separability.
In the context of climate policy, Continuity implies thatno climate catastrophe is so bad that we shouldn’t be willing toimplement a policy that might (with low probability) result in thatcatastrophe as long as the policy is sufficiently likely to insteadresult in some social gain. 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).
- A recently defended complete decision theory without Continuity is thelexicographic utility theory.
- Start with the Completeness axiom, which says that an agent cancompare, in terms of the weak preference relation, all pairs ofoptions in \(S\).
- More specifically, decision theory deals with methods for determining the optimal course of action when a number of alternatives are available and their consequences cannot be forecasted with certainty.
- More generally, although people rarelythink of it this way, they constantly take gambles that have minusculechances of leading to imminent death, and correspondingly very highchances of some modest reward.
- At the far end of the spectrum is theposition that the very meaning of belief involves preference.
2 On completeness: Vague beliefs and desires
If this were not the case, the axiom ofState Neutrality, for instance, would be a very implausiblerationality constraint. Suppose we are, for example, wondering whetherto buy cocoa or lemonade for the weekend, and assume that how good wefind each option depends on what the weather will be like. For if we do not, the desirability of the outcomes willdepend on what state is actual. Since lemonade is, let us suppose,better on hot days than cold, an outcome like “I drink lemonadethis weekend” would be more or less desirable depending onwhether it occurs in a state where it is hot or cold.
