Decision Theory

This means that if \(u\) is an ordinalutility function that represents the ordering \(\preceq\), then anyutility function \(u’\) that is an ordinal transformation of\(u\)—that is, any transformation of \(u\) that also satisfiesthe biconditional in (1)—represents \(\preceq\) just as well as\(u\) does. Hence, we say that an ordinal utility function isunique only up to ordinal transformations. This brings us to the Transitivity axiom, which says that if an option \(B\) is weakly preferred to \(A\), and\(C\) weakly preferred to \(B\), then \(C\) is weakly preferred to\(A\).

Comprehensive Min-Max Scaling Guide for Enhanced Data Normalization

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. To accommodateAllais’ preferences (and other intuitively rational attitudes torisk that violate EU theory), Buchak introduces a riskfunction that represents people’s willingness to tradechances of something good for risks of something bad. There are various non-expected utility theories that can accommodateAllais’ preferences without re-describing the outcomes.

If the theory is meant to describe the reasoning of a decision-maker,the first two interpretations would seem inferior to the third. Theproblem with the first two interpretations is that the decision-makermight be unaware of some of the logically possible states andoutcomes, as well as some of the states and outcomes that the modelleris aware of. (Having said that, one may identify the states andoutcomes that the agent is unaware of by reference to those of whichthe modeller is aware.) 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. A recently defended complete decision theory without Continuity is thelexicographic utility theory.

There are many false starts and punctuated by what Kuhn (2012) calls paradigm shifts. We think of our approach as opening a new window in a magnificent structure and a modest punctuation. In this chapter, we show its multidisciplinary heritage rooted in mathematics, cognitive psychology, social science and the practice. We want to show its punctuated continuity with, and its debt to, the achievements of the past. Our debt, notwithstanding, we also draw contrasts between our engineering decision-design methods and other traditional methods. Value in decision making is ultimately assigned according to what the people involved find to be of worth.

• The paper describes the point of departure of decision in complex, time-pressured, uncertain, ambiguous and changing environments.
• If we are interested inreal-world decisions, then the acts in question ought to berecognisable options for the agent (which we have seen isquestionable).
• Overview of decision theory; it analyzes alternatives and consequences for decision makers.View
• Moreover, now we see that one of Savage’srationality constraints on preference—the Sure ThingPrinciple—is plausible only if the modelled acts areprobabilistically independent of the states.
• The value of u (c ) measures the degree to which c would satisfy the agent’s desires and promote his or her aims.
• It encompasses various methodologies and perspectives, often drawing from economics, psychology, statistics, philosophy, and mathematics.

Furtherinterpretive questions regarding preferences and prospects will beaddressed later, as they arise. In 1738, Daniel Bernoulli published an influential paper entitled Exposition of a New Theory on the Measurement of Risk, in which he uses the St. Petersburg paradox to show that expected value theory must be normatively wrong. He gives an example in which a Dutch merchant is trying to decide whether to insure a cargo being sent from Amsterdam to St Petersburg in winter. In his solution, he defines a utility function and computes expected utility rather than expected financial value. Richard Jeffrey’s expected utility theory differs fromSavage’s in terms of both the prospects (i.e., options)under consideration and the rationality constraints onpreferences over these prospects.

The Limited Rationale in Decision Making, Impacts on the Evaluation of Artifacts in the Design Process

Some take the connection between rational preference and rationalbelief to run very deep indeed. At the far end of the spectrum is theposition that the very meaning of belief involves preference. Many question the plausibility, however, of equating comparativebelief with preferences over specially contrived prospects. A moremoderate position is to regard these preferences as entailed by, butnot identical with, the relevant comparative beliefs. A recent defender of this kind of pragmatism (albeitcast in more general terms) is Rinard (e.g., 2017). One well recognised decision-modelling requirement for Savage’stheory is that outcomes be maximally specific in every way thatmatters for their evaluation.

Decision theory draws tools from mathematics, philosophy, statistics and psychology in analysing how decisions are made. This theory also has to do with how choices are logically made based on probabilities and uncertain consequences. In summary, normative decision theory provides a framework for making rational and optimal decisions based on well-defined principles and criteria, regardless of how decisions are made in real-world situations. Theoretical studies have revealed that decision theory is a formal study of rational decision making formed largely by the joint efforts of mathematicians, philosophers, social scientists, economists, statisticians and management scientists (Jeffrey 1992).

Decision-Making Criteria

A conditionalising agent, by contrast,will never make 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 decision nodes. It is hard to deny that Ulysses makes a wise choice, given hispreferences, in being tied to the mast. Some hold, however, thatUlysses is nevertheless not an exemplary agent, since his present selfmust play against his future self who will be unwittingly seduced bythe sirens. While Ulysses is rational at the time of thefirst choice node, his is not rational over the time periodin question, since the sequence of choices that he inevitably pursuesis suboptimal.

Value Independence

A number of people have suggested models to represent agents who areaware of their unawareness (e.g., Walker & Dietz 2013, Piermont2017, Karni & Vierø 2017). Steele and Stefánsson(forthcoming-b) argue that there may not be anything especiallydistinctive about how a decision-maker reasons about states/outcomesof which she is aware she is unaware, in terms of the confidence shehas in her judgments and how she manages risk. That said, the way shearrives at such judgments of probability and desirability is worthexploring further.

Recall that the domain of the preferenceordering in Savage’s theory amounts to every functionfrom the set of states to the set of outcomes (what Broome 1991arefers to as the Rectangular Field Assumption). So if“I drink lemonade this weekend in hot weather” is one ofthe outcomes we are working with, and we have partitioned the set ofstates according to the weather, then there must, for instance, be anact that has this outcome in the state where it is cold! The moredetailed the outcomes (as required for the plausibility of StateNeutrality), the less plausible the Rectangular Field Assumption. Indeed, it isdifficult to see how/why a rational agent can/should form preferencesover nonsensical acts (although see Dreier 1996 for an argument thatthis is not such an important issue).

Therelation between descriptive decision theory and its normativecounterpart is then discussed, drawing some connections with a numberof related topics in the philosophical literature. Decision theory relates with how activities leading to decision making are understood. There are diverse types of decision making processes, hence, decision theory also covers diverse areas.

It is a branch of applied probability theory and analytic philosophy that involves assigning probabilities to various factors and numerical consequences to outcomes. The theory is concerned with identifying optimal decisions, where optimality is defined in terms of the goals and preferences of the decision-maker. Normative decision theory typically involves concepts like expected utility theory, which helps individuals make choices by weighing the probabilities of different outcomes against the desirability or utility of those outcomes. It also considers concepts like rationality, consistency, and coherence in decision-making processes. The central goal of rational choice theory is to identify the conditions under which a decision maker’s beliefs and desires rationalize the choice of an action. 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.

• Moreover, this book should be at the top of any instructor’s list of books to read in preparation for teaching a course on decision theory.
• Savage suggests that this definition ofcomparative beliefs is plausible in light of his axiom P4, which willbe stated below.
• Some of these branches lead to further choice points, oftenafter the resolution of some uncertainty due to new evidence.
• By contrast, if preferences areunderstood rather as mental attitudes, typically considered judgmentsabout whether an option is better or more desirable than another, thenthe doubts about Completeness alluded to above are pertinent (forfurther discussion, see Mandler 2001).

But this is a matter of emphasis, not any omission on Peterson’s part, and many instructors may share Peterson’s judgments of how space is best allocated in these chapters of the book. This section outlines the accessibility features of this content – including support for screen readers, full keyboard navigation and high-contrast display options. The following are direct quotes from (Keeney 1992a, b) except for our comments in italics. Every new idea and improvement is necessarily the result of standing on the shoulders of others. New knowledge, novel and useful practice are all part of evolving and connected strands of understanding, expertise and proficiency. The progression is cumulative and advancing to more insightful understanding and more effective practice.

Descriptive decision theory is concerned with characterising andexplaining regularities in the choices that people are disposed tomake. It is standardly distinguished from a parallel enterprise,normative decision theory, which seeks to provide an account of thechoices that people ought to be disposed to make. Decision theory is a logical study of how decisions are made in a structure or system where the decision environment is uncertain and the decision variables unknown.

Why should we assume that people evaluate lotteriesin terms of their expected utilities? The vNM theorem effectivelyshores decision theory is concerned with up the gaps in reasoning by shifting attention back to thepreference relation. In addition to Transitivity and Completeness, vNMintroduce further principles governing rational preferences overlotteries, and show that an agent’s preferences can berepresented as maximising expected utility whenever her preferencessatisfy these principles.

Gigerenzer et al. (1999) seek to replace the single all-purpose prescription to maximize expected utility by an ecological model of rationality in which decision makers employ a set of simple, highly localized decision heuristics. These heuristics efficiently generate choices that produce desirable consequences in the contexts where they tend to be employed, but they can go badly awry when used in out of context. Decision theory addresses risk and uncertainty by incorporating probabilities into the decision-making process. Under conditions of risk, probabilities of different outcomes are known or can be estimated, allowing for the calculation of expected values or utilities.