## Decision Theory Definition, Examples, Types, How it Works?

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I’ve always seen the distinction as decision theory is a game against nature,while game theory is about a game against an opponent who was also making a decision. I guess game theory is more about games, whereas decision theory could be me deciding whether to walk home or take the bus. It emphasised that in actual human decision-making “losses loom larger than gains”, people are more focused on changes in their utility states than the states themselves and estimation of subjective probabilities is severely biased by anchoring.

The vNM theorem effectively shores up the gaps in reasoning by shifting attention back to the preference relation. In addition to Transitivity and Completeness, vNM introduce further principles governing rational preferences over lotteries, and show that an agent’s preferences can be represented as maximising expected utility whenever her preferences satisfy these principles. In contrast, descriptive decision theory is concerned with describing observed behaviors often under the assumption that those making decisions are behaving under some consistent rules.

Although Tversky’s results were later replicated, it should be noted that there is ongoing controversy surrounding the level of empirical support for intransitive preference (see Regenwetter et al. 2011 for a recent literature review). There has been recent interest in yet a further challenge to expected utility theory, namely, the challenge from unawareness. In fact, unawareness presents a challenge for all extant normative theories of choice. To keep things simple, we shall however focus on Savage’s expected utility theory to illustrate the challenge posed by unawareness. It can actually be seen as a weak version of Independence and the Sure Thing Principle, and it plays a similar role in Jeffrey’s theory.

## FREQUENTLY ASKED QUESTIONS

Leonard Savage’s decision theory, as presented in his The Foundations of Statistics, is without a doubt the best-known normative theory of choice under uncertainty, in particular within economics and the decision sciences. In the book Savage presents a set of axioms constraining preferences over a set of options that guarantee the existence of a pair of probability and utility functions relative to which the preferences can be represented as maximising expected utility. Nearly three decades prior to the publication of the book, Frank P. Ramsey had actually proposed that a different set of axioms can generate more or less the same result. Nevertheless, Savage’s theory has been much more influential than Ramsey’s, perhaps because Ramsey neither gave a full proof of his result nor provided much detail of how it would go . However, the ingredients and structure of his theorem will be laid out, highlighting its strengths and weaknesses. Lara Buchak has recently developed a decision theory that can accommodate Allais’ preferences without re-describing the outcomes.

The theorem is limited to evaluating options that come with a probability distribution over outcomes—a situation decision theorists and economists often describe as “choice under risk” . According to Yudkowsky and Soares, standard decision theories are based on general arguments and brute intuitions. Moreover, they say, representation theorems reveal that the maximising of expected utility is derived from the basic constraints on rational preference and belief. Therefore, they reject the preference norms of the CDT and offer a representation theorem for FDT.

## Alternatives to Probability

For example, a bank may have $1 million to lend and knows that the longer it waits, the more loan applicants it can screen before deciding who to lend the money to. An optimal stopping rule in this case would specify conditions under which the bank would find it most profitable to stop screening further loan applications. Another example is determining when a bank should stop acquiring additional information about a borrower, and make a decision.

Let us nonetheless proceed by first introducing basic candidate properties of preference over options and only afterwards turning to questions of interpretation. As noted above, preference concerns the comparison of options; it is a relation between options. For a domain of options we speak of an agent’s preference ordering, this being the ordering of options that is generated by the agent’s preference between any two options in that domain. Some decisions are difficult because of the need to take into account how other people in the situation will respond to the decision that is taken. The analysis of such social decisions is the business of game theory, and is not normally considered part of decision theory, though it is closely related. However even with all those factors taken into account, human behaviour again deviates greatly from the predictions of prescriptive decision theory, leading to alternative models in which, for example, objective interest rates are replaced by subjective discount rates.

- Former university professor Leonard Jimmie Savage, the author of “The Foundations of Statistics,” outlined the different conditions of uncertainty that exist in modern-day decision-making theory.
- In the book Savage presents a set of axioms constraining preferences over a set of options that guarantee the existence of a pair of probability and utility functions relative to which the preferences can be represented as maximising expected utility.
- The analysis of such social decisions is often treated under decision theory, though it involves mathematical methods.
- Consequences are the features of a decision made that influence a decision-maker on a micro-level, i.e., whether an individual feels rested.
- It then follows that for any other proposition \(s\) that satisfies the aforementioned conditions that \(r\) satisfies, you should also be indifferent between \(p\cup s\) and \(q\cup s\), since, again, the two unions are equally likely to result in \(s\).

It is only by imposing overly strong conditions, as Savage does, that we can achieve this. Unfortunately, Bolker’s representation theorem does not yield a result anywhere near as unique as Savage’s. Even worse, the same preference ordering satisfying all these axioms could be represented as maximising desirability relative to two probability functions that do not even agree on how to decision theory is concerned with order propositions according to their probability. Perhaps there is always a way to contrive decision models such that acts are intuitively probabilistically independent of states. Recall that Savage was trying to formulate a way of determining a rational agent’s beliefs from her preferences over acts, such that the beliefs can ultimately be represented by a probability function.

## Utility measures of preference

On Buchak’s interpretation, the explanation for Allais’ preferences is not the different value that the outcome $0 has depending on what lottery it is part of. However, the contribution that $0 makes towards the overall value of an option partly depends on what other outcomes are possible, she suggests, which reflects the fact that the option-risk that the possibility of $0 generates depends on what other outcomes the option might result in. To accommodate Allais’ preferences , Buchak introduces a risk function that represents people’s willingness to trade chances of something good for risks of something bad. Savage showed that whenever these six axioms are satisfied, the comparative belief relation can be represented by a uniqueprobability function.

This approach encapsulates specification concerns by formulating a set of specific possible models and a prior distribution over those models. Below we raise questions about the extent to which these steps can really fully capture our concerns about model misspecification. Concerning , a hunch that a model is wrong might occur in a vague form that “some other good fitting model actually governs the data” and that might not so readily translate into a well-enumerated set of explicit and well-formulated alternative models g(y|x, d, α). Concerning , even when we can specify a manageable set of well-defined alternative models, we might struggle to assign a unique prior π(α) to them.

Kasper’s proposed DSS design theory for user calibration prescribes properties of a DSS needed for users to achieve perfect calibration, meaning that one’s belief in the quality of a decision equals the objective quality of the decision. In some special cases, however, like the economic theory of perfect competition, these complexities are absent. Has retained its early shape, but with continuing debate about the interpretation of probability and the means to be used for estimating it. The main line of development has led to the theory of maximizing expected utility. In addition, for some decision problems, maximization of the expected utility can only be accomplished through numerical or simulation methods (see Bielza et al.

Indeed, this may be one of the main reasons why economists have largely ignored Jeffrey’s theory. Economists have traditionally been skeptical of any talk of a person’s desires and beliefs that goes beyond what can be established by examining the person’s preferences, which they take to be the only attitude that is directly revealed by a person’s behaviour. For these economists, it is therefore unwelcome news if we cannot even in principle determine the comparative beliefs of a rational person by looking at her preferences. In order to understand the mechanisms underlying rational choice, it is helpful to first understand the nature of the certain-thing principle, which Savage developed. This principle says that we cannot rationally fear a risk, and so, we must always choose the most likely alternative. However, this principle is not always applicable, as many situations call for a different choice.

Rather, decision-makers must consult their own probabilistic beliefs about whether one outcome or another will result from a specified option. Decisions in such circumstances are often described as “choices under uncertainty” . For example, consider the predicament of a mountaineer deciding whether or not to attempt a dangerous summit ascent, where the key factor for her is the weather. If she is lucky, she may have access to comprehensive weather statistics for the region. Nevertheless, the weather statistics differ from the lottery set-up in that they do not determine the probabilities of the possible outcomes of attempting versus not attempting the summit on a particular day. Not least, the mountaineer must consider how confident she is in the data-collection procedure, whether the statistics are applicable to the day in question, and so on, when assessing her options in light of the weather.

In winter, when it is known that there is a 5% chance that the ship and cargo will be lost. In his solution he defines a utility function and computes expected utility https://1investing.in/ rather than expected financial value. A statistical decision rule that tells the decision-maker when to stop a sequential sampling process and make a decision.

## SPX vs SPY: Which is Better for Trading Options on the S&P 500?

Indeed, the probability of each \(p_i\) is explicitly conditional on the \(p\) in question. When it comes to evaluating acts, this is to say (in Savage’s terminology) that the probabilities for the possible state-outcome pairs for the act are conditional on the act in question. CBDT predicts that people choose by combining, in a specific way, the hypothetical memory with their personal assessment of similarity across types and locations of real estate. The theory’s axioms impose behavioral restrictions reflecting the consistency of similarity weights across decisions.

An important corollary to this is the recognition that accepting deontic constraints on choices makes good sense from the point of view of a modest view of one’s ability to anticipate the effects of choices, reflecting the value of prudence and the limits of predictive powers. Three different types of uncertainty can be found in decision-making theory – States, Consequences, and Actions. There are three different types of uncertainty that can be found in decision-making theory –States, Consequences, and Actions. The choices come with consequences and are usually discussed in two separate but distinct branches.

Richard Bradley defends a similar principle in the context of the more general Jeffrey-style framework, and so does Roussos ; but the view is criticised by Steele and Stefánsson (forthcoming-a, forthcoming-b) and by Mahtani . To the extent that decision theory can be reconciled with the full range of ethical theories, should we say that there are no meaningful distinctions between these theories? Brown and Dietrich and List demonstrate that in fact the choice-theoretic representation of ethical theories better facilitates distinctions between them; terms like “consequentialism” can be precisely defined, albeit in debatable ways. More generally, we can catalogue theories in terms of the kinds of properties that distinguish acts/outcomes and also in terms of the nature of the ranking of acts/outcomes that they yield . Another important thing to notice about Jeffrey’s way of calculating desirability, is that it does not assume probabilistic independence between the alternative that is being evaluated, \(p\), and the possible ways, the \(p_i\)s, that the alternative may be realised.