# Decision Theory

Decision theory is the study of an agent's choices. It is not a single theory, but a collection of models, each with its own assumptions and implications.

The most important thing to know about decision theory is that it doesn't give you a definite answer for every situation. Instead, it provides tools for thinking about problems in a clear and logical way.

There are many different ways to approach decision making, but all of them fall into one of two categories:

1. normative decision theory, which prescribes how you should make decisions, and
2. descriptive decision theory, which tries to describe how people actually make decisions.

## What is decision theory?

Decision theory is a logical study of how decisions are made in structures or systems where the decision environment is uncertain and the decision variables unknown. It is used to understand how customers make decisions and to make better business decisions. Decision theory has two forms: normative and descriptive. Its components include probability, uncertainty, the rational actor model, the expected utility model, and the Principle of Rationality. The rational actor model assumes that everyone seeks to maximize their own happiness when making a decision; the expected utility model assumes people will make decisions based on what they believe is most likely to happen; and finally, the Principle of Rationality states that people will act in ways consistent with their goals and beliefs.

## What are the philosophical foundations of decision theory?

The significance of decision theory is debated, but it is of great interest to philosophers of mind and psychology. Decisions theories have broad implications for debates in epistemology and philosophy of science, as it provides a framework for understanding individual choices and responding to new information. It can also be used to help us understand the behaviour of other people by reading their beliefs and desires from their chosen positions. Decision theory has a wide range of practical applications, including economics, mathematics, marketing, data and social sciences, biology, psychology politics and more.

## What are some examples of practical applications of decision theory?

1. Making real decisions

Decision theory is a tool used to understand and make difficult decisions. Examples of practical applications of decision theory can include choosing between two options, evaluating the consequences of a choice, and making investments or hiring decisions based on financial advice from professionals. Examples also include electing a political leader based on public opinion polls, buying products based on advertisements or testimonials, and selecting restaurants based on reviews by friends. Decision theory has applications in ethics, business, and law. Those looking for more information about decision theory can find additional resources such as journal articles and textbooks.

2. Sequential decisions

Decision theory can be applied to sequential decisions in order to understand the preferences of the decision-makers and assess which action should be taken at any given moment. This is done by comparing static and sequential decision models, with static models being better suited for certain types of decisions than the latter. Sequential decision problems involve a series of decisions that cannot adequately be modeled using static decision theory, so more complex scenarios must be handled using this approach instead. In addition, applying decision theory to sequenced decisions can help shed light on why people make the choices they do and why they may resist changing them.

3. Utilizing Jeffrey's theory

Jeffrey's theory is an approach to decision-making that takes into account the subjective perspective of the agent. It departs from Savage's theory by allowing for models based on how the agent perceives them, and does not separate between acts and outcomes or states of the world. Jeffrey's equation states that a proposition's desirability depends on the desirabilities of all possible ways in which it can be true, giving rise to a conditional expected utility formula which is used to evaluate a choice’s potential benefits and risks. This enables decisions to be made about what is desirable without differentiating between desired and undesired outcomes.

4. Utilizing Savage's theory

Savage's decision theory is a normative theory of choice under uncertainty which guarantees the existence of a pair of probability and utility functions. It allows for more flexibility and creativity in decision making because it considers outcomes to depend on the state of the world, rather than assigning probabilities to them. Savage's theory can be used to make decisions based on probabilities, when there is uncertainty or multiple options available. This is done by identifying different types of acts that lead to different outcomes and calculating the expected value (based on associated probabilities) for each option.

5. Understanding risk and regret attitudes

Risk and regret attitude is a way of thinking about decision making that takes into account the risks associated with different outcomes. It assumes that outcomes are separable and independent of one another. Savage's version of expected utility theory does not allow for value interactions between outcomes, which can lead to issues when making decisions. To address this problem, probability theory or value interactionism can be used to take into account how different outcomes might affect the probability or value of an option. Risk and regret are also important factors in Allais' paradox, as people often prefer options which generate it in violation of expected utility theory. The separability assumption fails when outcomes are dependent on each other, and people may have preferences based on the extra chance for one outcome over another (in this case, \$0). By taking risks and regrets into consideration, Allais' preferences become compatible with expected utility models when changes in indifference points occur due to risk or regret variables being taken into account.

6. Applying von Neumann and Morgenstern (vNM) representation theorem

The vNM representation theorem can be applied in decision-making to determine the optimal choice between different options. This is done by evaluating each option and determining how their expected utilities must be maximized in order to satisfy the vNM principle. For example, weighing a gamble where there is a 1-in-10 chance of losing one's life and a 10-in-100 chance of gaining \$10 against other options with similar outcomes can help make decisions based on an individual's preferences. Additionally, using the axioms of Independence and Continuity can help ensure that an agent’s preferences will not be affected by extraneous factors when making decisions.

7. Examining broader implications of Expected Utility (EU) theory

The implications of EU theory are that people should have consistent preference attitudes and prefer the means to their ends. This suggests that decision-makers should take into account a variety of factors, including their confidence in the data collection procedure as well as their intuition about the situation, when making decisions under uncertainty. Additionally, Savage's and Jeffrey's results demonstrate that satisfying preferences for an agent requires beliefs which are consistent with her tastes.

8. Utilizing cardinalizing utility

Cardinalizing utility can be used in decision-making to prioritize options by comparing desirabilities. This is done by taking into account the chance of each prize and the desirability of that prize when comparing two options. It is also possible to calculate the expected utility of a lottery when utilities are cardinal, which can help determine how desirable an option is. Using this method, a lottery with an expected utility of 3/4 would be considered desirable since it corresponds to the expected desirability or "expectedutility" of it.

9. Exploring ordinal utilities

An ordinal utility is a type of utility function which takes into account the order in which items are consumed. It can be represented by either an ordinal or interval-valued utility function and is used to construct weak orders of prospects or options. In decision theory, an ordinal utility can help make consistent decisions about how much of each resource to allocate and provide guidance on preferences over lotteries in risky prospects. The comparison of expectations between two sets of ordinal utilities is reversed when the highest utility in the second set is increased from 5 to 10.

10. Examining utility measures of preference

Utility is a measure of how desirable or preferable an option is to an individual. It can be measured through comparison with other options, typically using lotteries to determine the riskiness of the choice. The expected utility of a lottery is the sum of all possible outcomes and their respective utilities, taking into account which outcomes are combined together. In order to accurately represent someone's preferences using a utility function, more information than just preference ordering over three options must be revealed. Von Neumann and Morgenstern's 1944 paper provides an account on how one can use cardinality (interval-valued) utility representation for measuring preference orderings.

11. Analyzing challenges to EU theory

EU theory is an approach to decision-making that takes a stance on the structure of rational desire. Proponents of EU theory argue that it can be used to analyze challenges raised by critics who worry that the theory is either too permissive or not permissive enough with respect to what influences an agent's desires. Additionally, proponents argue that preferences which seem to violate Transitivity or Separability can be justified as long as the options being compared vary in their description depending on the other options under consideration. Nevertheless, some critics suggest that this notion of EU theory as a standard of rationality is vacuous and impotent because it stretches what are genuine properties of outcomes which can confer value or be desirable for an agent. In response, there are two main reactions: resisting the claim and embracing it; with each one assuming different constraints on content for an agents preferences (empirical constraints versus normative constraints). Ultimately, EU theory does not go far enough in capturing why an agent chooses one option over another; however Dietrich and List proposed a more general framework which captures different reasons for preference, while still allowing restrictions on what counts as a legitimate reason for preference.

12. Examining completeness: vague beliefs and desires

The significance of examining completeness in decision theory is that it allows for certainty about the ranking of options. By considering all relevant factors and allowing for epistemic uncertainty, a choice function that respects the agent's preferences can be determined, providing for a more comprehensive decision-making process. Additionally, Levi's condition on admissibility can be used to rule out EU-dominated options. Thus, by looking at completeness in this context, agents are able to make decisions with greater confidence and accuracy than without taking it into account.

13. Understanding rational belief

Decision theory helps us understand rational belief by providing a model for assessing the value of free evidence before making a decision. The theorem assumes that the person seeking free evidence knows the Bayesian learning rule, conditionalization, which is used to help them interrogate their degrees of belief and reflect on their pragmatic implications. Through this process, Bayesian decision theory also underpins a range of epistemic norms, such as the non-negative value of free evidence.

14. Understanding rational desire

Rational desire is a decision theory which takes a stance on the structure of an agent's desires, with the goal of making it more realistic and explanatory. It suggests that preferences are determined by a trade-off between fit and simplicity, taking into account minimal constraints based on different types of context-dependence in an agent's choices. Applying rational desire practically can help to clarify the normative basis for decision making by providing explicit restrictions on what counts as a legitimate reason for preference. Additionally, it allows for more detailed analysis of why an agent has certain preferences or desires, allowing them to make their own decisions based upon rationality rather than emotion or other external forces.