# Expected Utility

We always have to make decisions under uncertain conditions. For instance, pursuing a degree in economics may lead to employment with a large company, or unemployment and leave with crushing debt. Or a doctor's appointment may lead to early detection and treatment of serious diseases, or it could bet a total waste of money. Getting a job in a certain company can be the best way to live the life you want, or it can be the beginning of a stressful career path. All these scenarios show that life is always compelling use to make decisions. And it is like the two sides of a coin where you only have to choose one. But we make our decisions based on the best-expected outcome. Human beings are rational decision-makers who decide based on the risk level. Expected utility theory tells it explains how to choose rationally when one is not sure about the outcome of the action. This model can be summarized in the slogan "choose the action with the highest expected utility."

So, how do, or should people make decisions? Expected utility is a normative theory, which means it tries to answer this question. In other words, it is a model of how people SHOULD make decisions, how they DO make decisions, or a PREDICTIVE theory. This theory may not be the perfect model of the psychological approach to decision-making, but it offers a precise prediction of people's choices. The theory attempts to make faulty predictions about people's decisions in many real-life decisive situations (Kahneman & Tversky 1982). However, it does not tell whether decision-makers should base their choices on expected utility consideration.

In this unit, we shall be discussing the theory of expected utility. The expected utility of an action is seen as the weighted average of every possible outcome's utilities. Here, the utility of an outcome measures the extent of the outcome's preferability in relation to the alternatives. In other words, we make a decision according to the probability that the act will lead to the desired outcome. As rational decision-makers, human beings do not like to lose. Every decision they make has to look like the best choice for their lives. We shall be expounding on this basic definition using more terms that explain decision-making processes and the relationship between expected utility and choice. We will also briefly highlight the two main arguments used in explaining expected utility: representation theorems and long-term statistical arguments.

## What is the expected utility?

Expected utility is a concept that can be best explained using examples. Terms alone will not be enough to give you the right understanding. Let's say you want to go for a walk that might take several hours, and you need to either take and umbrella or leave it. How would you determine the best decision? They would rather not struggle under a sunny day for many people, but they would rather go into the rain with one than leaving. Hence, there are two acts you need to take here, one is taking the umbrella, and the second is leaving at home. Which one is the best course of action?

This is an informal recast that can get a bit more formally when three types of entities are involved:

- First, the outcomes are inevitable. These are the objects of non-instrumental preferences. In the example above, you can end up dry and without an umbrella, or dry and encumbered with a heavy, unstylish umbrella, or up wet because you left the umbrella.

- Second, the states. These are things outside that decision-makers cannot control and impacts the outcome of their decisions. In the example used above, it is either going to rain or not. This is the issue of the weather, which is outside your control.

- Third, acts. These are the objects of decision-makers' instrumental preferences. To some extent, these are things you can do as a decision-maker. In the example used, two actions are possibilities: one to bring the umbrella or to leave it at home.

Expected utility theory offers a way of listing and ranking the acts based on how choice-worthy they are. Higher expected utility is more considerable in choosing the act. This means it is better to choose the action with the highest expected utility, or at least one, where several acts are involved. Based on the general convention, we can make several assumptions about the relationship between acts, states, and outcomes.

- We can say that states, acts, and outcomes are propositions. This means there are sets of possibilities. A maximal set of possibilities, Ω, of which each state, act, or outcome is pre-set.

- The group of acts, groups of states, and a group of outcomes are all partitions on Ω. This is to say acts and states are all made individual, where the possibility in Ω, is one a point where the one state obtains and the agent performance on exact action to ensure a specific outcome.,

- We can also assume that acts are logically independents. No state rules out the performance of any act.

Looking at the example of the umbrella above, we can represent this argument in several matrices, where each entity is a representation of a different state on the world, the act, and the outcomes are performed in the real world. This is a framework that can be used to understand the real meaning of expected utility by stating that the expected utility of act A, (like taking the umbrella on your walk), is determined by two in critical features if the problem is faced:

- That the value of the individual outcome, which is measured by a real number, is called utility.

- And that probability of each outcome depends on A.

There are two terms used often explaining expected utility:

### Conditional probabilities

This factor represents how likely it is that the outcome will occur, supposing the agent chooses A. To get the right idea here, we must first look at which interpretations of probability is appropriate, and secondly, what it means to assign probability on the supposition that the decision-maker will settle on A. Theorists of expected utility often consider probability as a measurement of the individual degree of belief. In other words, preposition E is likely to the extent that the agent is confident of E. We can interpret formalism as an objective I chances.

### Outcome utilities

The term U (o) can be used to represent utility outcome o – roughly, how valuable o is. U is a function o that gives a real number to each of the outcomes in informal situations. The most important question here concerns the kind of value measured in utiles – typically not taken as currency units. According to Bernoulli (1738), money and more goods have a diminishing marginal utility, whereby, as the agent gets richer, every successful coin is less valued than the last.

Bentham (1738), classic utilitarian, defines utility as a measure of happiness or pleasure. Therefore, Saying that has a greater utility than B is to say that A brings out more pleasure or happiness. However, this interpretation has been faced by several objections, with one stating that there may not be a single good with rationality expected to gain.

Modern decision theorists look at utility as the measure of preference. This means saying A has a greater utility than B means the agent will choose A over B. We need to understand the Expected utility theory does not require that preferences are selfish or ego-centric. For instance, a decision-maker may be more willing to give money to a charity that spends the same on lavish dinners. Or a parent sacrificing their life by donating organs, then allow their child to die.

## Thoughts on expected theory

Sometimes understanding why someone can choose acts that maximize expected theory can be harder than you may think. But there are a few arguments that can help you get the right information much faster. One good solution is that the expected utility theory is rational bedrock. In simple terms, end rationality takes into account the maximization of expected utility. Some analysts, or even students, may find this argument less satisfying. But there are many other sources of justification that bring out the real picture. These are: long term arguments and representation theorems

Under LONG-RUN ARGUMENTS, maximization expects utility because it makes good policy in the long term. Feller (1968) looks at this argument from his own perspective, considering two mathematical facts about probability, which are the strong and weak laws of numbers. In each of these facts, there is a consideration of sequences of independent, identically in distributed trials. This is the kind of setup that comes from betting the same repeatedly on a sequence of spins games.

As expected, there are several objections to long arguments. For instance, many decisions cannot be repeated over indefinitely similar trials. Such decisions as which career one should pursue, whom they should marry, where to live, and many others are repeated over a small number of times. Also, when they are made more than once, the decisions cause different outcomes. Another argument is based on the independence assumptions, one or both, which may not be successful. One argument state that the probabilities of different trials are independent. This can be associated with casino gambles, though not true with other choices, like medical treatment choices. Another argument is that the strong and weak laws of large numbers are modally weak. None of the laws handles the fact that if a gamble were placed over and over indefinitely, the expected utility of the game would close with the average utility gain per trial.

REPRESENTATION THEOREMS is another argument of expected utility, which relies on. Zynda (2000), tries to formulate this argument by slightly modifying it to represent the role of utilities and probabilities. There are three sections of the argument: The rational condition, where the axioms of expected utility theory are used for rational preferences; representability, where one can be seen as having statuses of the belief that obey the laws of probability calculus if their preferences are in line with the axiom of expected utility theory; the reality condition, where a some represented as having degrees of belief in line with probability calculus can obey the laws of the calculus.

In this case, if decision-makers fail to pick acts with higher utility, they are seen to have violated axioms (or at least one) of rational preferences. If there is any truth in these preferences, then the argument means there is a problem with people whose preferences do not go with expected utility theory. Representation theorems are mathematical proofs of representability. No matter the axioms used, Rationality Condition and Reality Condition are controversial. Anyone trying to defend this argument can argue that what it is to have specific degrees of belief in and utilities is all about correspondent theorems.

Three influential representation theories differ from each other in three ways. One that these theorems disagree on the objects of preference, they differ in their treatment of probability. Each one proves that there are a probability and utility function representation for every suitable preference ordering. In these situations, it is easy to argue that representation theorems are perhaps the most suitable approach to understanding expected utility. Nevertheless, there are many arguments against is that it cannot be ignored.

#### Conclusion

Expected utility theory is a model that estimates the likely utility of a decision. There is uncertainty in the outcome, and the theory suggests that the rational choice is to pick the action with higher expected utility.

In economics and public policy, the expected utility gained traction in the 1940s and 50s across the US because it has the potential to provide a mechanism that would define the behavior of macro-economic variables. It has since become one of the main reference points when looking at consumer behaviors.

In public policy, it has several applications too. Harsanyi (1953), picks from this theory to explain how most socially just arrangements is one that maximizes combined welfare distribution among societies. It is, therefore, a critical subject to look at for economy students.

Author: James Hamilton