In day-to-day usage, utility refers to the subjective value, desirability, or satisfaction derived from choosing an alternative. Decision-making theories formalize utility as a real-valued function that represents a preference relation, commonly known as the utility function. This function, expressed in terms of real numbers, is crucial in the mathematical analysis of decision-making phenomena. The utility function assigns a numerical value to each possible outcome, reflecting the individual's preference or satisfaction level. More importantly, it allows us to predict and explain these phenomena, providing a sense of reassurance in the decision-making process. Expected utility theory is a central framework for decision-making under risk, using the concept of the expected value of utility, known as expected utility, to explain decision-making processes.

According to this theory, the following equation can be represented to illustrate the utility of carrying an umbrella when venturing outdoors:

EU (taking an umbrella when going out) = p1 (it rains) x u1 (going out with an umbrella when it rains) + p2 (it does not rain) x (u2 (going out with an umbrella when it does not rain).

In this equation, p1 and p2 are the probabilities of the respective events occurring, and p1 + p2 = 1 according to the axioms of probability.

For instance, imagine you are deciding between two risky investments.

Investment A: Offers a 50% chance of winning ₹10000 and a 50% chance of losing ₹2000.

Investment B: Offers a guaranteed gain of ₹3000

To compare these choices mathematically, we can define a utility function, U(x), that assigns a real number value to each possible outcome x. This value represents the level of satisfaction or benefit you get from that outcome. Here is a possible utility function:

U(x) = x0.5

where x is the amount of money gained or lost. This function assigns higher utility values to larger gains, but the increase in utility slows down as the gain increases (due to the square root).

We can calculate the expected utility of each investment by considering the probability of each outcome and its corresponding utility

Expected utility of investment A = (0.5 x U(₹10000)) + (0.5 x U(₹2000))

= (0.5 x 100) + (0.5 x 44.72)

= 50 + 22.36

= 72.36

Expected utility of investment B = U(₹3000)

= 54.77

By comparing the expected utilities of A and B, we can make a mathematically informed decision. In this case, if U(₹10000) > 2 x U(₹3000) (utility gain from a large win outweighs the potential loss and guaranteed gain), then investment A would have a higher

expected utility. This suggests the model prefers the riskier option A for potentially higher returns. This quantitative approach to utility is also implemented in decision-support technologies, including computer-based tools that are designed to provide decision-makers with the necessary assistance to achieve satisfactory outcomes.

The origins of expected utility theory can be traced back to the 18th century, when Daniel Bernoulli applied the theory to resolve the St. Petersburg paradox, a problem conceived by Nicolaus Bernoulli in 1713. The problem got its name from the place where it was first analyzed in detail. Daniel Bernoulli published his analysis of the problem in 1738 in the Commentaries of the Imperial Academy of Science of St. Petersburg. The publication brought widespread attention to the problem. Its paradoxical nature resulted in its name “St. Petersburg Paradox.” The paradox highlights a discrepancy between expected value and decision-making behavior.

The problem: Imagine a game of chance where you flip a fair coin repeatedly until it comes up heads. The payout of the game depends on the number of flips it takes to get heads. Specifically, if heads appear on the nth flip, the payout is 2n dollars. For instance, if heads appear on the first flip, you get $2; if it appears on the second flip, you get $4; on the third flip, $8, and so on.

The expected value (EV) of this game can be calculated by summing the products of the probabilities of each outcome and their corresponding payouts:

EV = 2 () + 4 () + 8 (1/8) + + n (1/n)

= ∑(12𝑛 x 2𝑛)𝑛−1

=∑1𝑛−1

=

Thus, the expected value of the game is infinite, which suggests that a rational person should be willing to pay any finite amount of money to play this game. However, in reality, most people would not pay a large amount to enter this game, which contradicts the theoretical prediction of infinite expected value. This discrepancy is known as the St. Petersburg paradox.

According to Daniel Bernoulli, the utility of money is not linear. Hence, instead of considering the expected value, we should consider the expected utility value. The expected utility value will consider the diminishing marginal utility of the money. If utility U is a logarithmic function of the money M, then

U(M) = log (M)

the expected utility will be

EU = ∑(12𝑛 x log (2𝑛))𝑛−1

=∑(12𝑛 x 𝑛log(2))𝑛−1

= log (2) ∑(𝑛2𝑛)𝑛−1

that converges to a finite value, resolving the paradox and showing that the expected utility is finite.

In reality, there are constraints such as limited resources, maximum payouts, and the finite nature of wealth, which make the infinite expected value impractical. People are inherently risk-averse. They prefer a certain smaller amount of money over a risky proposition with a higher expected value. The risk aversion also can be a reason that refrained people to pay a large amount to play the game.

Theories of decision-making are generally classified into three categories: normative, descriptive,

and prescriptive. Normative theories support the execution of rational decision-making. They offer

frameworks for the assessment of alternatives and the identification of the optimal choice. One

notable example is the expected utility theory, which is commonly used in the field of economics.

The expected utility of each alternative is determined by the likelihood of its occurrence and its

desirability. In order to accomplish this, it multiplies the utility of each outcome by its probability and

adds them all up. The most rational choice is the option with the highest expected utility. Utility

theory has numerous variations, and the majority of these are based on mathematically developed

principles and established axioms.

Descriptive theories, such as behavioral decision theory, play a significant role in understanding how

people make decisions, acknowledging both rational and irrational elements. Behavioral decision

theory, a significant example, encompasses a wide range of theoretical expressions, including those

developed mathematically and those explained solely in natural language. Psychologists, particularly

mathematical and experimental psychologists, have traditionally led the study of behavioral decision

theory, further divided into cognitive and social psychologists based on their research focus.

The discipline of decision-making has been traditionally dominated by normative theory. However,

the development of behavioral decision theory was necessitated by the growing recognition of the

gap between theoretical models and actual human behavior. Two major proponents of this theory

are Amos Tversky and Daniel Kahneman. They built upon the groundwork laid by W. Edwards's

seminal 1961 review article, "Behavioral Decision Theory." In recent years, behavioral decision

theory has gained significant momentum, influencing fields such as behavioral economics and

behavioral finance. This shift has displaced the one-sided emphasis on normative models, leading to

a rising trend of integrating insights from descriptive (behavioral) theory into normative frameworks,

thereby promoting a more sophisticated understanding of human decision-making.

The prescriptive decision-making approach is derived from prescriptions that are issued by

physicians. At the same time, it transcends mere compliance with physicians' directives. Its objective

is to facilitate rational decision-making by considering the specific circumstances of a given issue.

The prescriptive approach acknowledges that real-world complexities such as uncertainty,

ambiguity, and lack of information can render strict rules impractical, in contrast to normative

theory, which aims for universally optimal solutions. Although descriptive theory offers significant

insights into the actual decision-making process of individuals, relying solely on these observations

may not be sufficient to address complex issues. This is the point at which the prescriptive approach

becomes essential. It provides a framework for making optimal choices in real-world situations by

integrating knowledge from behavioral decision theory. The prescriptive approach is able to account

for potential biases and produce more effective decision-making by comprehending the manner in

which individuals think and behave under stress.

Decision-making broadly refers to the cognitive process of selecting an alternative from a set of available choices (Takemura, 2023). This process often involves evaluating different options, considering potential risks and benefits, and weighing our personal values and preferences. Selecting a preferred means of transportation, deciding which product to purchase based on features and budget, and determining which proposal to adopt after analysis are all examples of decision-making. Any alternative we choose will have an outcome, and the specific outcome may vary for the same alternative depending on the context or situation in which we make the decision. If A is the alternative, X is the outcome, and Θ is the event, then the decision-making function can be represented as

f: A x Θ = X.

There are situations when some outcomes are relatively more preferable than others. This preference, a crucial aspect of decision-making, is often based on a combination of factors, such as the desirability of the outcome itself and the probability of its happening. This concept of preference plays a key role in contrasting two main approaches to decision-making, the classical theory and the behavioral decision-making.

According to classical theory, we are perfectly rational decision-makers. We meticulously weigh the expected value of each outcome, considering both the desirability of the outcome and the probability of it happening. We then choose the alternative that leads to the optimal outcome, assuming we have access to all the necessary information and perfect cognitive abilities to process it. However, human decision-making often deviates from this classical idea. Our cognitive limitations and psychological factors, such as emotions, biases, and heuristics, can influence our choices in predictable and understandable ways. Behavioral decision theory acknowledges these limitations and explores how they can lead us to make suboptimal decisions compared to the classical model.

We must explore different decision-making forms to delve deeper into these contrasting approaches. These forms are categorized based on the level of information available about the outcomes of each alternative. Each form has its principles and assumptions, which influence how decisions are made and the resulting outcomes. Based on the information availability, the types of decisions are decision-making under certainty, risk, and uncertainty. In decision-making under certainty, we have complete knowledge about the outcomes of each alternative. This is a rare scenario in real-world decision-making. In decision-making under risk, we have some information about the probability of each outcome associated with each alternative. This is more common in our real-life situations. However, sometimes, the probabilities might not be precise in real-life situations.

In decision-making under uncertainty, we have limited or no information about the probabilities of the outcomes for each alternative. This can be further divided into decision-making under ambiguity and decision-making under ignorance. In decision-making under ambiguity, we know the possible outcomes but lack knowledge about their probabilities. In decision-making under ignorance, we are not even aware of all the possible outcomes.

Decisions are threads that weave our experiences, shaping our behaviours or actions. In practical contexts, behavioural decisions are influenced by a complex interplay of cognitive, affective, and social processes (Cristofaro, 2020; Sirigu & Duhamel, 2016). Our decisions are significantly influenced by our emotions. A range of emotions, from the desire for instant gratification to the weight of remorse, comprise affective processes and experiences (Ambrosino et al., 2008; Coricelli et al., 2007). Decisions to demonstrate a behavior are frequently influenced by the anticipation of pleasure or the fear of pain after that. Cognitive processes provide the tools to navigate through decisions. Information processing is the base of rational decision-making (Huber, 1980). (Goldsmith, 2015; Kahneman & Tversky, 2013)

The behaviours, standards, and anticipations of others frequently impact human decision-making (Ben-Akiva et al., 1999). Social processes include the influence of peers, authority figures, and cultural norms on our choices. We may conform to society, adopt the opinions of the individuals we respect, or adhere to societal expectations while making decisions (Claidière & Whiten, 2012). Looking in this way, behavioural decision-making will not happen in isolation. It is an outcome of multiple interactions between cognitive and affective factors and social processes factors, each playing a unique role in shaping the choices. The complexity of the decision, the availability of information, and potential consequences influence the extent to which behaviour we decide to demonstrate. 

Behavioural decision research is an ongoing investigation that aims to elucidate the complex mechanisms underlying human decision-making. By understanding the interplay of affective, cognitive, and social processes, we gain insights into the complexities of human behaviour and the factors that shape our decisions. This information provides us with the ability to make well-informed decisions, confront critical movements in life with heightened confidence, and recognize the complex interrelationships among elements that govern human behaviour.

References

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