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Bernoulli Utility

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Understanding Bernoulli Utility: Making Rational Decisions Under Uncertainty



We all make decisions daily, weighing potential gains against potential losses. But what happens when those gains and losses are uncertain? This is where the concept of Bernoulli utility comes into play. Developed by Jacob Bernoulli in the 17th century, Bernoulli utility provides a framework for understanding how we, as rational individuals, should make decisions when faced with risk. It essentially helps us quantify how much we value different outcomes, considering both the potential payoff and the likelihood of achieving it.

1. The Foundation: Utility and Diminishing Marginal Utility



At its core, Bernoulli utility focuses on the concept of utility. Utility represents the satisfaction or happiness derived from a particular outcome. Crucially, Bernoulli argued that the increase in utility from gaining an additional amount of something diminishes as you already possess more of it. This is known as the principle of diminishing marginal utility.

Imagine you're incredibly thirsty. The first glass of water brings immense satisfaction (high utility). The second glass still provides relief, but not as much as the first (lower marginal utility). By the fifth glass, the additional utility is minimal. This illustrates diminishing marginal utility – each additional unit provides less and less extra satisfaction.

2. Expected Utility: Weighing Probabilities and Payoffs



Bernoulli’s insight wasn't just about the utility of an outcome; it was about the expected utility. Expected utility considers both the potential utility of each outcome and the probability of that outcome occurring. It's calculated by multiplying the utility of each possible outcome by its probability and then summing these values.

Let's say you have a choice:

Option A: A guaranteed gain of $100 (utility = 100 units).
Option B: A 50% chance of gaining $300 (utility = 300 units) and a 50% chance of gaining nothing (utility = 0 units).

To determine the expected utility of Option B, we calculate: (0.5 300) + (0.5 0) = 150 units. In this simplified example, Option B has a higher expected utility (150) than Option A (100), suggesting a rational individual would choose Option B. However, the actual utility values depend on individual preferences.

3. Risk Aversion and the Shape of the Utility Function



The shape of the utility function visually represents the relationship between wealth and utility. For most people, this function is concave, reflecting risk aversion. A concave function implies that the increase in utility from an additional dollar decreases as wealth increases. Risk-averse individuals prefer a certain outcome to a gamble with the same expected value. They would choose the guaranteed $100 over the gamble, even though the gamble has a higher expected monetary value.

Conversely, a convex utility function represents risk-seeking behaviour, while a linear function indicates risk neutrality. Risk-neutral individuals only care about the expected monetary value, disregarding the risk involved.

4. Practical Applications: Beyond Gambling



Bernoulli utility isn't just about casino games. It has broad applications in numerous fields, including:

Finance: Investors use it to evaluate investment opportunities, weighing potential returns against risks.
Insurance: Insurance is a prime example of risk aversion in action. People pay a premium to avoid a potentially large loss, even if the expected value of the insurance is negative.
Healthcare: Decisions regarding medical treatments often involve weighing the potential benefits against risks and costs.
Economics: It informs models of consumer behaviour, predicting choices based on preferences and risk attitudes.

5. Actionable Takeaways and Key Insights



Understanding Bernoulli utility helps us make more rational decisions under uncertainty. By considering both the potential outcomes and their probabilities, and by acknowledging our individual risk preferences, we can make choices that better align with our goals and values. Recognising diminishing marginal utility allows us to make more informed decisions about resource allocation.

FAQs



1. Q: Is Bernoulli utility always accurate in predicting real-world decisions? A: No, it's a model, and real-world behaviour can be influenced by factors not considered in the model, like emotions, cognitive biases, and framing effects.

2. Q: How can I determine my own utility function? A: This is challenging, often requiring carefully designed experiments that involve choices under risk. However, introspection and observing your own behaviour can offer some insights.

3. Q: What is the difference between expected value and expected utility? A: Expected value is the average monetary outcome, while expected utility considers the subjective value (satisfaction) derived from each outcome, weighting it by probability.

4. Q: Does Bernoulli utility assume perfect rationality? A: Yes, the model assumes individuals are perfectly rational and can accurately assess probabilities and utilities. In reality, this is often not the case.

5. Q: Are there alternatives to Bernoulli utility? A: Yes, more sophisticated models like prospect theory address limitations of Bernoulli utility by incorporating cognitive biases observed in actual decision-making.


By understanding the principles of Bernoulli utility, we gain valuable tools for analyzing decisions under uncertainty and making choices that better align with our preferences and goals. It provides a framework for thinking critically about risk and reward in various aspects of life.

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Bernoulli Utility Function explained with modeling example. In this tutorial, you will learn basically what is a Bernoulli utility function, and how to use a Bernoulli Utility function in a decision tree. What is the Bernoulli Utility Function? Bernoulli suggests a form for the utility function in terms of a differential equation.

Probability - Daniel Bernoulli's Utility - Stanford University We focus on maximizing the part of utility that depends on wealth, namely Bernoulli’s utility. How do factors besides wealth, such as health, affect well-being? How should wealth be spent?

Worldwide Gen-Set & Cogeneration Directory 2024 - Issuu Contact: Jonas Ohrn tel: +46 46 385510 fax: +46 46 385519 www.bernoulli.se e-mail: [email protected] Manufacturing and sales of automatic self cleaning filters for liquids.

Lipschitz Bernoulli Utility Functions | Mathematics of 23 Dec 2022 · We obtain several variants of the classic von Neumann–Morgenstern expected utility theorem with and without the completeness axiom in which the derived Bernoulli utility functions are Lipschitz. Th...

Expected Utility Theory - Economics Online 31 Oct 2024 · Expected utility theory says that people make decisions to maximise their expected utility according to their risk tolerance. The historical background of expected utility theory has its roots in the work of a Swiss mathematician, Daniel Bernoulli, in the 18 th century.

Expected utility hypothesis - Wikipedia Bernoulli made a clear distinction between expected value and expected utility. Instead of using the weighted outcomes, he used the weighted utility multiplied by probabilities. He proved that the utility function used in real life is finite, even when its expected value is infinite.

Cardinal utility - Wikipedia In 1738, Daniel Bernoulli was the first to theorize about the marginal value of money. He assumed that the value of an additional amount is inversely proportional to the pecuniary possessions which a person already owns.

Expected Utility Theory - Economics Help Bernoulli in Exposition of a New Theory on the Measurement of Risk (1738) argued that expected value should be adjusted to expected utility – to take into account this risk aversion we often see.

Expected Utility Theory - SpringerLink Expected utility theory (EUT) originates from the eighteenth-century mathematician Daniel Bernoulli who in 1738 resolved an interesting paradox known as the St. Petersburg paradox (why were people only willing to pay a small amount for a risky gamble with an infinite expected monetary value?).

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von-Neumann-Morgenstern v. Bernoulli Utility Function The v.NM function maps from the space of lotteries to real number as it represents the preference defined on the lottery space while the Bernoulli is defined over sure amounts of money. Why is this distinction so important in the theory of expected utility?

(PDF) Lipschitz Bernoulli utility functions - ResearchGate 22 Apr 2021 · We obtain variants of the classical von Neumann-Morgenstern expected utility theorem, with and without the completeness axiom, in which the derived Bernoulli utility functions are Lipschitz.

1 Basic Concepts - Princeton University In other words, there is a utility function u defined over consequences, and a lottery is evaluated by the mathematical expectation or expected value of this utility. The underlying u function is sometimes called a Bernoulli utility function or a von Neumann-Morgenstern

Notes on Uncertainty and Expected Utility - UC Santa Barbara Expected utility theory has a remarkably long history, predating Adam Smith by a generation and marginal utility theory by about a century. 1 In 1738, Daniel Bernoulli wrote:

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Expected Utility Theories: A Review Note | SpringerLink 31 Jul 2018 · Although the seeds of expected utility theory were sown almost two and one-half centuries ago by Daniel Bernoulli (1738) and Gabriel Cramer, the first rigorous axiomatization of the theory was developed by John von Neumann and Oskar Morgenstern (1944).

Normative Theories of Rational Choice: Expected Utility 8 Aug 2014 · Bernoulli (1738) argued that money and other goods have diminishing marginal utility: as an agent gets richer, every successive dollar (or gold watch, or apple) is less valuable to her than the last.

Bernoulli's Hypothesis: What it Means, How it Works - Investopedia 30 Nov 2021 · Bernoulli's hypothesis states a person accepts risk both on the basis of possible losses or gains and the utility gained from the action itself. The hypothesis was proposed by mathematician...

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Expected Utility Theory: Unraveling Its Mysteries and Practical ... 28 Mar 2024 · Expected utility theory aids decision-making under uncertainty. Weighted average of utility and probability defines expected utility. Daniel Bernoulli’s contribution solved the St. Petersburg Paradox. Expected utility influences choices in scenarios like insurance and lottery ticket purchases.