Daniel Bernoulli (1700-1782) was a prominent Swiss-French mathematician and physicist, renowned for his contributions to various fields, including fluid mechanics, probability, and statistics. He was a key member of the famous Bernoulli family of mathematicians from Basel, Switzerland.
Here are some key points about Daniel Bernoulli:
Hydrodynamica: In 1738, he published his ground-breaking work titled “Hydrodynamica,” which laid the foundation for the study of fluid dynamics. This work introduced the Bernoulli principle, which describes the behaviour of fluid flow and the relationship between pressure and velocity.
Probability Theory: Bernoulli made significant contributions to probability theory, particularly through his work on the law of large numbers and the concept of expected value.
Kinetic Theory: He also contributed to the early development of the kinetic theory of gases, which explains the behaviour of gases in terms of the motion of their molecules.
Bernoulli’s work has had a lasting impact on various scientific disciplines, influencing fields such as engineering, economics, and biology.
Exposition of a New Theory on the Measurement of Risk (1738)
Daniel Bernoulli also addressed the limited willingness of people to engage in gambles based on utility maximization in his 1738 publication, Exposition of a New Theory on the Measurement of Risk. His insights are particularly encapsulated in his resolution of the St. Petersburg paradox, which highlights the discrepancy between expected value and actual decision-making behaviour, which suggested that people were not applying rational thinking to their financial decisions.
Here are some key points regarding Bernoulli’s contributions to this topic:
St. Petersburg Paradox: This paradox presents a gamble where the expected value is infinite, yet most people are unwilling to pay a high entry fee to participate. Bernoulli used this scenario to illustrate that individuals do not simply evaluate gambles based on expected monetary value. He also proposed that the value of a gambling prize is perceived differently based on the individuals economic and social status.
Utility vs. Wealth: Bernoulli proposed that people evaluate outcomes based on utility rather than just monetary value. He argued that the utility derived from wealth diminishes as wealth increases, meaning, for example, that a Prize of $500 has more value to a person earning $10,000 per annum, than a person earning $50,000. similarly, it has even more value, is that individual is in significant debt.
Expected Utility Hypothesis: He introduced the concept of maximizing expected utility instead of expected wealth. This means that individuals make decisions based on the expected utility of outcomes, which accounts for their risk preferences and the diminishing marginal utility of wealth. In short, people looking to appear wealthy, make decisions based on how they perceive others with view them, and that this can be used to make them pay more for products they regards as exclusive or status symbols.
Behavioural Implications: Bernoulli’s work laid the groundwork for modern expected utility theory, which is foundational in economics and decision theory. It explains why people might avoid certain gambles even when the expected value is favourable. This and other theories have been developed in order to allow people to exploited for financial gain by “higher thinker”, that later convinced most of America that they too could be rich, by spending all of their money on the right sales pitch.
It also highlighted that the majority of people seem to lack a basic understanding of logical and rational thinking, but tended to think they did, and would deliberately make poor choices, if it is suggested that their prior good choices were very poor in some way.
The St. Petersburg Paradox
The St. Petersburg Paradox is a fascinating concept in decision theory and economics that continues to be relevant today. Here’s a deeper look into it:
What is the St. Petersburg Paradox?
Origin: The paradox was introduced by Nicolaus Bernoulli in 1713. It presents a gambling scenario that challenges traditional notions of expected value.
The Gamble: In the game, a fair coin is tossed until it comes up heads. The pot starts at $2 and doubles with each toss. The player receives the amount in dollars corresponding to the number of tosses it takes to get heads. For example:
- If heads appears on the first toss, the player wins $2.
- If heads appears on the second toss, the player wins $4.
- If heads appears on the third toss, the player wins $8, and so on.
Expected Value Calculation: The expected value of this game is theoretically infinite because:
- The probability of getting heads on the first toss is 1/2 (win $2).
- The probability of getting heads on the second toss is 1/4 (win $4).
- The probability of getting heads on the third toss is 1/8 (win $8).
- This continues indefinitely, leading to an infinite expected value.
The Paradox: Despite the infinite expected value, most people would not be willing to pay a large entry fee to play this game. This discrepancy between expected value and actual willingness to pay is what makes it a paradox.
Modern Applications of the St. Petersburg Paradox
The St. Petersburg Paradox has significant implications in various fields today:
Risk Assessment: The paradox is used to illustrate how people evaluate risks and rewards, influencing modern gambling, insurance, and investment strategies.
Game Theory: The paradox continues to inspire new puzzles and insights in decision-making processes, contributing to ongoing research in game theory and strategic interactions.
Examples of the St Petersburg Paradox in Action
The St. Petersburg Paradox continues to be a significant concept in modern economics, decision theory, and behavioural finance. Here are some contemporary applications and examples that illustrate its relevance today:
Insurance and Risk Management
Example: When individuals purchase insurance, they often pay premiums that exceed the expected pay-out. This behaviour can be explained by the St. Petersburg Paradox, as people value the utility of peace of mind and financial security over the expected monetary value of the insurance policy.
Gambling and Betting
Example: In gambling scenarios, players may choose to participate in games with high potential pay-outs but low probabilities of winning. For instance, a lottery ticket might have a low chance of winning a large jackpot, yet many people buy tickets because the perceived utility of winning outweighs the cost of the ticket.
Investment Decisions
Example: Investors often face choices that resemble the St. Petersburg Paradox when evaluating high-risk, high-reward investments. For instance, investing in start-ups can yield substantial returns, but the probability of failure is high. Investors may still choose to invest based on the potential utility of a successful outcome.
Behavioural Economics
Example: The paradox is used to illustrate how people often make decisions that deviate from traditional economic models. For instance, individuals may avoid certain gambles even when the expected value is positive, reflecting their risk aversion and the diminishing marginal utility of wealth.
Game Theory and Strategic Decision-Making
Example: In strategic games, players may face decisions that involve uncertain outcomes with varying payoffs. The St. Petersburg Paradox helps explain why players might choose strategies that prioritize utility over expected value, influencing their choices in competitive environments.
Psychological Insights
Example: The paradox highlights cognitive biases, such as loss aversion, where individuals prefer to avoid losses rather than acquire equivalent gains. This insight is crucial in understanding consumer behaviour and marketing strategies.
The St. Petersburg Paradox remains a powerful tool for understanding human decision-making in uncertain situations. Its implications extend across various fields, from finance to psychology, illustrating the complexities of how we evaluate risk and reward.
The Free Effect
There is also a psychological phenomenon often referred to as the “free effect” or “freebie effect,” which suggests that when something is presented as “free,” it can significantly influence people’s willingness to spend money on related items or upgrades. Here’s how it works:
Key Concepts of the Free Effect
Perceived Value: When an item is offered for free, it creates a perception of high value. People often feel that they are getting a great deal, which can lead to increased spending on additional products or services.
Anchoring: The initial “free” offer serves as an anchor. Once a person perceives something as free, they may be more willing to spend money on related items, believing that the overall value is still favourable.
Loss Aversion: People tend to dislike losing out on opportunities. If they perceive that they can gain something for free, they may be more inclined to spend money to avoid missing out on additional benefits or upgrades.
Social Proof: When others are seen taking advantage of a free offer, it can create a bandwagon effect, encouraging individuals to spend more to be part of the group or to enhance their experience.
Examples of the Free Effect in Action
Freemium Models: Many software and app companies use a freemium model, where basic services are free, but premium features require payment. Users often start with the free version and later spend money to unlock additional features.
Promotional Offers: Retailers may offer a “buy one, get one free” deal. Customers may be more likely to purchase additional items because they feel they are getting a good deal.
Subscription Services: Many subscription services offer a free trial period. Once users experience the service, they may be more inclined to subscribe, even if the cost is significant.
The “free effect” illustrates how the presentation of a free element can lead to increased spending behaviour. This phenomenon is widely utilized in marketing and sales strategies to enhance consumer engagement and drive revenue.
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