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Measuring Preference: A Brief Summary of Utility Theory

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Utility: Some Definitions

- The desirability of preference that individuals or societies have for a given outcome
- Quantitative measure of the attractiveness of a potential outcome

Outlines

- Bernoulli Family
- Daniel Bernoulli’s Paper
- Utility and Utility Function
- Caius example
- St. Petersburg Paradox
- Extensions of Bernoulli theory
- von Neumann and Morgenstern utility
- Problems with Utility theory

Bernoulli Family

Daniel Bernoulli

Born: 8 Feb 1700 in Groningen, NetherlandsDied: 17 March 1782 in Basel, Switzerland

Simple Biography

The Victorian statistician Francis Galton describes him as “Physician, botanist, and anatomist, writer on hydrodynamics; very precocious”.

He was also a powerful mathematician and statistician, with a special interest in probability.

Exposition of a New Theory on the Measurement of Risk (1954)

- “Ever since mathematicians first began to study the measurement of risk, there has been general agreement on the following proposition:
- Expected values are computed by multiplying each possible gain by the number of ways in which it can occur, and then dividing the sum of these products by the total number of possible cases.”
- No characteristic of the persons themselves has been taken into consideration.

Why Utility

- Example: buying a lottery ticket
- All men cannot use the same rule to evaluate the gamble. There is no reason to assume that two persons encountering identical risks.
- “The determination of the value of an item must not be based on its price, but rather on the particular circumstances of the person making the estimate.”
- “It becomes evident that no valid measurement of the value of a risk can be obtained without consideration being given to its utility”

Utility and Utility Function

- The utility is dependent on the particular circumstances of the person making the estimate.
- U(X) = a+b*X
- U(X) = sqrt(X)
- U(X) = log(X)

U is utility

K is a constant of proportionality

X is the base amount of wealth

safe

5% will be lost.

storm

Caius example- Caius was contemplating whether he should take insurance on a shipment from Amsterdam to St. Petersburg.
- Shipment
- The Amsterdam underwriters want a full covered policy payment of 800 rubles, or 8% of the profit if no mishap occurs.
- Should Caius buy the policy?

Caius example

- If we go by simple ruble values, then he should self-insure.
- If Caius follows Bernoulli’s advice, he should buy the policy if his expected utility for insuring is greater than that for not insuring.

Caius example

- Suppose Caius’s capital is X rubles
- W.L.O.G. take k=1
- Caius’s expected utility of not insuring minus that of insuring is given by
- Newton’s Method
- If Caius has less than 5,042 rubles, he should (according to Bernoulli), buy the 800 ruble policy.

Caius example

- How much the underwriter (insurer) should have in hand in order to sell the 10,000 ruble policy for the amount of 800 rubles?
- If the underwriter has a stake of 14,242 rubles or more, Bernoulli’s rule tells us that selling the policy will improve the underwriter’s position.
- “But no one, however rich, would be managing his affairs properly if he individually undertook the insurance for less than five hundred rubles”

St. Petersburg Paradox

- From Nicolas Bernoulli’s letter
- Consider the following game
- Peter flips a coin and will give Paul:
- $2 if the first flip is a head
- $2k if the kth flip is the first head
- How much should Paul pay to play this game?
- A poor person might be unwilling to pay more than two dollars, the minimum possible pay-off of the game.

St. Petersburg Paradox (cont)

- The expectation is infinitely great but no one would be willing to purchase it at a moderately high price.

Applying the theory

- A relationship between one’s wealth and the amount one would pay to play this paradoxical game.
- If Paul owned nothing, his opportunity would be worth approximately two ducats
- 10 – 3 ducats
- 100 – 4 ducats
- 1000 – 6 ducats
- “We can easily see what a tremendous fortune a man must own for it to make sense for him to purchase Paul’s opportunity for twenty ducats.”

where α is Paul’s fortune

Applying the theory (cont)

- “The amount which the buyer ought to pay for this proposition differs somewhat from the amount it would be worth to him were it already in his possession. Since, however, this difference is exceedingly small if alpha is great.”

“His paper is one of the most profound documents ever written, not just on the subject of risk but on human behavior as well. Bernoulli’s emphasis on the complex relationships between measurement and gut touches on almost every aspect of life.”

“For the first time in history Bernoulli is applying measurement to something that cannot be counted. Bernoulli defines the motivations of the person who does the choosing. This is an entirely new area of study and body of theory. Bernoulli laid the intellectual groundwork for much of what was to follow, not just in economics, but in theories about how people make decisions and choices in every aspect of life.” (Bernstein, P.L. (1996). Against the Gods)

Bernoulli’s ContributionJeremy Bentham

- (1789) The Principles of Morals and Legislation
- Suggested measurement of pleasure and pain
- Utility: “…that property in any object, whereby it tends to produce benefit, advantage, pleasure, good, or happiness…when the tendency it has to augment the happiness of the community is greater than any it has to diminish it.”
- Utility became a tool for discovering how prices result from interactions between buyers and sellers
- Law of supply and demand

William Stanley Jevons

- (1871) The Theory of Political Economy
- “Value depends entirely on utility”
- Like Bernoulli, thought that the utility varies with the amount of commodity in one’s possession
- Reflected era’s zeal for measurement
- “Pleasure, pain, labor, utility, value, wealth, etc. are notions admitting of quantity”

Game Theory

- “The true source of uncertainty lies in the intentions of others”
- Invented by John von Neumann
- (1944) Theory of Games and Economic Behavior
- With Oskar Morgenstern
- Rigidly mathematical with an emphasis on numerical quantities
- Claim that this matches reality in the same way measurements have been attached to heat and light

Quantifying Preference

- Assume a man prefers milk to coffee and prefers coffee to tea
- Do you prefer a cup of coffee to a cup that has a 50-50 chance of being milk or tea?
- What about having $1 or $2
- Would you rather have $1 or a 50-50 chance of either winning $2 or nothing?
- Vary the probability to find the point that the man is indifferent between the sure thing and the gamble.

Expected Utility

- If the man is indifferent between $1 certain and the gamble:
- 50% preference means his preference for $1 over 0 is half as great as his preference for $2 over 0
- The utility of $2 is double the utility of $1
- Maximize expected utility based on rational behavior

von Neumann-MorgensternAxioms of Rational Behavior

- Transitivity: If the subject is indifferent between outcomes A and B and between B and C, he must be indifferent between A and C.
- Continuity of preferences: If A is preferred to B and B is preferred to no change, the there is a probability α such that the subject is indifferent between αA and B

von Neumann-MorgensternAxioms of Rational Behavior

- Independence:If one is indifferent between A and B then for any probability α, one is indifferent between αA and αB
- Desire for high probability of success: If A is preferred to no change and if α1 > α2, then α1A is preferred to α2A
- Compound probabilities: If one is indifferent between αA and B, and if α= α1α2, then one is indifferent between α1α2A and B

Problems with Utility Theory

- Entire risk profile cannot be captured with a single number (expected utility)
- Utile has no meaning to most people
- No natural utility function (i.e., when should we use log or square root?)
- People violate axioms
- Prospect Theory from Daniel Kahneman and Amos Tversky

Reference point

- Bernoulli: “The utility resulting from any small increase in wealth will be inversely proportionate to the quantity of goods previously possessed”
- The assessment of a risky opportunity is more dependent on the reference point from which the possible gain or loss will occur than on the final value of the assets that would result.
- “Our preferences can be manipulated by changes in the reference points” (Tversky)

Reference Point: Example

- Option 1: Everyone just won $30
- Flip a coin vs. no coin flip
- Heads: win $9
- Tails: lose $9
- 70% chose the coin flip
- Option 2: Everyone starts with $0
- Flip a coin vs. $30 for sure
- Heads: win $39
- Tails: win $21
- 43% chose the coin flip
- Students base their choice on the reference point

Failure of Invariance

- Invariance: If A is preferred to B and B preferred the C, then A is preferred to C
- Suppose you go to the theater with a ticket
- You lost your $40 ticket
- Do you buy another ticket?
- Suppose you plan to buy the ticket there
- You lost the $40
- Do you use another $40 to buy the ticket?
- Most people won’t replace the lost ticket, but will pay another $40

Conclusion

- Although does not work in practice, forms the foundation of
- Economic theory, supply and demand
- Theory of Utilitarianism
- Decision Making
- Game theory

References

- Bernoulli,D., “Exposition of a new theory on the measurement of risk”, Econometrica, Vol. 22, No. 1. (Jan., 1954), pp. 23-36.
- Bernstein, P. (1996). Against the Gods: The Remarkable Story of Risk. John Wiley & Sons: New York
- Stigler, G.J. (1968). “The Development of Utility Theory.” in Utility Theory: A Book of Readings. John Wiley & Sons: New York.
- Thompson,J.R. (2002). Models for Investors in Real world Markets. John Wiley & Sons: New York. pp.29-59
- Von Neumann, J. and Morgenstern, O. (1947) Theory of Games and Economic Behavior. Princeton University Press: Princeton.

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