http://harvardmagazine.com/2006/03/the-marketplace-of-perce.html
A fabulous read!
Sunday, April 27, 2008
Saturday, April 26, 2008
Microeconomics: Modelling Procrastination (2)
Hi all, pardon my brief departure. I have been working on an essay assignment about procrastination which is due in 3 days, and so have been slaving over that. This entry is about the main points of 2 articles I have read regarding procrastination. They are by O'Donoghue and Rabin (1999) and Fischer (1999) (I cannot link the articles here as they are in subscribed journals).
O'Donoghue and Rabin basically aim to model procrastination by showing present-biased preferences, where individuals are impatient and favour now over later. They use a mathematical formula and calculations to derive their results. Fischer also uses mathematical models to construct her ideas, but hers come with diagrams of marginal utility and what happens when the discount rate changes (and hence when the rate of time preference changes), and we can see procrastination visually. Both are technical papers, but the intuitive result is this: it is the individual's perception of their own impatience or their own subjective valuation of their discount rates that makes them procrastinate.
How interesting! The question that remains is, what exactly is our individual discount rates?!
O'Donoghue and Rabin basically aim to model procrastination by showing present-biased preferences, where individuals are impatient and favour now over later. They use a mathematical formula and calculations to derive their results. Fischer also uses mathematical models to construct her ideas, but hers come with diagrams of marginal utility and what happens when the discount rate changes (and hence when the rate of time preference changes), and we can see procrastination visually. Both are technical papers, but the intuitive result is this: it is the individual's perception of their own impatience or their own subjective valuation of their discount rates that makes them procrastinate.
How interesting! The question that remains is, what exactly is our individual discount rates?!
Wednesday, April 23, 2008
Microeconomics: Bernoulli and von Neumann-Morgenstern
These people did paramount work on utility functions and expected utility theory.
Rather than give their backgrounds, I thought that I would write their contribution instead.
First was Bernoulli, who formulated a utility function over an individual's wealth. Utility over wealth changes as the wealth level changes. Have a think at this: if you are broke, $100 is worth much more to you than if you were $1,000,000 rich, and were offered the same $100.
Bernoulli therefore put forth the notion that an individual's subjective valuation for a certain sum of money was not necessarily the same as the objective valuation.
What von Neumann and Morgenstern did was to extend Bernoulli's utility function to define expected utility over lotteries, but with several premises (see the previous post on the preference axioms). Their utility function is called the von Neumann - Morgenstern utility function, or more commonly, expected utility function.
In the first stages, one can only think of utility as being ordered (i.e. ordinal utility). For instance, if the utility of a certain commodity bundle, x (I shall use commodity bundles for the moment, as wealth must generally be mapped as a density function because of its continuous nature) is subjectively higher than the utility of another commodity bundle y, intuitively,
U(x) > U(y)
Fortunately, once we have the expected utility form (which satisfies the independence axiom, a.k.a. independence of irrelevant alternatives), we can then start to think of the notion of utility as being a cardinal form (i.e. one with real values). I will not go through the proofs here, but it is pretty easy to check them out on the web.
Hooray for the expected utility form!
Rather than give their backgrounds, I thought that I would write their contribution instead.
First was Bernoulli, who formulated a utility function over an individual's wealth. Utility over wealth changes as the wealth level changes. Have a think at this: if you are broke, $100 is worth much more to you than if you were $1,000,000 rich, and were offered the same $100.
Bernoulli therefore put forth the notion that an individual's subjective valuation for a certain sum of money was not necessarily the same as the objective valuation.
What von Neumann and Morgenstern did was to extend Bernoulli's utility function to define expected utility over lotteries, but with several premises (see the previous post on the preference axioms). Their utility function is called the von Neumann - Morgenstern utility function, or more commonly, expected utility function.
In the first stages, one can only think of utility as being ordered (i.e. ordinal utility). For instance, if the utility of a certain commodity bundle, x (I shall use commodity bundles for the moment, as wealth must generally be mapped as a density function because of its continuous nature) is subjectively higher than the utility of another commodity bundle y, intuitively,
U(x) > U(y)
Fortunately, once we have the expected utility form (which satisfies the independence axiom, a.k.a. independence of irrelevant alternatives), we can then start to think of the notion of utility as being a cardinal form (i.e. one with real values). I will not go through the proofs here, but it is pretty easy to check them out on the web.
Hooray for the expected utility form!
Tuesday, April 22, 2008
Microeconomics: Preference Axioms
My goodness, I am so pleased! While surfing the net for some information on the Bernoulli utility function (used in statistics and microeconomics), I stumbled upon a fabulous website that listed the preference axioms of expected utility theory perfectly. I have cut out and pasted a section here, and the entire page (with lots of really amazing stuff!) is from econport, at http://www.econport.org/econport/request?page=man_ru_basics3. It is just amazing how simple they have made the axioms!
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The Preference Axioms
In order to construct a utility function over lotteries, or gambles, we will make the following assumptions on people's preferences. We denote the binary preference relation "is weakly preferred to" by , which includes both "strictly preferred to", and "indifferent to".
Completeness: For any 2 gambles g and g' in G, either g g' or g' g. In English, this means that people have preferences over all lotteries, and can rank them all.
Transitivity: For any 3 gambles g, g', and g" in G, if g g' and g' g", then g g". In English, if g is preferred (or indifferent) to g', and g' is preferred (or indifferent) to g", then g is preferred (or indifferent) to g".
Continuity: Mathematically, this assumption states that the upper and lower countour sets of a preference relation over lotteries are closed. Along with the other axioms, continuity is needed to ensure that for any gamble in G, there exists some probability such that the decision-maker is indifferent between the "best" and the "worst" outcome. This may seem irrational if the best outcome were, say, $1,000, and the worst outcome was being run over by a car. However, think of it this way - most rational people might be willing to travel across town to collect a $1,000 prize, and this might involve some probability, however tiny, of being run over by a car.
Monotonicity: This big ugly word simply means that a gamble which assigns a higher probabilty to a preferred outcome will be preferred to one which assigns a lower probability to a preferred outcome, as long as the other outcomes in the gambles remain unchanged. In this case, we're referring to a strict preference over outcomes, and don't consider the case where the decision-maker is indifferent between possible outcomes.
Substitution: If a decision-maker is indifferent between two possible outcomes, then they will be indifferent between two lotteries which offer them with equal probabilities, if the lotteries are identical in every other way, i.e., the outcomes can be substituted. So if outcomes x and y are indifferent, then one is indifferent between a lottery giving x with probability p, and z with probability (1-p), and a lottery giving y with probability p, and z with probability (1-p). Similarly, if x is preferred to y, then a lottery giving x with probability p, and z with probability (1-p), is preferred to a lottery giving y with probability p, and z with probability (1-p). Note: This last axiom is frequently referred to as the Independence axiom, since it refers to the Independence of Irrelevant Alternatives (IIA).
The last axiom allows us to reduce compound lotteries to simple lotteries, since one can also be similarly indifferent between a a simple lottery giving an outcome x with a probability p, and compound lottery where the prize might be yet another lottery ticket, allowing one to participate in a lottery with x as a possible outcome, such that the effective probability of getting x was p.
-----------------------------------------------------------------------------------------------------------------
The Preference Axioms
In order to construct a utility function over lotteries, or gambles, we will make the following assumptions on people's preferences. We denote the binary preference relation "is weakly preferred to" by , which includes both "strictly preferred to", and "indifferent to".
Completeness: For any 2 gambles g and g' in G, either g g' or g' g. In English, this means that people have preferences over all lotteries, and can rank them all.
Transitivity: For any 3 gambles g, g', and g" in G, if g g' and g' g", then g g". In English, if g is preferred (or indifferent) to g', and g' is preferred (or indifferent) to g", then g is preferred (or indifferent) to g".
Continuity: Mathematically, this assumption states that the upper and lower countour sets of a preference relation over lotteries are closed. Along with the other axioms, continuity is needed to ensure that for any gamble in G, there exists some probability such that the decision-maker is indifferent between the "best" and the "worst" outcome. This may seem irrational if the best outcome were, say, $1,000, and the worst outcome was being run over by a car. However, think of it this way - most rational people might be willing to travel across town to collect a $1,000 prize, and this might involve some probability, however tiny, of being run over by a car.
Monotonicity: This big ugly word simply means that a gamble which assigns a higher probabilty to a preferred outcome will be preferred to one which assigns a lower probability to a preferred outcome, as long as the other outcomes in the gambles remain unchanged. In this case, we're referring to a strict preference over outcomes, and don't consider the case where the decision-maker is indifferent between possible outcomes.
Substitution: If a decision-maker is indifferent between two possible outcomes, then they will be indifferent between two lotteries which offer them with equal probabilities, if the lotteries are identical in every other way, i.e., the outcomes can be substituted. So if outcomes x and y are indifferent, then one is indifferent between a lottery giving x with probability p, and z with probability (1-p), and a lottery giving y with probability p, and z with probability (1-p). Similarly, if x is preferred to y, then a lottery giving x with probability p, and z with probability (1-p), is preferred to a lottery giving y with probability p, and z with probability (1-p). Note: This last axiom is frequently referred to as the Independence axiom, since it refers to the Independence of Irrelevant Alternatives (IIA).
The last axiom allows us to reduce compound lotteries to simple lotteries, since one can also be similarly indifferent between a a simple lottery giving an outcome x with a probability p, and compound lottery where the prize might be yet another lottery ticket, allowing one to participate in a lottery with x as a possible outcome, such that the effective probability of getting x was p.
Sunday, April 20, 2008
Statistics: Poker Probability
Have you played a game of poker and think, hmm... what are my chances of winning? Well, a friend of mine brought me to a page in Wikipedia that had a fabulous listing of poker probabilities. I copied the following picture from this link: http://en.wikipedia.org/wiki/Poker_probability. Have a look and a marvel here, or go directly to Wikipedia.
| Visual help | Hand | Frequency | Probability | Cumulative | Odds | Mathematical expression of frequency |
|---|---|---|---|---|---|---|
     | Royal flush | 4 | 0.000154% | 0.000154% | 649,739 : 1 | |
     | Straight flush (excluding royal flush) | 36 | 0.00139% | 0.00154% | 72,192.33 : 1 | |
     | Four of a kind | 624 | 0.0240% | 0.0256% | 4,164 : 1 | |
   | Full house | 3,744 | 0.144% | 0.170% | 693.2 : 1 | |
     | Flush | 5,108 | 0.197% | 0.367% | 507.8 : 1 | |
 | Straight | 10,200 | 0.392% | 0.76% | 253.8 : 1 | |
    | Three of a kind | 54,912 | 2.11% | 2.87% | 46.3 : 1 | |
   | Two pair | 123,552 | 4.75% | 7.62% | 20.03 : 1 | |
 | One pair | 1,098,240 | 42.3% | 49.9% | 1.37 : 1 | |
    | No pair / High card | 1,302,540 | 50.1% | 100% | 0.995 : 1 | |
| Total | 2,598,960 | 100% | 100% | 0 : 1 | |
![\left[{13 \choose 5} - 10\right]\left[{4 \choose 1}^5 - 4\right]](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_v1YbDibZ2LE2qAI5oW9ee5fZ7tqfY9KnQXFBB01plwIWlQxi58PbdjUPQqO6dkCeBgwExhOK8o4bUrww3Nv0x_etbw2Rmm1CGC9zQT-ReOfx1n0cAKOnfxnnp38sqh5w7IrciKttGgyjR0Ght1OQ=s0-d)
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