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.