# Paradoxes, Probabilities and How to Analyze Them

Content based on Kreps: “A Couse in Microeconomic Theory.”

Microeconomic theory serves to answer the question of how do rational actors make decisions in a way that creates an equilibrium. Microeconomic theory involves setting up maximization over choice functions where the preferences of the actors are given and an equilibrium is set up that creates a relationship between various inputs and outputs of the model assuming the market clears and supply is equal to demand. For example, one can perform what is called comparative statics where one changes the income for example, and then can see via the math whether people substitute more expensive goods for the inferior good. This type of math is known as real analysis where everything is made a function of everything else in an equation as an equation in this case is just a relationship between moving parts which are unknown quantities that yet must move to balance each other. It basically is an aid to thinking about questions such as what would happen if one part of the economy were to move against another part, and sometimes unforeseen results can be investigated by these equations, or at the very least, the modeling of concepts into these equations can uncover new concepts through the math in these equations. I compare the math modeling to computer programming. Solving the formulas eventually gets you a view of the world that runs very quickly and efficiently conceptually. Also, uncovering paradoxes within the formulas is like writing a computer program that doesn’t do what it is supposed to and the bugs are interesting in themselves for what it reveals about the logic in your code which is interesting in of itself beyond the task at hand. For example there is the Allais Paradox and the Ellsberg Paradox. The details of these paradoxes are what is interesting.

The Allais Paradox is a statement about the inconsistencies between the implied risk preferences of an individual who makes decisions about whether to accept two related gambles. The individual will take in the first choice the gamble that gives some chance of bringing in a result better than the certainty equivalent of a sure bet, even if there is a small chance of getting absolutely nothing. This is quite logical and one can reason from this that the individual is most likely ready to take risk and thus chooses the first gamble, thus underweighting the probability of a small probability event of losing it all. Yet in the second case, when there is a slightly smaller chance of winning, bur of winning a significantly larger sum, along with a slightly greater chance of losing it all, most people take it, which shows the small probability event of missing the target and of losing it all is ignored and thus underweighted. According to the literature, people have reconciled this inconsistency in risk preferences by saying people overall overweight small probability events which is counterintuitive. A professor said to me that small probability events do happen with regards to health, suggesting that I underweighted small probability events, but I think the overweighted small probability events just because they do happen. I’m not going to fight the last war like a general who reacts to observed events only but also to the events that could have happened but did not happen.

There is also the Ellsberg Paradox which involves pulling colored marbles from a jar. In the first case the individual selects the unambiguous payoff by choosing which color pays off when he draws a random ball. In the second case, he does the same thing by choosing which color does not pay off. The point of these constructed choices is to show that people tend to avoid ambiguity which is when the odds are not clear, and would rather bet on clear odds. Risk in other words is objective uncertainty while ambiguity is subjective uncertainty. Risk is running a marathon at risk of an injury while ambiguity is submitting a paper to a fickle teacher who grades it with a rubric he doesn’t share.

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