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Mathematical expectation and rational decision taking [split from "Defense in anarchy" thread]

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ToxicAssets Posted: Mon, May 6 2013 4:05 AM

 

malachi:
do you even know what mathematical expectation is?

Haha.. cute.

The problem though is that I'm not a chick and I'm no homo. So you should save your "negs" and cute little lines for someone else…

Anyways, I was trying to show you that even though the concept of mathematical expectation is at the heart of every decision under uncertainty, things are little bit more complicated than that.

It is not the mathematical expectation of the monetary return of a gamble that determines whether that gamble is worthwhile or not.

It is the mathematical expectation of the UTILITY FUNCTION of the returns.

Under normal conditions, we assume that people have concave utility functions, which can be interpreted as "risk-aversion".

It means that facing the alternatives of a gamble and an opportunity to get the average return of that gamble, they will generally choose the sure thing.

But on occasion the utility function of a certain individual might get convex, meaning that he's a "risk-taker".

Given a certain gamble, he'll prefer the small probability of earning a larger sum (and the risk of losing) than the sure thing of getting the average.

A reason why someone would have such a convex utility function with respect to a given gamble is that he's facing convex (non-diminishing) opportunities of investment for the returns he'll extract from that particular gamble.

Things like owing mobsters money and or facing some unique investment opportunity that will shut down soon may "convexify" your otherwise concave utility function, so the roulette game might be a rational decision for people in such a situation.

Individuals take decisions based on private and public information, and the shape of their utility functions takes both in considerartion.

Since most individuals face diminishing returns for their investments, we generally observe concave profiles for their utility functions.

That's why I was talking about pot-odds and implied pot-odds, assuming perhaps that you understood the fundamentals about poker.

You calculate your pot-odds by comparing how much you need to bet to take part in a given pot, but that's what everybody else knows. What other people don't know is what cards you have in the hole and how things are likely to turn out in your advantage (or against you) if certain scenarios unfold. And you use this information to estimate your implied pot-odds, which is going to serve as the basis of your decision. 

The mathematical expectation of a gamble is the public information everybody has, but it is not enough to corroborate a fully informed decision about taking or not taking the gamble.

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Student replied on Mon, May 6 2013 1:49 PM

 

A reason why someone would have such a convex utility function with respect to a given gamble is that he's facing convex (non-diminishing) opportunities of investment for the returns he'll extract from that particular gamble.
 
Things like owing mobsters money and or facing some unique investment opportunity that will shut down soon may "convexify" your otherwise concave utility function, so the roulette game might be a rational decision for people in such a situation.
 
Not sure I understand this bit. 
 
Concave means the marginal utility of income is increasing (each extra dollar provides less and less utility) and convex means the marginal utility of income is increasing (each extra dollar provides more and more utility). Everything I have read has simply taken whether the utility function is concave or convex as given. So I  am having a hard time seeing how it is dependent on the opportunities of investment.
 
Would you mind elaborating? Are you pulling from behavioral literature? If so, t would be interesting to see some citations. 
 
 

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Malachi replied on Mon, May 6 2013 4:47 PM

ToxicAssets:

 

malachi:
do you even know what mathematical expectation is?

Haha.. cute.

The problem though is that I'm not a chick and I'm no homo. So you should save your "negs" and cute little lines for someone else…

Anyways, I was trying to show you that even though the concept of mathematical expectation is at the heart of every decision under uncertainty, things are little bit more complicated than that.

It is not the mathematical expectation of the monetary return of a gamble that determines whether that gamble is worthwhile or not.

It is the mathematical expectation of the UTILITY FUNCTION of the returns.

Under normal conditions, we assume that people have concave utility functions, which can be interpreted as "risk-aversion".

It means that facing the alternatives of a gamble and an opportunity to get the average return of that gamble, they will generally choose the sure thing.

But on occasion the utility function of a certain individual might get convex, meaning that he's a "risk-taker".

Given a certain gamble, he'll prefer the small probability of earning a larger sum (and the risk of losing) than the sure thing of getting the average.

A reason why someone would have such a convex utility function with respect to a given gamble is that he's facing convex (non-diminishing) opportunities of investment for the returns he'll extract from that particular gamble.

Things like owing mobsters money and or facing some unique investment opportunity that will shut down soon may "convexify" your otherwise concave utility function, so the roulette game might be a rational decision for people in such a situation.

Individuals take decisions based on private and public information, and the shape of their utility functions takes both in considerartion.

Since most individuals face diminishing returns for their investments, we generally observe concave profiles for their utility functions.

That's why I was talking about pot-odds and implied pot-odds, assuming perhaps that you understood the fundamentals about poker.

You calculate your pot-odds by comparing how much you need to bet to take part in a given pot, but that's what everybody else knows. What other people don't know is what cards you have in the hole and how things are likely to turn out in your advantage (or against you) if certain scenarios unfold. And you use this information to estimate your implied pot-odds, which is going to serve as the basis of your decision. 

The mathematical expectation of a gamble is the public information everybody has, but it is not enough to corroborate a fully informed decision about taking or not taking the gamble.

 

It is the mathematical expectation of the UTILITY FUNCTION of the returns.

how do you derive a mathematical expectation from a utility function?

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Wheylous replied on Mon, May 6 2013 11:57 PM

If you have U(x) = something, then the expected utility is EU(x) = P1*U(x1) + P2*U(x2)

Where P1 is the probability of payoff x1.

Although I think that this is an incorrect explanation of how humans behave (I don't think we use straight-up math probabilities. We probably use some fuzzy function)

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@Student

First of all, we don't have fixed uses for money. The opportunities we have for investing any income we have change with time, so it's natural that our perception of utility of income adapts to these changing conditions. 

But what is important to keep in mind is that the concept of utility function optimization is a very stylized model for how humans take decisions.

Individual utility functions are never directly observable, they are inferred by something called "revealed preferences", that is, the patterns of decision taking that an individual engage.

In real life some decisions are taken "sequentially" and others follow a "set and go" procedure. "Set and go" decisions are those where all the optimization is done with information available before the action starts. Once the initial decision is made, there's no more coming back. "Sequential" decisions are those decisions that are taken in a step by step process, each decisional step using the bits of information that unfolded up until it.

Decisions can be also of "incremental" or "yes/no (or discrete)" types. "Incremental" decisions are those that allow for arbitrarily small changes in a spectrum of choices. "Discrete" decisions are those where only a certain number of options are open, and there are no compromise positions between them.

Another important distinction among decisions is the "private" or "public" character of the information being used by the decision maker, when he interacts with other decision makers.

And finally, there's the "time factor". Some decisions can be planned very carefully and time itself is not a very scarce resource to be economized in the process. Other decisions might need urgent solutions, even if better alternatives could be found after more deliberation.

There are other categorical distinctions between decisions, but those above are perhaps the most relevant here, so we will restrict ourselves to them.

The assumption of a concave utility function is very consistent to the behavior observed when decision takers face decisions that are "set and go", "incremental", based on "public information" and with abundant time for deliberation. Let's call this kind of decision "regular".

That's because the optimization procedure is very simple: you invest the next unit of money to the best yet unsatisfied alternative, and you keep doing that until you're done.

Since every investing decision is incremental, all information is available before hand, and the decision maker has access to the same information as the observer of the decision maker behavior, Here the decision taker is always in condition to increment the opportunity that is the best  marginally. 

The "revealed preferences" profile of his inferred utility function will look concave, as his returns diminish after each dollar is spent.

However, if the decision making process is not "regular", the "revealed preferences" will generally show inconsistencies with a concave utility profile.

For instance, if you happen to know privately that a certain game is rigged, or maybe that you're privileged with a mysterious lucky factor when playing crabs (your "private information"), your revealed behavior might be inconsistent with risk aversion feature of concave utility functions.

And similarly, if decision making time is limited, if alternatives are limited, etc. That's why I was talking about owing money to gangsters, or having some medical condition, or some once-in-a-life-time oportunity of investment. All these situations disrupt the regularity of your decision making process, and thus your profile of utility.

Also, somewhat related to the discussion above, is the pricing of options. Options are like bets, they pay you a random amount of money that depends on certain observable events in the market. Theoretically they should have a value according to the players utily function/risk preferences. But since options and the underlying assets are tradeable and since there are no-oportunities of arbitrage, and decisions can be updated, their price does not depend on individual utility function profiles. One can show (it's quite technical though) that under certain regularity assumptions there's a probability measure where the market price is given by the expectation, with no risk premium.

The best book about these distinctions in decision making is "Knowledge and Decisions" by Thomas Sowell. This book (like most Sowell's books) is not likely to be particularly distasteful to austrianites, since it is actually based on the paper "The use of knowledge in society", by Hayek.

Another good book I recommend that deals with similar problems is Vernon Smith's "Rationality in Economics". This book looks a bit more technical than Sowell's (in the neoclassical sense of having equations and such), but it actually provides a very good conciliation between Hayek's and Simon's approaches to decision making. 

Also, as you've said, there is a lot of material on behavioral economics showing these reveled preferences "violations" of rational expectations assumptions. One that is particularly accessible is "Predictably Irrational" by Dan Arieli. This book, on the other hand, will be very hard to swallow by hardcore austrians. 

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Malachi:

It is the mathematical expectation of the UTILITY FUNCTION of the returns.

how do you derive a mathematical expectation from a utility function?

"do you even know what mathematical expectation is?" - Malachi

 

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@Wheylous

Yes, that's correct for the case where your random payoff takes only two values.

As I've stated above, this is not an explanation of the internal cognitive process of a decision taker.

It is rather a stylized model that is consistent with certain profiles of reveled preferences in rational decision taking.

And the same goes for fuzzy logics. But I agree that decision protocols based on fuzzy logics A.I. algorithms are generally more flexible and apt to simulate lifelike decision takers than those based on simple optimization of pre-specified utility functions.

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baxter replied on Tue, May 7 2013 5:18 AM

>the concept of mathematical expectation is at the heart of every decision under uncertainty

I see strawberry and chocolate ice cream at the store. It's uncertain how good each of them will taste.

I decide on the strawberry ice cream.

Where is the mathematical expectation here???

>Under normal conditions, we assume that people have concave utility functions

What is the meaning of "concave" and "convex" for functions that are subject to arbitrary, monotonically non-decreasing transformations?

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Malachi replied on Tue, May 7 2013 5:33 AM

ToxicAssets:

Malachi:

It is the mathematical expectation of the UTILITY FUNCTION of the returns.

how do you derive a mathematical expectation from a utility function?

"do you even know what mathematical expectation is?" - Malachi

 

 

so, you dont know and cannot demonstrate it. got it. then why did you start this thread?

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Malachi replied on Tue, May 7 2013 5:34 AM

Wheylous:

If you have U(x) = something, then the expected utility is EU(x) = P1*U(x1) + P2*U(x2)

Where P1 is the probability of payoff x1.

Although I think that this is an incorrect explanation of how humans behave (I don't think we use straight-up math probabilities. We probably use some fuzzy function)

 

what is the mathematical expectation of that?

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I see strawberry and chocolate ice cream at the store. It's uncertain how good each of them will taste.

I decide on the strawberry ice cream.

Where is the mathematical expectation here???

The fact that you can create a useful mathematical model to describe the movement of the Moon around the Earth doesn't mean that the Moon is making computations either.

What is the meaning of "concave" and "convex" for functions that are subject to arbitrary, monotonically non-decreasing transformations?

This is not true for Expected utility theory, which was the subject of our thread.

According to Von Neumann-Morgenstern utility theorem, there's a unique utility function compatible with a consistent set of preferred lotteries, and the uniqueness is only up to an additive constant or a scalar multiplication. Affine transformations preserve convexity/concavity profiles, which are related to preferences for risk.

So, in the end, you can't subject utility functions (in expected utility theory) to arbitrary monotonically non-decreasing transformations. 

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Malachi:

ToxicAssets:

Malachi:

It is the mathematical expectation of the UTILITY FUNCTION of the returns.

how do you derive a mathematical expectation from a utility function?

"do you even know what mathematical expectation is?" - Malachi

 

so, you dont know and cannot demonstrate it. got it. then why did you start this thread?

 

Ok, sorry about this, I told Jon Irenicus that I would be nice and cut you some slack, and now I'm being mean to you again.

If one day, after you finish your college, you decide to study probability theory in a graduate school, you will learn that expectation is in fact a functional operator on a probability space.

It associates measurable real/complex-valued functions (a.k.a. random variables) on a given probability space to a number, real or complex.

So the question "how do you derive the mathematical expectation of a utility function" becomes trivial.

You just apply calculate E[U] =  \int_\Omega  U(w) dP(w)

Say you have a coin toss, and let's call "heads" 0 and "tails" 1.

Applying the formula above, you get E[U]=U(0)*P(0)+U(1)*P(1).

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Malachi replied on Tue, May 7 2013 5:32 PM

yes, you have continued to be rude but its irrelevant to me because you havent given me any reason to care what your opinion of me is, indeed you havent even shown yourself capable of discussing these ideas coherently.

And similarly, if decision making time is limited, if alternatives are limited, etc. That's why I was talking about owing money to gangsters, or having some medical condition, or some once-in-a-life-time oportunity of investment. All these situations disrupt the regularity of your decision making process, and thus your profile of utility.

so are you trying to address human decision making or not? its hard to tell because you wont explain yourself or your propositions.

One can show (it's quite technical though) that under certain regularity assumptions there's a probability measure where the market price is given by the expectation, with no risk premium.

if you mean to say that when faced with a certain outcome, people tend to expect what they actually get, then ok. but whats your point? we dont need stylized abstracted versions of how people dont think so we can mess with the variables until we achieve approximately the same result as what we already knew.

Also, as you've said, there is a lot of material on behavioral economics showing these reveled preferences "violations" of rational expectations assumptions.

those decision makers whose preferences were revealed must not have known how to calculate mathematical expectation of a utility function.

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Malachi replied on Tue, May 7 2013 5:42 PM

If one day, after you finish your college, you decide to study probability theory in a graduate school, you will learn that expectation is in fact a functional operator on a probability space.

mathematical expecation is what you think you might get if you repeated the trial a bunch of times.

It associates measurable real/complex-valued functions (a.k.a. random variables) on a given probability space to a number, real or complex.

if you can accurately calculate mathematical expectation, you can use it to compare courses of action and find out which one is most profitable, or least profitable, or somewhere in between.

So the question "how do you derive the mathematical expectation of a utility function" becomes trivial.

of course it is.

at least you admit that this has nothing to do with how people actually make decisions, rationally or otherwise.

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ToxicAssets:

I see strawberry and chocolate ice cream at the store. It's uncertain how good each of them will taste.

I decide on the strawberry ice cream.

Where is the mathematical expectation here???

The fact that you can create a useful mathematical model to describe the movement of the Moon around the Earth doesn't mean that the Moon is making computations either.

What is the meaning of "concave" and "convex" for functions that are subject to arbitrary, monotonically non-decreasing transformations?

This is not true for Expected utility theory, which was the subject of our thread.

According to Von Neumann-Morgenstern utility theorem, there's a unique utility function compatible with a consistent set of preferred lotteries, and the uniqueness is only up to an additive constant or a scalar multiplication. Affine transformations preserve convexity/concavity profiles, which are related to preferences for risk.

So, in the end, you can't subject utility functions (in expected utility theory) to arbitrary monotonically non-decreasing transformations. 

 

 

Positive scalar multiplication. Otherwise concave becomes convex.

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