How rational are you?

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ohai's picture

With a risk neutral decision maker, standard deviation usually (always?) doesn’t determine preferences though. 

“Be aware of this 40 year old men in our country.They suffer from mid life crisis and are always trying to be in the good books of young girls.” -Rahul Roy

jmh530's picture

Since when is it irrational to be risk averse?

ohai's picture

No one said it is irrational to be risk averse. Most people who answered the original question assumed risk neutrality, which I guess should have been stated. My statement, that dispersion of outcomes is irrelevant to a risk neutral decision maker, does not say anything about the rationality or irrationality of risk aversion. Also, as others have stated, risk aversion does not affect preference of one game in the original question over the other, although the absolute value of each game is affected. 

“Be aware of this 40 year old men in our country.They suffer from mid life crisis and are always trying to be in the good books of young girls.” -Rahul Roy

ohai's picture

I did, however, state that “aversion to ambiguity” is irrational. This is not the same as risk aversion. Ambiguity can always be expressed in probabilities. Thus, it is irrational to think that ambiguity is anything but a different set of probabilities. 

“Be aware of this 40 year old men in our country.They suffer from mid life crisis and are always trying to be in the good books of young girls.” -Rahul Roy

bchad's picture

jmh530 wrote:

@Systematic

The uniform distribution still has a standard deviation. Check wikipedia for the formula.

I think Systematic’s argument is that the standard deviations aren’t relevant, not that they don’t exist.  And for a single play with binomial outcomes, there is an argument for that.

You want a quote?  Haven’t I written enough already???

bchad's picture

ohai wrote:

I did, however, state that “aversion to ambiguity” is irrational. This is not the same as risk aversion. Ambiguity can always be expressed in probabilities. Thus, it is irrational to think that ambiguity is anything but a different set of probabilities. 

It’s irrational in a mathematical sense.  But it is not irrational in a strategic sense.  What I mean is that if you don’t know how something is randomized (i.e. the probability distribution), you are effectively not all that sure about the rules of the game.  If the judge in game 2 is allowed to change the probability, p, as he/she pleases, it is possible that they can use that information to defraud you.  So if the ambiguity is not so much about what the actual probability distribution is, but whether there are parts of the game that can be used to put you at a disadvantage, it is not irrational to have a higher discount rate for those possibilities.

One can argue that that is just the same as having a different overall probability distribution, but if you have no idea how that distribution is actually distributed, then you are going to need to puff up your risk premium substantially to account for that.

You want a quote?  Haven’t I written enough already???

BrownianBridge's picture

bchadwick wrote:

ohai wrote:

I did, however, state that “aversion to ambiguity” is irrational. This is not the same as risk aversion. Ambiguity can always be expressed in probabilities. Thus, it is irrational to think that ambiguity is anything but a different set of probabilities. 

It’s irrational in a mathematical sense.  But it is not irrational in a strategic sense.  What I mean is that if you don’t know how something is randomized (i.e. the probability distribution), you are effectively not all that sure about the rules of the game.  If the judge in game 2 is allowed to change the probability, p, as he/she pleases, it is possible that they can use that information to defraud you.  So if the ambiguity is not so much about what the actual probability distribution is, but whether there are parts of the game that can be used to put you at a disadvantage, it is not irrational to have a higher discount rate for those possibilities.

One can argue that that is just the same as having a different overall probability distribution, but if you have no idea how that distribution is actually distributed, then you are going to need to puff up your risk premium substantially to account for that.

The question states that that the distribution in game 2 is chosen randomly. However, even if the judge gets to select whichever distribution he desires, this does not change the fact that the odds for game 2 are still 50/50 and that the expected value of the earnings are equal to that of game 1.

Say that this game has been played previously and that people have a tendency to select the color purple 57% of the time (and that both the judge and the players are aware of this). The judge knows this, and can use this information when selecting the distribution. So he may select 100 white balls and zero purple balls, for example. But you know that people have a tendency to select the color purple 57% of the time and that the judge knows this, so you can therefore select your color based on the fact that he may try to fool you. But the judge knows that you know this, and you know that the judge knows you know this, etc. After this infinite recursion, you are still left with 50/50 odds.

brain_wash_your_face's picture

Complete information - I know that you know that you know that I know that I know that you know that you know that i know that you know that I know that you know that I know I know that you know that you know that I know that I know that you know that you know that i know that you know that I know that you know that I know I know that you know that you know that I know that I know that you know that you know that i know that you know that I know that you know that I know I know that you know that you know that I know that I know that you know that you know that i know that you know that I know that you know that I know that you have syphiilis.  I choose to take my chances.

It was me.  I acted alone.  It was meticulously planned.  I let the dogs out.

sdevine42's picture

[/quote]

so you can therefore select your color based on the fact that he may try to fool you. But the judge knows that you know this, and you know that the judge knows you know this, etc. After this infinite recursion, you are still left with 50/50 odds.

[/quote]

….so i can clearly not choose the wine in front of me….

“In war there is no substitute for victory.” - Douglas MacArthur

Turd Fergeson's picture

“How rational are you?”

Are there degrees of rationality?

KISS MY CONVERSE.

Mobius Strip's picture

Y is the random variable associated with you picking the winning color. It is equal to 1 with probability p and 0 with probability 1-p (bernoulli distribution). If p itself is a random variable X as in the second case, then the conditional expectation and conditional variance of Y are as follows:

E[Y|X]=X, Var[Y|X]=X(1-X)

By the law of total expectation, we have:

E[Y]=E[E[Y|X]]=E[X]

By the law of total variance, we have:

Var[Y]=E[Var[Y|X]]+Var[E[Y|X]]=E[X(1-X)]+Var[X]=E[X]-E[X^2]+E[X^2]-E[X]^2=E[X]*(1-E[X])

For a risk-averse player, use some sort of Sharpe-like ratio to measure which choice is more attractive:

ratio = E[Y]/Var[Y]=E[X]/{E[X]*(1-E[X])=1/(1-E[X])

so as long as the you don’t have any knowledge that certain color will be favored, i.e. X is a symmetrical distribution, then E[X]=1/2, and your ratio is 1/(1-1/2)=2, regardless of what the variance of X is. You should be indifferent between case 1 (X=1/2=constant, VarX=0) and case 2 (say X is uniform, or normal, or any distribution centered around 1/2 with positive variance Var[X]>0).

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