Risk Probability

Rhyn

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Nov 20, 2012
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Hi everyone. Recently came across this question in an upper level class and to be honest I am stuck on it. I am not completely sure if this is the correct forum but here it is. Any guidance is appreciated.

Consider a purely probabilistic game that you have the opportunity to play. Each time you play there are n potential known outcomes x1, x2, ..., xn (each of which is a specified gain or loss of dollars according to whether xi is positive or negative). These outcomes x1, x2, ..., xn occur with the known probabilities p1, p2, ..., pn respectively (where p1 + p2 + ... + pn = 1.0 and 0 <= pi <= 1 for each i). Furthermore, assume that each play of the game takes up one hour of your time, and that only you can play the game (you can't hire someone to play for you).

Let E be the game's expected value and S be the game's standard deviation.

1. In the real world, should a rational player always play this game whenever the expected value E is not negative? Why or why not?

2. Does the standard deviation S do a good job of capturing how risky this game is?
Why or why not?

3. If you personally had to decide whether or not to play this game, how would
you decide?



~Rhyn
 
Hi everyone. Recently came across this question in an upper level class and to be honest I am stuck on it. I am not completely sure if this is the correct forum but here it is. Any guidance is appreciated.

Consider a purely probabilistic game that you have the opportunity to play. Each time you play there are n potential known outcomes x1, x2, ..., xn (each of which is a specified gain or loss of dollars according to whether xi is positive or negative). These outcomes x1, x2, ..., xn occur with the known probabilities p1, p2, ..., pn respectively (where p1 + p2 + ... + pn = 1.0 and 0 <= pi <= 1 for each i). Furthermore, assume that each play of the game takes up one hour of your time, and that only you can play the game (you can't hire someone to play for you).

Let E be the game's expected value and S be the game's standard deviation.

1. In the real world, should a rational player always play this game whenever the expected value E is not negative? Why or why not?

2. Does the standard deviation S do a good job of capturing how risky this game is?
Why or why not?

3. If you personally had to decide whether or not to play this game, how would
you decide?



~Rhyn
A "rational" player is defined how? Is it "rational" to be a risk taker? Is it "rational" to be a risk avoider? Is it rational to be risk neutral? Is a utlility function provided? Are those people who buy both lottery tickets and homeowners' insurance rational?

What are the stakes? What can you lose, etc?

If the outcomes and probabilities are:

+100 80%

+210 10%

-1,000 10

The expected value is 80 + 21 - 100 = 1 > 0. Do you like that bet? Might someone like that bet?

How about outcomes and probabilities of:

+500 1%

-1 99%

Lots of people like that kind of bet. Is much of the world insane?

Personally, I think the problem is inadequately specified. Was it posed in a math class or an economics class? Economists frequently sneak, advertently or inadvertently, some normative assumptions into problems no matter how much they claim to "Wertfrei." In extreme cases, economics is religion pretending to be a science.
 
A "rational" player is defined how? Is it "rational" to be a risk taker? Is it "rational" to be a risk avoider? Is it rational to be risk neutral? Is a utlility function provided? Are those people who buy both lottery tickets and homeowners' insurance rational?

What are the stakes? What can you lose, etc?

If the outcomes and probabilities are:

+100 80%

+210 10%

-1,000 10

The expected value is 80 + 21 - 100 = 1 > 0. Do you like that bet? Might someone like that bet?

How about outcomes and probabilities of:

+500 1%

-1 99%

Lots of people like that kind of bet. Is much of the world insane?

Personally, I think the problem is inadequately specified. Was it posed in a math class or an economics class? Economists frequently sneak, advertently or inadvertently, some normative assumptions into problems no matter how much they claim to "Wertfrei." In extreme cases, economics is religion pretending to be a science.

It was introduced to us in a class that helps students prepare for technical questions in interviews for careers such as investment banking, private equity, hedge funds, etc.

The second bet is definitely worth taking.

As for the "rational" person, I think the question is trying to gauge each individual's risk tolerance and a "rational" person therefore is subjective since everyone has a different level of risk tolerance.

No utility function is provided but I am thinking that money is not linear.


I also think the question is not worded very well.
 
It was introduced to us in a class that helps students prepare for technical questions in interviews for careers such as investment banking, private equity, hedge funds, etc.

The second bet is definitely worth taking.

As for the "rational" person, I think the question is trying to gauge each individual's risk tolerance and a "rational" person therefore is subjective since everyone has a different level of risk tolerance.

No utility function is provided but I am thinking that money is not linear.


I also think the question is not worded very well.
You're right about the second bet. But I wrote it down wrong. It should have been 0.1% for $500 and - 1 at 99.9%. The expected value of THAT bet is 0.50 - 0.999 = -0.499, and people who play the lottery make that kind of losing bet every day.

OK It is probably a question purposely underspecified, designed to make you think about what additional inforrmation is required to give a sensible answer.
 
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