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[sharechat] An exercise re risk aversion


From: "Capitalist" <capitalist@paradise.net.nz>
Date: Mon, 23 Sep 2002 16:35:08 +1200


A forward from another group that may interest the mathematically minded, since there has been some talk of risk/reward. BTW I personally have my doubts it is inductive proof.
 
>>The discussion on risk made me think of the following
inductive proof on risk aversion I learned a few years
back. 
 
Imagine you are offered the following game:  You will
flip a coin.  If it's tails, the game is over, and you
get nothing.  If it's heads, you win two dollars and
keep going.  You flip again, and if it's tails, the
game's over and you leave with your $2.  If it's
heads, you win $4 (total, not in addition to your
previous winnings) and keep going.  Again, if it's
tails, you leave with your $4, and if it's heads you
win $8 and keep going.  This continues until you
finally get a tail.

You cannot lose money from this game, only win it.
However, imagine that the person running it charges
you a set fee in order to play.  How much would you be
willing to pay to play this.  Quick, throw out a
number before reading farther.

...

To figure out how much you should pay, you should
figure out the expected value of the game.  There's a
1/2 probability you'll get nothing.  (If the first
toss is tails.)  There's a 1/4 probability you'll get
exactly $2.  (Heads then tails.)  There's a 1/8
probability you'll get exactly $4.
(Heads-heads-tails)  And so on.  So the expected value
of playing the game is:

(1/2)*0 + (1/4)*2 + (1/8)*4 + (1/16)*8 + (1/32)*16 +
...

This equals:

0 + (2/4) + (4/8) + (8/16) + (16/32) + ...

Which of course equals:

0 + 1/2 + 1/2 + 1/2 + 1/2 + ...

Which doesn't converge.  The expected value of the
game is infinite.

Head scratcher time.  How many of you said they'd pay
an infinite amount of money to play this game?  I
don't see too many hands raised.  Which makes sense.
Would you really bet everything you own, leaving
yourself penniless and naked on the street, to play a
game where there's a 50% chance you'll be left with
nothing?  And where there's only a 1 in a thousand
chance you'll get more than a thousand dollars?

My suspicion is that very few people said they'd pay
more than $10 to play, and almost nobody offered more
than $100.  Even now, that you've seen the math and
know the expected value is infinite, I doubt you'd pay
more than $100 to play.

People are risk averse, QED.

=====
--------------------------
Steve Friedman

 
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