
Printable version 
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. (Headsheadstails) 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 
