Thursday, November 8, 2007

Dan the Man does it again!

Yesterday, I talked about Daniel Bernoulli's breakthrough idea called Diminishing Marginal Returns. It is helpful to read it before reading on. Today we'll extend the idea to Bernoulli's idea of quantifiable risk management.

In the NFL, a Fair Game is defined as one that does not involve the New England Patriots and illicit video cameras. In probability theory, a fair game is defined in more general terms: the expected outcomes are the same for each participant. For instance , to start and NFL game, the referee flips a "fair" coin to determine who gets the option of possession or choice of field. Presumably, each time is just as likely to win the coin toss, as the probability of each option, Heads or Tails, is the same: 1/2.

If we assume Daniel Bernoulli's assumption of Diminishing Marginal Utility, an individual who defined his utility of wealth according the the natural logarithm will actually REJECT a fair game. Bernoulli, himself used the following example to illustrate his point.

Suppose two individuals with $100 (or 100 Ducats, as Bernoulli actually used), each offer up $50 to participate in a fair game. What was this intriguing game which cost $50 to play? A simple coin toss, winner takes all money in the pot, $100. So now, each player has a 50-50 chance of walking away with only $50 if they lose, or $150 if they win. Got it?

Now we need to know a little about what Expected Value is and how to calculate it. Expected value is one of the most important
concepts in probability theory. It is the sum of the probability of each possible outcome in a situation multiplied by the outcome value (or payoff.) We interpret it as the amount of money one would "expect" to end up with as the result of a fair game that is played many times. This important number can help an individual decide if he wants to play the game at all.

Back to our $50 roll of the dice. The Expected Value of this game is $100, and is calculated as:

$50 x 0.50 + $150 x 0.50 = $100

Now since each player started out with $100 to begin with, they can only hope to "break even" in the long run if they play this game many, many times. As a result, this is also the expected value of NOT playing the game, so it would not make sense for them to play the game, unless of course the just like the activity itself.

But what is the Expected UTILITY of the game? We find it by adding the utility value (the natural log value) of the $50 payoff times the probability of it occurring to the utility value of the $150 payoff time the probability of IT occurring. The table bellow shows the results.

The utility of NOT playing the game is the natural log of $100, or ln(100) = 4.605

So, although the Expected value of playing or not playing is the same, the Expected utility is actually higher for NOT playing (4.605) than it is for playing (4.461). This means the player, according the Bernoulli and irrefutable math, will reject the game even though it is fair. Stated more intuitively, since he only has the chance of breaking even in the long run, it is not worth the time or effort to play the game. This is the same reason someone would reject digging a giant hole in the ground all morning with a shovel only to fill it back up with the same dirt later that afternoon. The blisters just aren't worth it unless we planted a tree, or buried treasure, or struck oil.

This fine example of opting out of game means that we, as humans, have an aversion to risk. It is a nice innate safety feature, if you think about it. In fact, Bernoulli himself called this risk aversion as "nature's admonition to avoid the dice."

Of course, one can still play with dice without any risk at all--"Yahtzee!"

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