Utility of Money and the St. Petersburg Paradox
Consider the following game:
We will flip a fair coin, until a tail appears for the first time.
- If the tail appears in the first throw, you win $2^1=2$ dollars.
- If the tail appears in the second throw, you win $2^2=4$ dollars.
- If the tail appears in the third throw, you win $2^3=8$ dollars.
- ...
- If the tail appears in the $n$-throw, you win $2^n$ dollars.
What is the amount of money that someone should risk to enter this game? (This question works best when given to a person that claims to never play a lottery, roulette, or any gambling game, because the expected return is lower than the bet.)
Computing the expected return of this game, we have:
$E=\frac{1}{2}\cdot 2+\frac{1}{4}\cdot 4 + \frac{1}{8}\cdot 8 + \cdots = 1+1+1+ \cdots =\infty$
In other words, the expected utility is infinity, and a rational player should be willing to gamble an arbitrarily large amount of money to enter this game.
Can you find anyone willing to bet $1,000 to play this game? Or $10,000? Or even $100? Yes, I did not think so.
Can you find anyone willing to bet $1,000 to play this game? Or $10,000? Or even $100? Yes, I did not think so.
This paradox is called the St. Petersburg Paradox, posed in 1713 by Nicholas Bernoulli and solved in 1738 by Daniel Bernoulli. Since then, a number of potential explanations appeared.
The most common approach is to use expected utility theory. In this case, we introduce a utility function $U(x)$, which describes the "satisfaction" that someone would get by having $x$ amount of money.
The most common approach is to use expected utility theory. In this case, we introduce a utility function $U(x)$, which describes the "satisfaction" that someone would get by having $x$ amount of money.
Utility of Money: The basic idea is that people do not bet based on the absolute amounts of the return but rather based on the utility of the award. The value of an additional $100 when I have $100 in the bank is much higher compared to the case when I have $1,000,000 in the bank. This means that the "utility of money" function is a concave function of the available funds.
Just for demonstration, below you can see such a concave utility-of-money function that we have computed as part of a research project:
This concavity also partially explains the "risk aversion" that most people have: they prefer certainty over uncertainty. This means that they will reject even a reasonable bet with positive expected return. Why? Notice that the utility gained by winning is smaller than the decrease in utility that results from losing the bet. The higher the concavity, the higher the risk aversion.
If you want to read more about utility of money and its applications to portfolio management, insurance, and analysis of other cases, take a look at this book chapter.
So, next time that someone claims never to engage in any bet with a negative expected return, give the setting of the Bernoulli paradox with the positive expected return and observe the reactions...
11 comments:
maybe you mean concave ?
Never drink and derive :-)
yes, concave...
For me it is very easy: I do not bet. Period. I've worked in horse racing and have learned "ο παίζων χάνει και ο πίνων μεθά" (he who drinks gets drunk; he who gambles loses).
But then again I am not the general case of non-gamblers and I admin trying to calculate prior reading the rest of the post :)
"I never enter a game with a negative profit expectancy" doesn't imply "I am willing to enter one with a positive expectancy".
p=>q does not imply ¬p=>¬q
rather, "positive/negative expected profit", I should've written. Expectancy is 0 to 1.
Stazybo: Of course "not betting on negative expected profit bets" does not imply that someone will bet on bets with positive expected profit. But often people that claim not to bet due to the negative expected return of the bet, they will base their argument solely on the expectation of the return. (Which is easy to show that is not sufficient.)
It is a good example for introducing the concept of "risk aversion", i.e., the need to take into consideration the variance of the bet and not only the expectation.
It's quite worthwhile to consider the OTHER notable way of analyzing the results of a bet, which is to consider the probability of making money on the bet.
And THAT offers conclusions of quite a different shape.
Thus, consider, for a wager, N, what is the probability P(N) that you break even given that wager?
We actually only need to consider wagers of the form $2^n$, where $n$ is a positive integer.
- If you wager 2 dollars, then it is certain that you will get a return of at least 2 dollars, the worst case being where the tail appears in the first throw.
- If you wager 4 dollars, then your chances of breaking even are 1/2
- If you wager 8 dollars, the probability of breaking even is 1/4
And this relationship continues. The larger the wager, the lower the probability of breaking even.
And this is so, irrespective of the fact that the expected payoff is infinite.
It would certainly be worth my while to pay 2 dollars to participate in this "lottery." But the value of engagement at higher prices is not nearly so obvious.
Expected value is by no means sufficient to support correct conclusions.
Nice post.
I will be thankful if u care to explain why the following analysis is wrong :
The number of throws in the game is a geometrically distributed random variable , with 1/2 as its parameter. Hence on expectation the number of throws is 2.
Now, based on this why is it wrong to say that on expectation the amount of money you win is 2^2 = 4$ .
Thanks,
Giridhar.
This is the flaw: You assume that the game will have exactly as many rounds as the expectation and then you compute the payoff for that game. You need to weight each round with the payoff.
The main trick in this paradox is that you assign exponentially high payoffs to games that are exponentially unlikely to happen. So while it is highly unlikely to reach a large number of rounds, the payoff from these games is going to increase tremendously your average payoff.
Just to elaborate on Chris Browne's point.
The probability that you win 2$ is 50%.
The probability that you win 4 or more $ is 1/4 + 1/8 + 1/16 ... = 50%
The probability that you win 8 or more $ is 1/8 + 1/16 + 1/32 ... = 25%
and so on.
So, if you bet 4$, you lose 2 with 50% chance but you win 2+ with 50% chance, so you may as well bet.
If you bet 8$ however, you lose money with 75% chance and win with only 25%, so it's not an advantageous bet.
The amount sm should risk to enter is 4$.
Dimitris
I am not entirely convinced that the "probability of making money on a bet" is always the best approach for analyzing bets.
Consider for examples the "bets" that Venture Capitalists put on start-up companies. More than half of the investments will fail. A few will break even, giving some very modest return, and a couple of them will generate the 10x - 100x returns that are the ones that will carry the rest of the investments. You can hardly call this approach irrational, even though the VC's lost money in more than 50% of the investments.
The basic point is to start considering the variance of the bet, together with the expectation. (Potentially, we should also consider higher moments as well.) This gives a much better picture than focusing exclusively on the expectation.
Post a Comment