On diminishing returns
It is intuitively true that the marginal utility of a dollar decreases as you have more money. If you earn $2000 dollars a month, the first $1000 prevents you from living on the streets, while the next $1000 pays for less essential needs. Because you can freely allocate your money, you will generally allocate it to the most important things first, and less important things second.
This is important in anything that involves chance, such as investment or gambling. For example, suppose you have $1000, and you have the opportunity to bet it all for a 50% chance of winning an additional $1000. You would prefer not to take that bet, because:
U($1000) > 0.5 U($2000) + 0.5 U($1000)
where U is the utility function. Even though the average payoff in dollars is zero; the average payoff in utility is negative. This leads to other nice results, including the notion that risk is bad, and the notion that giving money to poor people is good.
But what exactly is the functional form of U?
Clearly there’s no single right answer. The utility of money depends on your personal preferences and what you’d use it for. For purposes of abstract discussion we just want a simple function with the desired qualitative properties. One possibility is that U is a logarithmic function, but plenty of other functions could serve the same purpose.
What interests me is that there is apparently some theoretical justification for logarithmic utility. Furthermore, the argument is not based on your ability to freely choose how to allocate your money. That argument is made in a 1956 paper by J. L. Kelly, Jr., which I report on here.
The Kelly Criterion
Suppose that you’re given the opportunity of a favorable bet. Perhaps you like to bet on horse racing, and you have some inside information which gives you an edge over the bookkeepers. Suppose that you get this inside information on a regular basis, and have the ability to reinvest your winnings each time. What fraction of your money should you bet?
If you’re trying to maximize your expected money, you would bet everything. But this seems rather pathological, particularly when the bet is repeated. Eventually you’d lose all your winnings. The only reason this is a “good” strategy is because there’s a tiny chance of a staggeringly huge payout.
Kelly suggests an alternate strategy where you bet a fixed fraction of your money. The amount you should bet is given in mathematical form by the Kelly Criterion.
I will omit statement of the Kelly Criterion, because what I’m interested in is not the mathematical expression itself, but where it comes from. Where it comes from is a logarithmic utility function. If you use logarithmic utility and wish to optimize your expected utility, you will naturally arrive at the Kelly Criterion.
But then, where is this logarithmic utility function coming from?
In which I criticize Kelly
I was disappointed to find that Kelly does not have any solid justification for logarithmic utility, and mostly relies on what I consider to be question-begging and arguments by illustration.
Confusingly, Kelly is not coming from an economic perspective, but from a communication theory perspective. Suppose that you’re trying to communicate a series of 0s and 1s, but the communication channel is noisy and some of the numbers come out wrong. How do you measure the amount of information being communicated? If you had to choose between more noise and a slower communication rate, how do you weigh your options?
The answer depends on the specific application of the information. In particular, it depends on how much you dislike getting incorrect information. To analyze the problem quantitatively, Kelly imagines a scenario where the information has well-defined payoffs. Thus the scenario where you’re betting on horse races based on inside information.
I find this rather odd, since the more obvious application of the Kelly Criterion is not in gambling, but investing. Generally, investments have an expected positive payoff, whereas gambling has a negative payoff. Because Kelly uses gambling as his example, he needs to construct an elaborate scenario where you have inside information on the bet, and that’s why the bet is favorable. It also leads to a discussion of “track take”, which is an obscure horse racing term that I was content to be ignorant of.
Anyway, the argument is that you can use this information to exponentially multiply your money, and you’d like to maximize the expected rate of exponential growth. But why? Why not maximize the expected payoff instead? I agree with Kelly’s conclusion, but I feel this is begging the question. Most of Kelly’s argument appears to be along these lines.
Justifying logarithmic utility
I will highlight what I believe to be Kelly’s best argument for logarithmic utility. Given the scenario illustrated, the Kelly strategy is highly likely to beat any other strategy. As the number of bets goes to infinity, the probability that the Kelly strategy will produce more money than any other strategy approaches 1.
This argument is interesting, because it implies a particular ordering of strategies. Namely, strategy X is better than strategy Y if X has over 50% chance of producing greater payoff than Y. It is unnecessary to state any particular utility function to perform this calculation, because this calculation is not based on expected utility. All you need are ordinal preferences. That means you must order different outcomes in terms of your preference, but you don’t need to say how much you prefer one outcome over another. You need to be able to say that more money is better than less money, but you don’t need to say whether $2 is twice as good as $1.
However, upon closer investigation, the calculation falls apart. It’s possible to construct a set of strategies such that X is preferable to Y, Y is preferable to Z, and Z is preferable to X. Here is one such construction:
Strategy X: 2/3 probability of $1, 1/3 probability of $4
Strategy Y: 2/3 probability of $3, 1/3 probability of $0
Strategy Z: always pays off $2
Luckily, the horse racing scenario considered by Kelly is more straightforward, and you can ignore such pathological scenarios. The Kelly strategy indeed beats out every other strategy in the long run. So it is possible to “derive” a logarithmic utility function using ordinal preferences.
Of course, this assumes that you have repeated opportunities to make these bets, and at each step you may reinvest your winnings, and you’re not allowed to go into debt. These assumptions are reasonable in the world of investment, but I’m skeptical of the wider application of the Kelly strategy. In most cases, using a logarithmic utility function is simply a matter of convenience, and is not in any sense rigorous.