Image

Consider a two horse race– with probability 3/4 Duchess Camilla wins, with probability 1/4, it’s Princess Anne. You have $1 to bet. Camilla pays $2 for a win, and Anne $1– the precise numbers don’t matter. You are risk-neutral. Which horse should you pick? Camilla, of course, who pays $1.50 in expectation, whereas Anne pays only $0.25. Now suppose this race is run twice and you reinvest your winnings from the first race as a bet in the second. How you should now bet? You will bet on Camilla in the second race, for the same reason as before. Indeed, it is not hard to see that you should bet on Camilla in the first race as well, since you should maximize the expected amount of money you’ll have available to bet on the second. This is kind of boring. If there is any finite number of repetitions, bet on Camilla always.

But what if there is an infinite number of these races, where you reinvest all your winnings, and you care about your wealth in the far distant future? Now there’s a problem with always betting on Camilla– she will lose eventually, and you will be broke as soon as that happens. You must bet some of that $1 on Anne. How much should you bet? If you bet a fraction f on Camilla and 1-f on Anne, your wealth at date T, w(T), say, will be a product of n(T) factors of 2f and T-n(T) factors of 1-f, where n(T) is the number of times that Camilla wins. It follows that (1/T)ln(w(T)) = (n(T)/T)ln(2f) + ((T-n(T))/T)ln(1-f). The law of large numbers implies that (1/T)ln(w(T)) tends to (3/4)ln(2f) + (1/4)ln(1-f). You should maximize this long run limiting growth rate. In this case, it follows that the optimal f = 3/4.

As a strategy for investment, this log criterion was proposed by Kelley in 1956, as Lones Smith reminded me yesterday. He also said this Kelly criterion led to “Beat the Dealer”, the MIT Blackjack team, and eventually the Kevin Spacey movie “21”.

Kelley was roundly criticized by PA Samuelson in 1971, who objected to the criterion of long run wealth maximization, but Kelley was rehabilitated by Blume and Easley in several relatively recent papers.

Without needing to take a stance on the merits of the objections, they do not apply with a biological reinterpretation. In particular, the criterion of long run population maximization is evolutionarily convincing. The corresponding result is that the evolved utility function should be the expectation, with respect to aggregate risk, of the natural log of the expected offspring conditional on each aggregate state.

Individuals should be more averse to aggregate risk than they are to idiosyncratic risk. Preferences do not satisfy “probabilistic sophistication,” since it is not enough to list the outcomes and the associated probabilities. Preferences are interdependent, since the impact of gambles on others matters. Individuals may be induced to take idiosyncratic gambles in the face of aggregate risk.

Advertisements