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.

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July 18, 2012 at 3:24 pm

Lones SmithI love the citation of trendy leading economists! ^_^

As a matter of aesthetic, I am at the forefront of a worldwide movement in our field to stop using ln, and go back to log. Euler invented logs and base e was implied. Then for the longest time, slide rules were all the rage. And base 10 was in. And of course, computer science came along with its fascination with binary. But dammit, our whole profession means base e when it writes log (with a few weird exceptions that no doubt someone will point out). As such, let us agree to use the suggestive word “log” rather than the needless ln — which our wonderful high school calculus teacher used to call “lawn”.

All I am saying is: On all your papers, give peace a chance! And log too! Let us kill the lawn beast once and for all.

Oh, and nice post Arthur. ^_^

July 18, 2012 at 3:52 pm

E^ Agreed. About log, and that it’s a nice post.

July 19, 2012 at 9:10 pm

IFBTO (@FranInfoNews)I hadn’t thought about this before. But your clear presentation of the Kelly criterion brings to mind, for me, the matching probability paradox.

“That is, they allocate their responses to the two options in proportion

to their relative payoff probabilities.

Thus suppose that a monetary payoff of fixed size is given with probability p=0.7 for choosing left and with probability 1-p=0.3 for choosing right.

Probability matching refers to behavior in which left is chosen on about 70% of trials and right on 30%.

Such responding violates rational choice theory because the optimal strategy in such tasks, after an initial period of experimentation and assuming that the payoff probabilities are stationary, is always to select the

option associated with the higher probability of payoff.

On any trial, the expected payoff for choosing left is higher than the expected payoff for choosing right.”

Perhaps this matching behavior is a nod to the Kelly criterion?