How Portfolio Theory Parallels Evolutionary Biology and Predicts Design by Coin-flipping

                                                        by
                               Wm S. Cooper, Prof. Emeritus

                               University of California, Berkeley

Dark against white?

    The subject of evolutionary theory might seem far removed from the analysis of financial investments.  Indeed it is, but there are some close parallels in the underlying mathematics.  Because of these resemblances, insights about how to invest funds wisely carry over as insights into population biology.  The lessons to be learned from them about organismic design run deep.  For instance, they bear on the question:  If Evolution is the designer of all living things, how much of the design does it leave to chance?

    The conceptual leap to be made is that the way organisms multiply in a population is analogous to the way investments multiply in a brokerage account or portfolio.   The analogy hangs on a simple mathematical fact -- an oddity governing the long-run increase of both populations and portfolios.  In what follows I'll first present the mathematical oddity, then delve into its implications for the growth of an investment portfolio, and finally draw out the consequences for the evolutionary process.

The Mathematical Oddity

   Suppose we multiply a number by itself a few times, say ten times over, e.g.

       2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 1,024

Now let's replace each number with other numbers in such a way that the average term size remains the same.  For example,

      1 x 3 x 1 x 3 x 1 x 3 x 1 x 3 x 1 x 3  =  243

The resulting product is much reduced.  Someone inexperienced with product series might have reflexively expected that since the average (arithmetic mean) of the terms has not changed, the product should remain about the same.  But no, introducing variation in this way always lowers the product, while smoothing out variation by moving the entries closer to their mean always raises it.

Application To Investing

    That mathematical quirk has profound consequences for the theory of financial investment.  Suppose a hypothetical investor wishes to invest a sum of money somewhere in the international stock market and let it compound there indefinitely.  He analyzes the market in Country A and finds that funds invested there increase in abrupt starts and stops.  For the sake of having definite numbers in hand let us imagine they grow by a multiplicative factor of 3 in expansive years but only 1 in steady years.  These two annual growth factors occur randomly and unpredictably but with equal frequency over the long run.  He looks at neighboring Country B and finds a similar situation.  So it looks like no matter which of the two countries he chooses, if past patterns continue his money can be expected to increase by a factor of  243 in a typical 10 year period.  Not too shabby, but he wonders if somehow he could do even better.

    He then notices something.  There happens to be a perfect negative correlation between the returns in the two countries.  Because of their competitive positions, when one country flourishes the other languishes.  One might have for example over a representative ten year period in which the good and the steady seasons happen to alternate,

     Country A:  1 x 3 x 1 x 3 x 1 x 3 x 1 x 3 x 1 x 3  =  243

     Country B:  3 x 1 x 3 x 1 x 3 x 1 x 3 x 1 x 3 x 1  =  243

What should the investor do?  He cannot predict ahead of time which country will be successful in any given year.  So he astutely divides his money equally between the two countries.  Each year his funds will double, since (1+3)/2 = 2 is its annual multiplicative increase.  Because the variation has been smoothed out, in his combined portfolio he will receive the much improved return of

    50-50 mixture of A and B:  2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 1,024.

To get this extravagant return, at the end of each year he will have to transfer some funds to restore the 50-50 balance between the two countries.  But it is well worth the trouble.

    It is hard to find investments that have a strong negative correlation through time.  However, there doesn't have to be a negative correlation.  The effect will still be present with no correlation.  Even if there is some positive correlation, with economies A and B in lock step oftener than not, splitting the funds will still be the thing to do.  Here is what happens when the economies grow in concert 60% of the time:

     Country A:  1 x 3 x 1 x 3 x 1 x 3 x 1 x 3 x 1 x 3  =  243

     Country B:  1 x 1 x 1 x 3 x 3 x 1 x 1 x 3 x 3 x 3  =  243

     50/50 mix:  1 x 2 x 1 x 3 x 2 x 2 x 1 x 3 x 2 x 3  =  432

Still an impressive improvement.  A perfect positive correlation would naturally eliminate the mixing advantage entirely, but for anything short of that, diversification is bound to help.

     Even if Country B has distinctly poorer prospects than A, it can still be advantageous to include a lesser share of it in the portfolio.  Suppose Country B's growth spurts are only doublings, not triplings.  It still pays to put a quarter of the funds into it:

     Country A:   1 x 3 x 1 x 3 x 1 x 3 x 1 x 3 x 1 x 3  =  243

     Country B:   2 x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2 x 1  =   32

     75/25 mix:  1.25 x 2.5 x1.25 x 2.5 x 1.25 x 2.5 x 1.25 x 2.5 x 1.25 x 2.5  =  298

So, counterintuitive as it may seem, a little dilution by an inferior investment can actually improve a portfolio's long-run performance.

    Such is the magic of diversification.  The profitability of spreading one's funds around is well established in standard works on portfolio theory and asset allocation.  Diversify, diversify!

Application to Evolutionary Theory

       The mathematics of portfolio theory carries over into population biology.  Imagine a hypothetical population of animals -- rabbits, say -- inhabiting a territory in which the dark ground is covered with snow in half the winters.  Over time, snowy and snowless winters occur at random with equal frequency.  In snowy winters it is advantageous for a rabbit to have a white coat; in snowless winters dark is better.  Dark is fine for summer.  There is no way the rabbits can tell ahead of time whether the oncoming winter will be snowy or not.  What should be their winter camouflage strategy?

      For specificity assume the population, if all of them are white every winter, would increase in size by a multiplicative factor of 3 in years with snowy winters and remain unchanged in snowless ones.  For a population all of which are dark in winter it is the other way around.  Consider a representative ten year stretch in which snowy and snowless winters happen to occur equally often.  Comparing the possibilities we see:

      All white:          1 x 3 x 1 x 3 x 1 x 3 x 1 x 3 x 1 x 3  =    243 

      All dark:            3 x 1 x 3 x 1 x 3 x 1 x 3 x 1 x 3 x 1  =    243

      50/50 mixture:  2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2  =  1,024

Clearly, a population that could somehow maintain itself in a state where, at the onset of each winter, half its members are white and half dark, would grow dramatically faster over the long run than a population of uniform winter coloration.  As a mixture it could outcompete any similar unmixed competitor.  For that matter it could beat even a mixed population whose mixture did not stay close to 50-50.  It is advantaged because its "portfolio" is optimally balanced.  In Ecology such an advantage is sometimes called a Portfolio Effect.

    But how could a population make use of the portfolio effect?  You would think that over the ages evolution would surely have found a way to exploit this universal mathematical principle.   However it is not easy to think of a standard genetic population model, Mendelian or otherwise, that would do so.  The problem lies in the rebalancing of the portfolio.   How could a mixed population, after a season in which selection has altered the statistical distribution of its traits, reset itself to its initial distribution in preparation for the next season?  Indeed the very idea of such rebalancing seems at odds with the usual notions of selection and genetic inheritance.      

     So if ordinary genetic determination of traits will not do the trick, what's left?  Absence of genetic determination.  It has been called "Coin-flipping".  Under the coin-flipping (strategy-mixing) regime, the white and dark subpopulations are not genetically differentiated.  Coloration is not under strict genetic control. Rather, all individuals in the population have a genetic inheritance that leaves the white/dark choice undetermined.  Instead of genetic predetermination, chance factors of some sort determine the color independently for each individual.  For each member there is a 50% probability of white and the same for dark, resulting usually in an approximately 50-50 color distribution throughout the population (assuming it is sizable).  If this optimal probability distribution should stray from the ideal balance, selective forces will tend to pull it back. Metaphorically, the selective pressures regulate the weighting of the "coin".

    Ingenious, but what might such a coin be like?  In the case of a behavioral trait, it could conceivably be a random environmental clue that the animal experiences and acts upon instinctively.  That is what humans consciously contrive to do when they make a decision by flipping a coin.  But with a physical trait such as fur color, the deciding event would presumably be something random within the developmental process.  The probability of occurrence of this random event would be under genetic control even though the specific outcome in particular cases would not.  This is a tricky concept.  There is still genetic influence but it is incomplete.  The genes do not rigidly determine the trait; they only determine a probability distribution over the available traits.   The probability distribution, though it may be the same for every individual, results in different observable outcomes for different individuals.  The upshot in the case of the rabbits is that an observer would normally see a winter population that is approximately half white, half dark.

     Coin-flipping works especially well for our hypothetical animals because the growth spurts of the two traits are negatively correlated through time.  When white flourishes, dark does not.  Unlike the stock market, in ecological systems negative correlations are commonplace.  It is unsurprising when different alternative traits are found to be superior under different environmental conditions.  But what if there is a positive correlation?

    There are species of migratory birds in which a portion of the population does not migrate but instead overwinters in the home territory.  Consider a model in which, in seasons of favorable winter conditions at the migrating destination, the migrators triple in number, but when unfavorable they merely maintain their numbers.  Similarly for those that overwinter.  In 60% of the years, conditions are alike at home and at the migration destination -- either both are favorable or both are unfavorable.  One might have e.g.

     All migrate:     1 x 3 x 1 x 3 x 1 x 3 x 1 x 3 x 1 x 3  =  243

     All stay put:    1 x 1 x 1 x 3 x 3 x 1 x 1 x 3 x 3 x 3  =  243

     50/50 mix:      1 x 2 x 1 x 3 x 2 x 2 x 1 x 3 x 2 x 3  =  432

So there can be a mixing advantage even with a positive correlation.  Again it could be awkward to explain how the optimal mixture could be maintained with simple determinist genetics, but coin-flipping would do it.

    Even if a trait has a distinctly inferior long-run growth rate, it can still be of benefit to the population to maintain some of it.   Returning to the simpler rabbit model for an illustration, here's what happens when in favorable years the darks only double.

     All white:     1 x 3 x 1 x 3 x 1 x 3 x 1 x 3 x 1 x 3  =  243

     All dark:       2 x 1 x 2 x 1 x 2 x 1 x 2 x 1 x 2 x 1  =   32

     75/25 mix:  1.25 x 2.5 x1.25 x 2.5 x 1.25 x 2.5 x 1.25 x 2.5 x 1.25 x 2.5  =  298

Evidently, though dark by itself is less fit than white by all ordinary measures of individual fitness, the presence of some dark can enhance the prospects of the population as a whole.  Along with the fittest there can be survival of the less fit.

     A coin weighted as shown, i.e. 3 to 1 in favor of white, happens to maximize population growth in this example.  Selection will tend to draw the actual weighting toward this optimal value.  Should it stray, any weighting giving better than 50-50 odds to white is sufficient to preserve the mixing advantage until the optimum is neared again.  In this way the portfolio effect gives rise to a mechanism for the maintenance of variation.  

    The darks are the losers of a forced gamble.  As individuals they must put up with inferior life prospects for the good of the population as a whole. This would appear to be a form of altruism in the sense of individual self-sacrifice for the general welfare.  Unlike other forms of altruism, this kind doesn't depend on kinship, reciprocity, or group selection.  It might be called "Coin-flipping altruism" or "Portfolio altruism".  Altruistic acceptance of the risky coin flip is an evolutionarily stable strategy that cannot be invaded by a selfish defector to white-for-sure.

      Generalizing to three or more alternative traits, one could speak of a portfolio effect based on "roulette wheel spinning", or in the continuous case "dart throwing".  Clearly, the opportunities for the portfolio effect to influence the course of evolution are ubiquitous.

Conclusions

    The mathematics of investment theory is wonderfully predictive of greater organismic variation throughout the biosphere.  It is a vindication of a certain permissive "looseness" in organismic design -- of regulated sloppiness if you will.  Due to the portfolio effect, tight genetic guidance deserves less credit, and beneficent accident more of the credit, for the diversity of nature than might otherwise have been supposed.  The reach of genetic governance is to some extent foreshortened. The diversifying influence of the portfolio effect, though often weak, is nonetheless pervasive.  It has the ability to sustain innumerable characteristics that could not otherwise be maintained.   Nature is the richer for it.

Sources

     The foregoing is an attempt to simplify and distill the essential points of a line of research begun in the 1980's.  Key papers include

    W. S. Cooper, R. H. Kaplan, "Adaptive `Coin-Flipping': A Decision-Theoretic Examination of     Natural Selection for Random Individual Variation."  Journal of Theoretical Biology 94: 135-151. 1982.

    R. H. Kaplan, W. S. Cooper, "The Evolution of Developmental Plasticity in Reproductive Characteristics: An Application of the `Adaptive Coin-Flipping' Principle."  American Naturalist 123: 393-410. 1984. 

The ideas have been applied in various contexts since then.