Bet Hedging, Risk Aversion, Sex, and the Unit of Selection

A man sits down at a roulette wheel with $100. How should he
distribute his bets?

It is a trick question. The answer is that the man should not
sit down at the table in the first place. Roulette (in America)
provides an expected return of $36 for every $38 bet.

So, lets change the problem. Say that the man HAS TO bet. He
has gone to Las Vegas on one of those package deals which includes
$100 in marked chips which must be used to gamble at the tables.
He hates gambling, but he has always wanted to see Sigfreid and Roy.
He needs to turn as many as possible of those chips into real money
for show tickets and meals.

Ok, now one solution to the problem is to divide the stake equally
among the 38 numbers. One of them will pay off, so he will receive
$35 in real money plus $1 in marked chips, for every $38 in marked
chips invested. Guaranteed.

How is this superior to simply placing the whole stake on Black, or
even on double-zero. Well, it isn't superior if you take the goal
as that of maximizing expected return. The expected return is the
same regardless of how the bets are distributed. But the suggested
solution has the virtue that it minimizes the variance in the
return.

A gambler who is willing to sacrifice some amount of expected
return in order to hold down the variance in the return is said
to be 'risk averse'. Is aversion to risk 'rational'? Sure, and
so is its opposite 'risk seeking'. Game theorists are willing
to describe any system of objectives as rational. The only thing
'irrational' is a strategy which does not further those objectives.

One strategy that can be useful in achieving a 'risk averse'
objective is 'bet hedging' - betting on both sides of an alternative,
in effect, betting against yourself. In our roulette wheel example,
it provides an expected loss, but the loss is guaranteed to be small.
In some stock-market scenarios, it can provide an expected gain,
though one which is also guaranteed to be small.

Two things are needed in order to hedge bets. One is that the
total stake must be divisible so that it can be spread among the
alternatives. The other is that the alternatives must not be
independent - it would not have done our Las Vegas junketer any
good to place his 38 bets one at a time on 38 successive roles
of the wheel. Same expected return, but now he is taking a risk.

What does all this have to do with evolutionary biology? Well,
some theorists have claimed that the units of natural selection
behave as if they were risk averse maximizers of fitness (i.e.
reproductive success). For example, Lewontin and Cohen
On Population Growth in a Randomly Varying Environment
R. C. Lewontin, D. Cohen
PNAS Vol 62, No 4, (Apr 15, 1969), 1056-1060
used the mathematical theory of "Gambler's Ruin' to argue that
in a randomly varying environment, the unit's goal is not
the maximization of the expectation of absolute fitness, but
rather the maximization of the expectation of the logarithm of
absolute fitness. The expectation is taken over the space
of environmental variation.

Now, there is some subtlety here. If W is the absolute fitness,
then ln W = r where r is the Fisher growth-rate metric of fitness.
There is an important difference between maximizing r = E(ln W) and
maximizing r' = ln (E(W)). The idea is that an organism can and
should 'hedge its bets' by having some offspring adapted to one
environment and other offspring adapted to different environments.

But is this even possible? An organism only HAS a few offspring.
The 'stake' cannot be evenly divided among the bets. However
risk averse the organism is, its opportunities for bet hedging
seem severely limited. Furthermore, bet hedging only makes sense
if the bets are NOT independent. An organism with two offspring
adapted to two different environments is only hedging if both
of those offspring encounter the same environment.

So much for the background - now it is time for some wild conjectures.
1. Sex exists because it provides a mechanism for bet hedging.
2. But this only makes sense if the unit of selection is seen
as the gene-clone, as in a gene's eye view justification
of Hamilton's rule. A gene clone can spread its bets
evenly among the alternatives - it has a 'stake' that is
divisible. Organisms, for the most part, do not.
3. As suggested in the paper by Bergstrom and Lachmann
The Fitness Value of Information
Carl Bergstrom and Michael Lachmann
http://arxiv.org/PS_cache/q-bio/pdf/0510/0510007.pdf
the process of natural selection can be given an information
theoretic interpretation in which there is an identity between
fitness (Fisher's r) and information acquired about the actual
distribution of environments.

My rudimentary grasp of probability and statistics doesn't allow
me to express this stuff in a rigorous model yet. The difficulty
lies in separating the environmental variation into temporal variation
(affecting all organisms the same) and spatial variation (affecting
organisms differently). The interplay between these two kinds of
variation seems crucial. I suspect that I need two distinct E
(expectation) operators for the two kinds of variation, and that
one of these operators 'commutes' with the logarithm operator
while the other does not.



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Relevant Pages

  • Re: Bet Hedging, Risk Aversion, Sex, and the Unit of Selection
    ... > behave as if they were risk averse maximizers of fitness (i.e. ... > On Population Growth in a Randomly Varying Environment ... > environment and other offspring adapted to different environments. ... An organism only HAS a few offspring. ...
    (sci.bio.evolution)
  • Re: pre-tuning to baldwin effect
    ... mean that the fitness level of an organism is hardly determined by the ... You seem to want 'fitness' to mean 'ability to evolve progressively'. ... the ability to cope with different types of environment ... the environment doesn't change and there isn't much evolution, ...
    (sci.bio.evolution)
  • Re: DNA carries information
    ... the distinction between three things, ... The Fitness Value of Information ... always a complete organism) as a black box, ... tasting, making any measurement on the environment, nothing at all ...
    (talk.origins)
  • Re: What is the concept of fitness ?
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    (talk.origins)
  • Re: What is the concept of fitness ?
    ... need to overcome fluid friction in the first case, ... and c, we can expect that an increase in d will increase fitness, at ... if you have the priors to make the rate of expectation ... the fitness is the rate at the time in a given environment at which it ...
    (talk.origins)