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Ted Bergstrom, UCSB
Storage for Good Times and Bad:
Of Squirrels and Men
A fable of food-hoarding,
As in Aesop and Walt Disney…
 This fable is told with animal characters,
but has more ambitious intentions.
 What can evolution tell us about the
evolution of preferences toward risk?
 For the moral of the story, we look to the
works of another fabulist…
 Arthur Robson

Preferences toward risk
Robson (JET 1996) : Evolutionary theory
predicts that:
 For idiosyncratic risks, animals should seek
to maximize arithmetic mean reproductive
success. (Expected utility hypothesis.)
 For aggregate risks, they should seek to
maximize geometric mean survival
probability.

A Simple Tale
Squirrels must gather nuts to survive
through winter.
 Gathering nuts is costly—predation risk.
 Squirrels don’t know how long the winter
will be.
 How do they decide how much to store?

Assumptions:
There are two kinds of winters, long
and short.
 Probability distribution of winters is iid
with probability pS of a short winter
and pL =1-pS of a long winter.
 Two strategies, S and L. Store enough
for a long winter or a short winter.
 Probability of surviving predators: vS
for Strategy S and vL=(1-h)vS for
Strategy L.

Survival probabilities
A squirrel will survive and produce ρ
offspring iff it is not eaten by predators
and it stores enough for the winter.
 If winter is short, Strategy S squirrel
survives with probability vS and Strategy L
with probability vL<vS.
 If winter is long, Strategy S squirrel dies,
Strategy L squirrel survives with prob vL

No Sex Please
Reproduction is asexual (see Disney and
Robson). Strategies are inherited from
parent.
 Squirrels reproduce at rateρif they survive
through the winter.
 If pure strategies are the only possibility,
short-winter squirrels will be wiped out
after first long winter.
 Eventually all squirrels use Strategy L.
 Does this meant that evolution will produce
only long-winter squirrels?

Can Mother Nature Do Better?
This seems inefficient if long winters are
very rare.
 How about a gene that randomizes its
instructions.
 Gene “diversifies its portfolio” and is
carried by some Strategy S and some
Strategy L squirrels.
 In general, such a gene will outperform
the pure strategy genes.

Random Strategy
A randomizing gene tells its squirrel to
use Strategy L with probability πL and
Strategy S with probability πS.
 The survival rate of squirrels carrying this
gene will be

◦
◦
SS(π)= vS πS+vL πL, if the winter is short.
SL(π)=vL πL
if the winter is long
Long run reproduction rate
Since reproduction is multiplicative from
year to year, if there are k short winters
and T-k long winters over a period of T
years, then the expected number of
offspring of a π-strategist will be
ρTSS(π) kSL(π) T-k
and the average annual growth rate will be
ρ SS(π) k/TSL(π)(T-k)/T

Predicted Random Strategy
By the law of large numbers, as T becomes large,
k/T will almost certainly be very close to pS .
 Therefore the average annual growth rate of
the population of π strategists will be close to
ρSS(π) pSSL(π) pL


The evolutionary process would result in a
population of π strategists that maximize this
long growth rate.
Maximal-growth strategy
The mixed strategy π=(πS,πL) that
emerges will be the one that has the
highest long run average growth rate.
 Thus we seek π= (πS,πL) that maximizes
pSlog(SS(π))+pL log(SL(π))
subject to the constraints that
πS + πL=1, πS ≥0, and πL≥0.

The solution
Recall that h is the hazard rate from collecting
enough nuts for a long winter.
If pL<h, the mixed strategy that maximizes
long run growth is as follows:,
 πL = pL/h and πS =1- pL/h
SS = pSvS and SL=pLvL/h
SL/SS= (pL/1-pL)÷(h /1-h)<1
If pL>h, long run growth is maximized by the
pure strategy πL =1, πS =0, with SS=SL=vL
Some implications
If long winters are rare enough, and food
gathering hazardous enough, the most
successful strategy is a mixed strategy.
 Probability matching. Probability of Strategy L
is pL /h , proportional to probability of long
winter.
 For populations with different distributions of
winter length, but same feeding costs the die-off
in harsh winters is inversely proportional to
their frequencies.

Further Implication
(Recall that short-winter-squirrels survive
with probability (1-pL)vS and long-wintersquirrels with probability (1-h)vS.)
 If pL<h, then survival-probabilitymaximizing squirrels would prefer the
short winter strategy.
 Evolution must somehow motivate some
squirrels to take the long winter strategy,
even though it gives them lower survival
probability.

Ask the Psychiatrists

Do some people systematically act against
their own self-interest?
◦ Psychiatrists claim that about 5% of US
population afflicted by “compulsive hoarding
disorder.”
◦ They claim that 1.5% suffer from
“pathological” gambling problems and 4.8%
have “subclinical levels of gambling problems.”
An evolutionary explanation?
Compulsive hoarding and compulsive
gambling seem too prevalent to be simple
biological malfunctions.
 Maybe in evolutionary past, compulsive
hoarders made it through rare and
extreme famines.
 Maybe compulsive gamblers who “got
lucky” were the only ones to get through
other tough times.

Evolutionary bet hedging.

If this is the case, it may be that a gene
that gives you some compulsive gamblers
and some compulsive hoarders might
have a higher long term growth rate than
a gene that induces prudent, expected
survival maximizing behavior in all of its
carriers.
Some Twin study evidence
Using standard twin-study methodology,
researchers find risk preferences to be 2/3
genetically and 1/3 environmentally
determined, but found “no effect of
shared (observed) environmental factors”
 Another twin study finds 50% of variance
in compulsive hoarding to be genetically
determined and 50% environmentally
determined.

Heredity vs environment?

Maybe the differences that the twin
studies attribute to differences in
environment are not really due to
significant differences in environment, but
are evolutionary bet-hedging of the kind
we have modeled.
Generalizations
Model extends naturally to the case of
many possible lengths of winter.
 Let pt be the probability that winter lasts
for t days, for t=1,…W.
 Choose probabilities πt of storing enough
for t days.
 Let vt=v1(1-h)t be probability of avoiding
predators if you collect t days’ food.
 Let St(π) be expected survival rate of type
if winter is of length t.

A Result
If the probability distribution of the
length of the winter is log concave, all
squirrels attempt to gather at least
enough nuts to maximize private
survival probability.
 Some will gather more-enough for
each possible winters length.
 For long winters, survival rates will be
proportional to product of frequency
winter length and probability of
surviving predation when storing for
that length of winter.

Do Genes Really Randomize?
Biologists discuss examples of phenotypic
diversity despite common genetic heritage.
 Period of dormancy in seed plants—
Levins
 Spadefoot toad tadpoles, carnivores vs
vegans.
 Big variance in size of hoards collected by
pikas, golden hamsters, red squirrels, and
lab rats (both in the field and in
laboratories)--Vanderwaal

Is Gambling Better Than Sex?
Well, yes, this model says so.
 Alternative method of producing variation—
sexual diploid population, with recessive gene
for Strategy S.
 Whats wrong with this? Strategy proportions
would vary with length of winter.
 But gambling genes would beat these genes by
maintaining correct proportions always.

Gambling and income distribution
Suppose that instead of diversifying incomes
by differential effort, a ``central authority’’
was able to redistribute nut holdings of
squirrels in such a way as to maximize
expected long run growth.
 This possibility separates diversification of
outcomes from diversification of production
strategies.
 Efficiency would have all of them seek to
collect Y days’ food where Y maximizes
expected collection vYY, and then they
would receive consumption by lottery.

Casinos or theft
Humans, although they lack bushy tails and
the ability to scamper from tree to tree are
able to organize redistribution by means of
lotteries.
Both humans and squirrels (especially
humans) can also manage redistribution by
theft.
Squirrel Redistribution?
A growth-maximizing distribution
Suppose that the amount of nuts collected by
each squirrel who survives predation is enough
to last for Y days, where Y<W.
 If there are N surviving squirrels, let Nt be the
number of squirrels that are allocated t days’
worth of nuts.
 The set of feasible allocations consists of all
allocations such that
 1N1+2N2+…+tNt+…WNW=NY
Equivalently, where πt=Nt/N, this constraint is
St πt=Y

The maximization problem
If the winter is of length t, then the only
survivors will be those who were
allocated t or more nuts.
 Thus where π=(π1+…+πW) is the
distribution of wealth in the population,
the distribution of survival probabilities is
S=(S1…SW) with St=Sj≥t pj
 Then πt=St -St+1

Posing the problem
The maximum growth rate is achieved by
means of the distribution S of survival
probabilities that maximizes
St=1W ptlog(St)
subject to the constraints that
St=1W St=Y and
1≥S 1≥S 2≥S3 …≥SW

About the Solution
Suppose that the distribution of length of
winter is unimodal with mode m.
 There is some number v of days food
supply that minimizes food cost per unit
of survival probability.
 Nobody will receive a positive amount of
food less than v.
 If total amount of food is not enough to
give everybody v, then some get 0 and
rest get v or more.

More about the solution

Where total amount of food is large
enough, all get at least an amount r≥v
and some get exactly enough food for
each possible length of winter, with the
fraction receiving exactly t>r days supply
being proportional to the frequency of
winters of length t.
A theory of Income distribution??

Could this maximal growth distribution
tell us anything about income
distributions that arise in human societies
as a result of gambling, coercion, and
theft?
That’s all, folks…