Download Storage for Good Times and Bad: Of Squirrels and Men

Storage for Good Times and Bad:
Of Squirrels and Men
Ted Bergstrom, UCSB
A fable of food-hoarding,
• As in Aesop and Walt Disney…
• The fable concerns squirrels, but has more
ambitious intentions.
• What can evolution tell us about the evolution of
our preferences toward risk?
• For the moral of the story, we look to the works of
another great fabulist…
• Art Robson
Preferences toward risk
• Robson (JET 1996) : Evolutionary theory
predicts that:
• For idiosyncratic risks, humans 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.
• Climate is cyclical; cycles of length
k=kS+kL, with kS short and k L long winters.
• 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.
• Suppose pure strategies are the only
possibility.
• Eventually all squirrels use Strategy L.
• But what if long winters are very rare?
Can Mother Nature Do Better?
• 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 reproduction rate of this gene will be
–
–
SS(Π)= vS ΠS+vL ΠL, if the winter is short.
SL(Π)=vL ΠL
if the winter is long.
Optimal Random Strategy
• Expected number of offspring of a random
strategist over the course of a single cycle is
ρkSS(Π) kSSL(Π) kL
• Optimal strategy chooses probability vector
Π=(ΠL ,ΠS ) to maximize above.
• A gene that does this will reproduce more rapidly
over each cycle and hence will eventually
dominate the population.
Describing the optimum
• There is a mixed strategy solution if
aL=kL/k<h.
• Mixed solution has ΠL =aL/h and
SL/SS= aL(1-h)/(1- aL)h.
• If aL>h, then the only solution is the pure
strategy L.
Some lessons
• If long winters are rare enough, the most
successful strategy is a mixed strategy.
• Probability matching. Probability of Strategy L is
Is aL /h , proportional to probability of long winter.
• For populations with different distributions of
winter length, but same feeding costs the die-off in
a harsh winter is inversely proportional to their
frequency.
Generalizations
• Model extends naturally to the case of
many possible lengths of winter.
• Replace deterministic cycle by assumption
of iid stochastic process where probability
of winter of length t is at
• Choose probabilities Πt of storing enough
for t days. Let St(Π) be expected survival
rate of type if winter is of length t.
Optimization
• Then the optimal mixed strategy will be the
one that maximizes the product
S1(Π) a1S2(Π) a2 … SN(Π) aN.
• Standard result of “branching theory.”
Application of law of large numbers. See
Robson, JET.
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—Vander Wall
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.
Casino Gambling
• Humans are able to run redistributional
lotteries. What does this do?
• This possibility separates diversification of
outcomes from diversification of
production strategies.
• If some activities have independent risks,
individuals can choose those that maximize
expected risks, but then gamble.
A Squirrel Casino
• Suppose squirrels can gamble nuts that they have
collected in fair lotteries.
• Let v(y) be probability that a squirrel who collects
y days supply of nuts is not eaten by predators.
• Expected nuts collected is yv(y).
• Optimal strategy for gene is to have its squirrels to
harvest y* where y* maximizes yv(y) and then
gamble.
Human Gamblers
• Humans are able to run redistributional
lotteries. What does this do?
• This possibility separates diversification of
outcomes from diversification of
production strategies.
• If some activities have independent risks,
individuals can choose those that maximize
expected risks, but then gamble.