Storage for Good Times and Bad: Of Squirrels and Men

1
Storage for Good Times and BadOf Squirrels and
Men

• Ted Bergstrom, UCSB

2
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

3
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.

4
A Simple Tale

• Squirrels must gather nuts to survive through
  winter.
• Gathering nuts is costlypredation risk.
• Squirrels dont know how long the winter will be.
• How do they decide how much to store?

5
Assumptions

• There are two kinds of winters, long and short.
• Climate is cyclical cycles of length kkSkL,
  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.

6
Survival probabilities

• A squirrel will survive and produce ? 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 vLltvS.
• If winter is long, Strategy S squirrel dies,
  Strategy L squirrel survives with prob vL

7
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?

8
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.

9
Random Strategy

• A randomizing gene tells its squirrel to use
  Strategy L with probability ?L and Strategy S
  with probability ?S.
• The reproduction rate of this gene will be
• SS(?) vS ?SvL ?L, if the winter is
  short.
• SL(?)vL ?L if the
  winter is long.

10
Optimal Random Strategy

• Expected number of offspring of a random
  strategist over the course of a single cycle is
• ?kSS(?) kSSL(?) kL
• Optimal strategy chooses probability vector
  ?(?L ,?S ) to maximize above.
• A gene that does this will reproduce more rapidly
  over each cycle and hence will eventually
  dominate the population.

11
Describing the optimum

• There is a mixed strategy solution if
• aLkL/klth.
• Mixed solution has ?L aL/h and
• SL/SS aL(1-h)/(1- aL)h.
• If aLgth, then the only solution is the pure
  strategy L.

12
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.

13
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 ?t of storing enough for t
  days. Let St(?) be expected survival rate of
  type if winter is of length t.

14
Optimization

• Then the optimal mixed strategy will be the one
  that maximizes the product S1(?)
  a1S2(?) a2 SN(?) aN.
• Standard result of branching theory.
  Application of law of large numbers. See Robson,
  JET.

15
Do Genes Really Randomize?

• Biologists discuss examples of phenotypic
  diversity despite common genetic heritage.
• Period of dormancy in seed plantsLevins
• Spadefoot toad tadpoles, carnivores vs vegans.
• Big variance in size of hoards collected by
  pikas, golden hamsters, red squirrels, and lab
  ratsVander Wall

16
Is Gambling Better Than Sex?

• Well, yes, this model says so.
• Alternative method of producing variationsexual
  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.

17
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.

18
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.

19
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.