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Storage for Good Times and Bad: Of Squirrels and Men Ted Bergstrom, UCSB A fable of food-hoarding, As in Ae sop and Walt Disney… Sometimes it is easier to understand human foibles by observing them in animals. The fable concerns squirrels, but has more ambitious intentions.

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a fable of food hoarding
A fable of food-hoarding,
  • As in Aesop and Walt Disney…
  • Sometimes it is easier to understand human foibles by observing them in animals.
  • The fable concerns squirrels, but has more ambitious intentions.
  • What can evolution tell us about the evolution of our preferences toward risk?
preferences toward risk
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
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?
  • There are two kinds of winters, long and short.
  • Climate is cyclical; cycles of length k=kS+kL, with kSshort and k Llong 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
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
reproduction of strategies
Reproduction of strategies
  • 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
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
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 ΠLif the winter is long.
optimal random strategy
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
Describing the optimum
  • There is a mixed strategy solution if


  • 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
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.
  • 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.
nature vs nurture controversy
Nature vs Nurture controversy
  • Usual assumption: Common genetics and different behavior implies different nurture.
  • Maybe not.
  • Diversity may be genetically mandated.
  • 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
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
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
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
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.