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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… The fable concerns squirrels, but has more ambitious intentions. What can evolution tell us about the evolution of our preferences toward risk?

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Storage for good times and bad of squirrels and men l.jpg

Storage for Good Times and Bad:Of Squirrels and Men

Ted Bergstrom, UCSB


A fable of food hoarding l.jpg
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


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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.


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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?


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Assumptions

  • 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.


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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


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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?


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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.


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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.


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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.


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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.


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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.


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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.


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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.


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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


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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.


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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.


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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.


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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.


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