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### 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 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

- 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
- 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 ΠLif 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

- 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.

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