Storage for Good Times and Bad: Of Squirrels and Men

1. Storage for Good Times and Bad:Of 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 costly—predation risk. • Squirrels don’t 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 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.

6. 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

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 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.

10. 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.

11. 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.

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 Πt of storing enough for t days. Let St(Π) 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(Π) a1S2(Π) a2… SN(Π) 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 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

16. 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.

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.