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=k S +k L, with k S 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: v S for Strategy S and v L =(1-h)v S 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 v S and Strategy L with probability v L <v S. If winter is long, Strategy S squirrel dies, Strategy L squirrel survives with prob v L

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 Π L and Strategy S with probability Π S. The reproduction rate of this gene will be – S S (Π)= v S Π S +v L Π L, if the winter is short. – S L (Π)=v L Π L if the winter is long.

Optimal Random Strategy Expected number of offspring of a random strategist over the course of a single cycle is ρ k S S (Π) kS S L (Π) kL Optimal strategy chooses probability vector Π=(Π L, Π S ) 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 a L =k L /k<h. Mixed solution has Π L =a L /h and S L /S S = a L (1-h)/(1- a L )h. If a L >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 a L /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 a t Choose probabilities Π t of storing enough for t days. Let S t (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 S 1 (Π) a1 S 2 (Π) a2 … S N (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.