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… This fable is told with animal characters, but has more ambitious intentions. What can evolution tell us about the evolution of preferences toward risk? For the moral of the story, we look to the works of another fabulist… Arthur Robson

3. Preferences toward risk Robson (JET 1996) : Evolutionary theory predicts that: For idiosyncratic risks, animals 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. Probability distribution of winters is iid with probability p S of a short winter and p L =1-p S of a long winter. 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.

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 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

7. No Sex Please Reproduction is asexual (see Disney and Robson). Strategies are inherited from parent. Squirrels reproduce at rate ρ if they survive through the winter. If pure strategies are the only possibility, short-winter squirrels will be wiped out after first long winter. Eventually all squirrels use Strategy L. Does this meant that evolution will produce only long-winter squirrels?

8. Can Mother Nature Do Better? This seems inefficient if long winters are very rare. 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 π L and Strategy S with probability π S. The survival rate of squirrels carrying 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

10. Long run reproduction rate Since reproduction is multiplicative from year to year, if there are k short winters and T-k long winters over a period of T years, then the expected number of offspring of a π -strategist will be ρ T S S ( π ) k S L ( π ) T-k and the average annual growth rate will be ρ S S ( π ) k/T S L ( π ) (T-k)/T

11. Predicted Random Strategy By the law of large numbers, as T becomes large, k/T will almost certainly be very close to p S. Therefore the average annual growth rate of the population of π strategists will be close to ρ S S ( π ) pS S L ( π ) pL The evolutionary process would result in a population of π strategists that maximize this long growth rate.

12. Maximal-growth strategy The mixed strategy π =( π S, π L ) that emerges will be the one that has the highest long run average growth rate. Thus we seek π = ( π S, π L ) that maximizes p S log(S S ( π ))+p L log(S L ( π )) subject to the constraints that π S + π L = 1, π S ≥0, and π L ≥0.

13. The solution Recall that h is the hazard rate from collecting enough nuts for a long winter. If p L <h, the mixed strategy that maximizes long run growth is as follows:, π L = p L /h and π S =1- p L /h S S = p S v S and S L =p L v L /h S L /S S = (p L / 1 -p L )÷(h / 1 -h)< 1 If p L >h, long run growth is maximized by the pure strategy π L =1, π S =0, with S S =S L =v L

14. Some implications If long winters are rare enough, and food gathering hazardous enough, the most successful strategy is a mixed strategy. Probability matching. Probability of Strategy L is p L /h, proportional to probability of long winter. For populations with different distributions of winter length, but same feeding costs the die-off in harsh winters is inversely proportional to their frequencies.

15. Further Implication (Recall that short-winter-squirrels survive with probability ( 1 -p L )v S and long-winter- squirrels with probability ( 1 -h)v S.) If p L <h, then survival-probability- maximizing squirrels would prefer the short winter strategy. Evolution must somehow motivate some squirrels to take the long winter strategy, even though it gives them lower survival probability.

16. Ask the Psychiatrists Do some people systematically act against their own self-interest? ◦ Psychiatrists claim that about 5% of US population afflicted by “compulsive hoarding disorder.” ◦ They claim that .5% suffer from “pathological” gambling problems and 4.8% have “subclinical levels of gambling problems.”

17. An evolutionary explanation? Compulsive hoarding and compulsive gambling seem too prevalent to be simple biological malfunctions. Maybe in evolutionary past, compulsive hoarders made it through rare and extreme famines. Maybe compulsive gamblers who “got lucky” were the only ones to get through other tough times.

18. Evolutionary bet hedging. Evolutionary bet hedging. If this is the case, it may be that a gene that gives you some compulsive gamblers and some compulsive hoarders might have a higher long term growth rate than a gene that induces prudent, expected survival maximizing behavior in all of its carriers.

19. Some Twin study evidence Using standard twin-study methodology, researchers find risk preferences to be 2/3 genetically and 1/3 environmentally determined, but found “no effect of shared (observed) environmental factors” Another twin study finds 50% of variance in compulsive hoarding to be genetically determined and 50% environmentally determined.

20. Heredity vs environment? Maybe the differences that the twin studies attribute to differences in environment are not really due to significant differences in environment, but are evolutionary bet-hedging of the kind we have modeled.

21. Generalizations Model extends naturally to the case of many possible lengths of winter. Let p t be the probability that winter lasts for t days, for t=1,…W. Choose probabilities π t of storing enough for t days. Let v t =v 1 (1-h) t be probability of avoiding predators if you collect t days’ food. Let S t ( π ) be expected survival rate of type if winter is of length t.

22. A Result If the probability distribution of the length of the winter is log concave, all squirrels attempt to gather at least enough nuts to maximize private survival probability. Some will gather more-enough for each possible winters length. For long winters, survival rates will be proportional to product of frequency winter length and probability of surviving predation when storing for that length of winter.

23. 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 (both in the field and in laboratories)--Vanderwaal

24. 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.

25. Gambling and income distribution Suppose that instead of diversifying incomes by differential effort, a ``central authority’’ was able to redistribute nut holdings of squirrels in such a way as to maximize expected long run growth. This possibility separates diversification of outcomes from diversification of production strategies. Efficiency would have all of them seek to collect Y days’ food where Y maximizes expected collection v Y Y, and then they would receive consumption by lottery.

26. Casinos or theft Humans, although they lack bushy tails and the ability to scamper from tree to tree are able to organize redistribution by means of lotteries. Both humans and squirrels (especially humans) can also manage redistribution by theft.

27. Squirrel Redistribution?

28. A growth-maximizing distribution Suppose that the amount of nuts collected by each squirrel who survives predation is enough to last for Y days, where Y<W. If there are N surviving squirrels, let N t be the number of squirrels that are allocated t days’ worth of nuts. The set of feasible allocations consists of all allocations such that 1N 1 +2N 2 +…+tN t +…WN W =NY Equivalently, where π t =N t /N, this constraint is  t  π t =Y

29. The maximization problem If the winter is of length t, then the only survivors will be those who were allocated t or more nuts. Thus where π =( π 1 +…+ π W ) is the distribution of wealth in the population, the distribution of survival probabilities is S=(S 1 …S W ) with S t =  j≥t  j Then π t =S t -S t+1

30. Posing the problem The maximum growth rate is achieved by means of the distribution S of survival probabilities that maximizes  t=1 W  p t log(S t )  subject to the constraints that  t=1 W  S t =Y and 1≥S 1 ≥S 2 ≥S 3 …≥S W

31. About the Solution Suppose that the distribution of length of winter is unimodal with mode m. There is some number v of days food supply that minimizes food cost per unit of survival probability. Nobody will receive a positive amount of food less than v. If total amount of food is not enough to give everybody v, then some get 0 and rest get v or more.

32. More about the solution Where total amount of food is large enough, all get at least an amount r≥v and some get exactly enough food for each possible length of winter, with the fraction receiving exactly t>r days supply being proportional to the frequency of winters of length t.

33. A theory of Income distribution?? Could this maximal growth distribution tell us anything about income distributions that arise in human societies as a result of gambling, coercion, and theft?

34. That’s all, folks…