Evolutionary, Family Ties, and Incentives

1. Evolutionary, Family Ties, and Incentives Ted Bergstrom, UCSB

2. Evolutionary Foundations of Classic Family Dramas Love and conflict • Between siblings • Between mates • Between parents and offspring

3. Games between siblings • Symmetric two-player game • Payoff function M(x1,x2) • Degree of relatedness r. • r is the probability that if you are a mutant, your opponent is like you. • If normals use strategy x and mutant uses y, expected payoff to mutant is V(y,x)=rM(y,y)+(1-r)M(y,x)

4. Semi-Kantian approach • If nature forces you to play with people who act like you, then “it pays to be nice”. • With sexual reproduction, if you are a mutant, the probability is ½ that your sibling has same mutation. • In game with siblings, if normals do x and mutant uses y, expected payoff to mutant is V(y,x)=1/2 M(y,y)+1/2 M(y,x) • In games with cousins, r=1/8.

5. Equilibrium in strategies • Individuals hard-wired for strategies. • Reproduction rate determined by payoff in two player games • Strategy x is equilibrium if V(y,x)≤V(x,x) for all y. That is, if x is a symmetric Nash equilibrium for the game with payoff function V(y,x)=rM(y,y)+(1-r)M(y,x)

6. Reaction functions or utility functions? • For humans, set of possible strategies is enormous • Would have to encode response functions to others’ strategies • Beyond memory capacity • Preferences and utility functions an alternative object of selection. • Individuals would need notion of causality and ability to take actions to optimize on preferences.

7. Ethics or brotherly love? • Semi-Kantian utility functions • V(y,x)=rM(y,y)+(1-r)M(y,x) would be stable against mutant utilities. • How about love? • Biologist Wm Hamilton proposes “inclusive fitness utilities” H(x,y)=M(x,y)+sM(y,x) and claims that selection will result in s=r. • Hamilton’s rule: • Love thy kin r times as well as thyself

8. Can love do the trick? • Yes, if M is a concave function. • Then x is a symmetric Nash equilibrium for V if and only V1(y,x)=r M1(y,y)+rM2(y,y)+(1-r)M1(y,x) = M1(y,x)+ rM2(x,x) when y=x • The same first order condition makes x a symmetric Nash equilibrium for H with H1(y,x)=M1(y,x)+rM2(x,y) when y=x.

9. Equivalence • With concave payoff functions if there are inclusive fitness functions H(y,x)=M(y,x)+sM(x,y), the equilibrium sympathy levels under natural selection will be s=r. • If M is not a concave function this is not necessarily true.

10. An implicit assumption • We have assumed here that preferences are private information. • Alger and Weibull propose an alternative theory in which each player is aware of the other’s utility function.

11. Alger-Weibull Theory: Transparent sympathies Alger and Weibull propose that • evolution acts on degrees of sympathy • Individuals know each other’s degree of sympathy • Outcomes are Nash equilibria for game with sympathetic preferences. • With sympathies, s1,s2, equilibrium strategies are x(s1,s2), x(s2,s1) • Selection is according to payoff V*(s1,s2)=V(x(s1,s2),x(s2,s1))

12. Household production • Alger and Weibull suggest a household public goods model M(y,x)=F(y,x)-c(y) where F is a weakly concave symmetric production function and c(y) is the cost of exerting effort y. Assume c’’(y)>0.

13. Sympathy and joint production • With sympathy s, person 1’s utility function is U(x,y)=M(x,y)+sM(y,x) =(1+s)F(x,y)-c(y) • Equivalent to U*(x,y)=F(x,y)-c(y)/(1+s) • For this game, sympathy and low aversion to work are equivalent.

14. Results • Sign of cross partial dX2(s1,s2)/ds1 is same as that of cross partial of production function • If efforts of two workers are complements, then in equilibrium increased sympathy by one person increases equilibrium effort of the other. • If substitutes, then increased sympathy decreases equilibrium effort of the other.

15. Implication • If complementarity (substitutability) in production, then equilibrium sympathy level exceeds (is less than) coefficient of relatedness.

16. Conjugal Love: An Arboreal Allegory • Alice and Bob live on fruit and berries. • They get cold at night. • Alice is a skilled fire-builder. Bob is not.

17. Primitive cooperation • Alice divides her time between gathering food and building fire. • Bob doesn’t try building fires. He spends all of his time gathering food and he huddles next to Alice’s fire. • And wishes she would build a bigger fire. • Bob leaves some food by the fire for Alice. • He benefits because Alice makes a bigger fire. (income effect of food Bob leaves)

18. Too Little Fire • No love or altruism is involved. Both benefit from Bob’s gifts to Alice. • But there is still an undersupply of fire. • Alice accounts only for her own benefit when deciding how much fire to build. • A scheme where Bob pays Alice a food wage that depends on the size of fire would make both better off. • But this requires monitoring that may not be possible.

19. Case of common interests • Suppose that all that Alice and Bob really care about is the size of the fire. • They want food only because it gives them strength to do their work. • Then Alice and Bob have dominant strategies. • Bob eats enough to maximize the amount that he can give to Alice. • Alice eats enough to maximize the size of fire that she can build. • Both agree about what each should do. Outcome is efficient.

20. How are children like fire? • Suppose the household good is children, who share genes of two parents. • Evolutionary theory predicts selection for behavior that maximizes surviving descendants. • Consumption of goods not an end in itself, but an instrument for reproductive success.

21. Monogamy • Lifelong monogamous couples share identical reproductive goal. • Each is a perfectly motivated agent of the other’s reproductive success. • Common interest is the evolutionary foundation of conjugal love.

22. Snakes in Eden • Adultery • Divorce • Death and remarriage • In-law problem

23. Formal Model • Expected number of surviving children for Alice and Bob is Y=F(xA,xB) where xA and xB are resources devoted to childcare. • Let cA and cB be own consumption by Alice and Bob. • Utilities are UA (cA, Y) and UB(cB, Y).

24. Budgets • Where g is gifts from Bob to Alice, budgets are xB + cB +g=mB(cB) xA + cA=mA(cA)+g The functions mi(ci) reflect effect of own consumption on earnings capacity.

25. Harmonious interests • Suppose that • Alice and Bob care only about reproductive success • Their reproductive interests coincide • Then UA (cA, Y)=Y and UB(cB, Y)=Y where Y=F(xA,xB) =F(mA(cA)-cA-g, mB(cB)-cB+g). Both will want mi’(ci)=1 and F_1=F_2.

26. Partial conflict • Suppose that they care about their own consumptions as well as number of offspring: • Own consumption includes expected reproductive success with other partners. • e.g. UA (cA, Y)= cA (1-r)Yr and UB(cB, Y)=cB (1-r)Yr • In Nash noncooperative solution: • Less than Pareto efficient Y • Less specialization than is efficient.

27. Cooperative solution • Both parents could improve success by cooperation. • If monitoring possible, may achieve cooperation as Nash equilibrium of repeated game. • Efficient outcomes would maximize some family utility function of form wUA (cA, Y)+(1-w)UB(cB, Y)

28. Fairness or Love? • (Fairness) Maybe Bob and Alice agree that they will both try to maximize wUA (cA, Y)+(1-w)UB(cB, Y) • (Love) We could express Alice’s utility as UA (cA, Y)+((1-w)/w)UB(cB, Y) and Bob’s as UB (cB, Y)+(w/(1-w))UB(cB, Y)