Evolutionary, Family Ties, and Incentives

Ted Bergstrom, UCSB

Evolutionary Foundations of Classic Family Dramas

Love and conflict

– Between siblings– Between mates– Between parents and offspring

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)

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.

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)

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.

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

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.

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.

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.

Alger-Weibull Theory: Transparent sympathies

Alger and Weibull propose that 1) evolution acts on degrees of sympathy2) Individuals know each other’s degree of

sympathy 3) Outcomes are Nash equilibria for game with

sympathetic preferences. 4) With sympathies, s1,s2, equilibrium strategies

are x(s1,s2), x(s2,s1)5) Selection is according to payoff V*(s1,s2)=V(x(s1,s2),x(s2,s1))

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.

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.

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.

Implication

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

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.

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)

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.

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.

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.

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.

Snakes in Eden

• Adultery

• Divorce

• Death and remarriage

• In-law problem

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).

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.

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.

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.

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)

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)