Kin selection and Evolution of Sympathy

1. Kin selection and Evolution of Sympathy Ted Bergstrom

2. Games between relatives • Symmetric two-player game • Payoff function M(x1,x2) • Degree of relatedness r. • 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)

3. 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 game with payoff function V(y,x)=rM(y,y)+(1-r)M(y,x)

4. 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.

5. What would utility functions be? • Could be the functions V(y,x)=rM(y,y)+(1-r)M • Could be sympathetic utilities: H(x,y)=M(x,y)+sM(y,x) Biologist William Hamilton’s calls this “inclusive fitness” and proposes s=r.

6. 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))

7. Alternative Theory:Opaque sympathies • Selection is for utility and sympathy, not strategies (as in Alger-Weibull theory). • Individuals cannot determine sympathies of others, can only observe actions. • Mutants act as if probability that their opponent is like them is r. • Normals almost never see mutants. They act as if opponent is sure to be a normal.

8. Equilibrium with opaque sympathies • If M(y,x) is a concave function in its two arguments then equilibrium sympathy levels between persons with degree of relatedness r is s=r. • This is Hamilton’s rule. • Proof First order conditions for symmetric Nash equilibrium are same for games with payoffs V and H. • Second order conditions satisfied if M concave.

9. Transparent sympathies • Alger Weibull 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.

10. 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.

11. Results • Sign of cross partial dX_2(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.

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