Kin selection and Evolution of Sympathy Ted Bergstrom

Games between relatives • Symmetric two-player game – Payoff function M(x 1, x 2) • 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)=r. M(y, y)+(1 -r)M(y, x)

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)=r. M(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.

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

Alger-Weibull Theory: Transparent sympathies Alger and Weibull propose that 1) evolution acts on degrees of sympathy 2) Individuals know each other’s degree of sympathy 3) Outcomes are Nash equilibria for game with sympathetic preferences. 4) With sympathies, s 1, s 2, equilibrium strategies are x(s 1, s 2), x(s 2, s 1) 5) Selection is according to payoff V*(s 1, s 2)=V(x(s 1, s 2), x(s 2, s 1))

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.

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.

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.

Sympathy and joint production • With sympathy s, person 1’s utility function is U(x, y)=M(x, y)+s. M(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 d. X_2(s 1, s 2)/ds 1 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.