Download Kin selection and Evolution of Sympathy

Kin selection and Evolution of
Sympathy
Ted Bergstrom
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)
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)+(1r)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)=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.
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, 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))
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)+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 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.
Implication
• If complementarity (substitutability) in
production, then equilibrium sympathy
level exceeds (is less than) coefficient
of relatedness.