Preference evolution
Ingela Alger and Jörgen Weibull
May 8, 2014
1
Introduction
Preferences ) behaviors ) material payo¤ consequences ) evolutionary
selection pressure on preferences [indirect evolution, Güth and Yaari (1992)]
Question: how could preferences that di¤er from material payo¤ maximization survive?
Literature on preference evolution has so far shown that there are two
mechanisms whereby evolution by way of natural selection leads to nonsel…sh preferences
First mechanism: e¤ect of own preferences on others’behaviors [Schelling
(1960)]
– Inequity-averse responders do well in ultimatum bargaining
Preference evolution under complete information [Fershtman and Judd
(1987), Bester & Güth (1998), Bolle (2000), Possajennikov (2000), Koçkesen, Ok & Sethi (2000), Sethi & Somanathan (2001), Heifetz, Shannon
and Spiegel (2007)]: non-sel…sh preferences
Preference evolution under incomplete information [Ok & Vega-Redondo
(2001), Dekel, Ely & Yilankaya (2007)]: sel…sh preferences
Second mechanism: assortative matching
A long-standing tradition in biology [Hamilton (1964), Hines and Maynard
Smith (1979), Grafen (1979), Bergstrom (1995, 2003)]
Literature on preference evolution [Alger (2010), Alger and Weibull (2010,
2012, 2013)]
Result: preferences that induce non-sel…sh behaviors are selected for, and
sel…sh preferences are selected against
Assortativity is positive as soon as there is a positive probability that interacting parties have inherited their preferences or moral values from a
common “ancestor” (genetic or cultural)
In biology: genetics, kinship and “inclusive …tness” (Hamilton, 1964)
In social science: culture, education, ethnicity, geography, networks, customs and habits
Homophily [McPherson, Smith-Lovin, and Cook (2001), Ruef, Aldrich, and
Carter (2003), Currarini, Jackson, and Pin (2009, 2010), Bramoullé and
Rogers (2009)]
This morning:
1. Evolutionary stability of strategies in a population where individuals are
uniformly randomly matched into pairs to interact
2. Evolutionary stability of preferences (within the parametric class of altruistic
preferences) in a population where siblings interact in pairs; in sibling interactions there is assortativity : a mutant is more likely than a resident to interact
with a mutant
What’s next?
A general model of evolutionary stability of traits in a population where individuals are randomly (but perhaps assortatively ) matched into n-player groups
to interact + applications
2
The general model
A continuum population
Individuals are randomly (but not necessarily uniformly) matched into nplayer groups
Each group plays a symmetric game in material payo¤s
Material payo¤ from playing xi 2 X against x i 2 X n 1:
Normal form (material) game
= hX; ; ni
(xi; x i)
Each individual carries some heritable trait
behavior in the material game
2
which determines his/her
For our stability analysis we consider populations with at most two types
present, and , in arbitrary proportions 1 " and "
If " is small and positive,
trait
is called the resident trait and
the mutant
We study the type distribution’s robustness to small and rare random
shocks
The matching process is exogenous and random
For a given population state s = ( ; ; "):
– let Pr ( j ; ") be the probability that, for a given resident, another
group member (uniformly randomly drawn from the group) is a resident
– let Pr ( j ; ") be the probability that, for a given mutant, another
group member (uniformly randomly drawn from the group) is a resident
Let
(") = Pr [ j ; "]
Let lim"!0
Pr [ j ; "] and call
(") = , for some
– Uniform random matching )
the assortment function
2 [0; 1], the index of assortativity
=0
– Sibling interactions when types are inherited from parents )
– “Cultural parents” and homophily:
2 (0; 1)
= 1=2
Statistical issue for n > 2: potential conditional dependence (given the
type of the individual at hand, between pairs of other members)
We assume that conditional dependence vanishes in the limit as " ! 0
Thus, for a mutant, the type distribution among the other n
converges to Bin ( ; n 1) as " ! 0
1 players
Assume: an individual’s trait uniquely determines her average material
payo¤
Let F ( ; ; ") and G ( ; ; ") denote the average material payo¤ to an
individual with trait and trait , respectively
Assume: F ( ; ; ) and G ( ; ; ) are continuous
De…nition 1 A trait 2 is evolutionarily stable against a trait
there exists an " > 0 such that for all " 2 (0; " ):
2
if
F ( ; ; ") > G ( ; ; ") :
is an evolutionarily stable trait (EST) if it is evolutionarily stable against
all traits 6= in .
A su¢ cient condition for
2
to be an EST is that, for all
lim F ( ; ; ") > lim G ( ; ; ")
"!0
Let H :
2!
"!0
6= ,
(1)
R be the function de…ned by
H ( ; ) = lim G ( ; ; ")
"!0
H ( ; ) = lim"!0 G ( ; ; ") = lim"!0 F ( ; ; ") = lim"!0 F ( ; ; ") implies that (1) may be written:
H( ; )>H( ; )
Proposition For 2
to be an EST, ( ; ) must be a Nash equilibrium of
the two-player game in which the common strategy set is
and the payo¤
function is H . A su¢ cient condition is that ( ; ) is a strict Nash equilibrium
of this game.
2.1
Strategy evolution
An individual’s strategy depends only on his/her trait; formally, let
=X
If x is the resident strategy and y the mutant strategy,
G (x; y; ") =
n
X
n
m
m=1
1
1
!
[Pr (yjy; ")]m 1 [Pr (xjy; ")]n m
y; y(m 1); x(n m)
and
H (y; x) =
n
X
m=1
n
m
1
1
!
m 1 (1
)n m
y; y(m 1); x(n m)
For n = 2:
H (y; x) = (1
)
(y; x) +
(y; y )
For n = 3:
H (y; x) = (1
)2
(y; x; x) + 2
(1
)
(y; y; x) + 2
(y; y; y )
Proposition Suppose that
is continuously di¤erentiable and that X is an
open set. Then, if x
^ 2 X is an evolutionarily stable strategy,
^) +
1 (x
(n
1)
^)
n (x
= 0;
where x
^ is the n-dimensional vector whose components all equal x
^.
A canonical public-goods situation ( 2 (0; 1] and c > 0):
0
1
n
X
1
xj A
(xi; x i) = @
n j=1
H (y; x) =
n
X
m=1
n
m
m
y+ 1
n
1
1
!
c 2
xi
2
m 1 (1
m
x
n
)n m
c 2
y
2
H1 (y; x)jy=x = 0 is necessary and su¢ cient for x to be an ESS
Proposition The unique ESS is:
2
x
^=4
+
1 (1
n
c
)
3
5
1
2
2.2
Preference evolution under complete information
Each trait
uniquely determines a utility function u : X n ! R
2
Letting (n) ( ; ; m=n) be the equilibrium material payo¤ to a -individual
in a group with a share m=n of -individuals:
G ( ; ; ") =
n
X
n
m
1
1
!
[Pr ( j ; ")]m 1 [Pr ( j ; ")]n m
m=1
(n) ( ; ; m=n)
and
H( ; )=
n
X
m=1
n
m
1
1
!
m 1 (1
)n m (n) ( ; ; m=n)
2.2.1
Altruism
Trait: degree of altruism
Utility for an individual i with degree of altruism
u (xi; x i) =
(xi; x i) +
X
:
xj ; x j
j6=i
Set of potential traits:
Let
= [ 1; 1]
be the resident trait and
H( ; )=
n
X
m=1
n
m
1
1
!
the mutant trait:
m 1 (1
)n m (n) ( ; ; m=n)
The public goods example again:
u (xi; x i) =
(xi; x i) +
= [1 + (n
0
X
xj ; x j
j6=i
0
1
n
1) ] @ xi +
X
c @ 2
xi +
2
j6=i
1
1
1X A
xj
n j6=i
x2j A
If there are m -altruists and (n m) -altruists, a Nash eq. strategy pro…le is
a n-dimensional vector with m components equal to y and n m components
equal to x, where (x; y ) solves:
8
>
<
>
:
h
1
hn
1
n
+ 1
+ 1
1
n
1
n
i h
my +
i hn
my +
n
1
1
i
m x
n
i
m x
n
1
1
cx = 0
cy = 0
Proposition The unique locally evolutionarily stable degree of altruism is
^=
1+ 1
1 (1
n
1
n
) (1
(1
) (1
)
)
2.3
Preference evolution under incomplete information
Each trait
uniquely determines a utility function u : X n ! R
2
De…nition 2 In any state s = ( ; ; ") 2 S , the (assumed unique) ( Bayesian)
Nash Equilibrium is the strategy pair (x ; y ) 2 X 2 satisfying
(
x 2 arg maxx2X U
y 2 arg maxy2X U
where
U =
nX1
n
n
X
n
m=0
U =
m=1
1
m
!
[Pr ( j ; ")]n m 1 [Pr ( j ; ")]m u
x; y (m); x (n m 1)
1
m
!
[Pr ( j ; ")]n m [Pr ( j ; ")]m 1 u
y; y (m 1); x (n m)
Given s = ( ; ; "), let x("); y(") denote the unique BNE. Then:
G ( ; ; ") =
n
X
n
m
m=1
1
1
H( ; )=
m=1
n
m
[Pr ( j ; ")]m 1 [Pr ( j ; ")]n m
(m 1)
y("); y (")
n
X
!
1
1
!
(n m)
; x (")
m 1 (1
)n m
(m 1)
y(0); y (0)
(n m)
; x (0)
Let
X denote the the best-reply correspondence,
:X
(y ) = arg max u
x2X
and X
x; y(n 1)
X the set of …xed points under
X = fx 2 X : x 2
Let
8y 2 X
,
(x)g
be the set of behavioral clones:
=
n
2
: 9 x 2 X such that
o
N
E
(x; x) 2 B
( ; ; 0)
Theorem Suppose the behavior of homo moralis, in the absence of mutants,
is uniquely determined. Then:
(a) Homo moralis with degree of morality
types that are not its behavioral clones.
is evolutionarily stable against all
(b) All types that are not its behavioral clones are evolutionarily unstable if the
type set is rich.
So, what, exactly, is a homo moralis?
For each x 2 X n and 2 [0; 1], and any player i, let x
~ i be a random
vector with statistically independent components x
~j (j =
6 i) where
h
i
Pr x
~ j = xi =
h
i
and Pr x
~j = xj = 1
8j
De…nition 3 A homo moralis is an individual with utility function
u (xi; x i) = E [ (xi; x
~ i)]
for some
8x 2 X n:
2 [0; 1], the individual’s degree of morality.
For 0 < < 1, the individual’s goal is to choose a strategy xi that, if used
with probability by other players, would maximize her material payo¤.
(How would it be if, with probability , each individual would do what I
do?)
For n = 2:
u (x; y ) = (1
)
(x; y ) +
(x; x)
For n = 3:
u (x; y; z ) = (1
+
)2
(1
(x; y; z ) +
)
(1
(x; y; x) + 2
)
(x; x; z )
(x; x; x)
Corollary Any BNE strategy x in a monomorphic population of homo moralis
with = is also an ESS. Moreover, if a strategy is a ESS for some , it is
also a BNE strategy in a monomorphic population of homo moralis with degree
of morality = .
Evolutionarily stable strategies may be viewed as emerging from preference evolution when individuals are not programmed to strategies but are
rational and play equilibria under incomplete information.
3
Implications
Applications to
– environmental economics
– moral hazard, principal-agent relations
(Alger and Ma (2003), Alger and Renault (2006,2007)
– bargaining
– participation and voting in elections
4
Conclusions
Our analysis suggests that sel…shness is evolutionarily stable only in special
circumstances, while homo moralis with degree of morality equal to the
index of assortativity is always evolutionarily stable.
Moral preferences may thrive, even under incomplete information and even
in very large groups
Lots of new challenges: extensions, applications, tests in laboratory experiments...
THE END