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1.
Evolutionary Stabilities in
Games with Continuous Strategy Spaces:
Characterizations with Examples
Woojin Lee
Department of Economics
University of Massachusetts
Amherst, MA 01002
1
2.
Evolutionary game theory
– EGT is a growing business.
-- It inspired some discussion about social norms and conventions.
Binmore (1994, 1998); Bowles (2004); Sethi and Somanathan (2001);
Young (1998)
-- But it covers a wide variety of different models.
infinite versus finite population games
Eshel (1996) discusses 18 different terms of evolutionary stability.
what is common: the idea of stability of a strategy (or a strategy profile) against
small perturbations
what is different: all other details
2
3.
Evolutionary game theory
-- EGT studies mainly symmetric uni-population games with discrete strategy spaces.
Gintis (2000); Hammerstein and Selten (1994); Hofbauer and Sigmund (1998);
Samuelson (1997); Weibull (1995)
-- Bi-population games or games with continuous strategy spaces are much harder to
analyze, mainly because dynamic analysis is very difficult with a general setting.
-- But a distinguishing feature of evolution is that the change of aggregate behavior is
gradual (continuous) rather than abrupt due, perhaps, to some inertia, adjustment
costs, informational imperfections or bounded rationality. cf. punctuated equilibrium
-- Often the obtained results from general analysis do not replicate what are obtained
from symmetric uni-population games or games with discrete strategy spaces.
3
4.
Evolutionary stabilities
-- A fundamental axiom of EGT: any consensus strategy (or a strategy profile)
established in the population is never fully fixed, and is always subject to erratic,
small deviations from it; such small deviations can occur in at least two ways.
(1) they may occur due to some substantial change on the part of a certain
‘aberrant’ minority (a mutant in biology and an experiment in social science); or
(2) they may reflect small continuous shifts of the entire population from the
consensus strategy.
-- Correspondingly, there are at least two senses in which a strategy (or a strategy
profile) can be said to be evolutionarily stable.
(1) it must be resistant to invasion by a rare mutant strategy: uninvadability;
(2) a slightly shifted population from an evolutionarily stable state must actually
be able to evolve back to it: dynamic attainability.
4
5.
Evolutionary stabilities
-- The first sense of evolutionary stability does not necessarily imply the second sense
of stability.
-- An x * -population being resistant to invasion in the first sense would be immune to
a mutant, but a shifted x -population living near x * might not be immune to the same
mutant without the tendency of recovery in the second sense; in that case, natural or
social selection might amplify any small deviation of the entire population from the
original state.
-- In the standard mixed extension of the games with discrete strategy spaces, payoff
functions are bi-affine and the condition of resistance to invasion implies the condition
of actual recovery from small perturbations and vice versa, but this is not true in
general non-linear games with continuous strategy spaces.
5
6.
Two different proposals for the dynamic attainability in uni-pop games
-- Local m-stability (or convergence stability): small perturbations of the entire
population from an uninvadable strategy should end up with selective advantage to
strategies that render the population back to it but not to strategies towards the other
side.
Eshel (1983)’s continuously stable strategy (CSS) is an uninvadable strategy that is
locally m-stable.
-- Local superiority (or a neighborhood invader strategy): an uninvadable strategy
should be able to invade all established communities populated by players using
strategies that are sufficiently similar to it.
Apaloo (1997)’s evolutionarily stable neighborhood invader strategy (ESNIS) is
an uninvadable strategy that is locally superior.
6
7.
Two different proposals for the dynamic attainability in uni-pop games
-- These two conditions are different not only from the condition of resistance to
invasion but also from each other, although they are all identical if payoff functions are
bi-affine.
-- These proposals were originally made in the context of symmetric uni-population
games with one-dimensional strategy spaces. We extend these proposals into a
general setting, covering bi-population games with multi-dimensional strategy spaces,
and provide various characterizations for these conditions with examples.
-- We cover games with continuous pure strategy spaces, and as special cases mixed
extensions of discrete strategy games, in which the payoff functions are bi-affine and
the strategy spaces are k-dimensional simplexes. But we do not study mixed
extensions of continuous pure strategy games.
7
8.
Symmetric uni-pop. games with continuous strategy spaces
-- Two individuals are repeatedly drawn at random from an infinitely large single
population to play a symmetric two person game.
-- They do not know whether they are player 1 or 2 and thus cannot condition their
strategies on this information.
-- They share the same strategy space and the payoff function.
p (x ',x ) is the payoff of a player who chooses x ' Î S against an opponent who
chooses x Î S .
Strategies take values on a convex subset of the k-dimensional real Euclidean
space.
8
9.
Symmetric uni-pop. games with continuous strategy spaces
-- Suppose that all individuals are initially programmed to play a certain consensus
strategy (say, x ), and then we inject into this x -population a small population share
of individuals who are likewise programmed to play some other strategy (say, x ' ).
-- Then j (x ',x ;x ) = p (x ',x ) - p (x ,x ) is the net payoff gain of a player playing the minority
strategy x ' relative to a player playing the consensus strategy x . We say:
(1) x ' strongly invades x -population if j (x ',x ;x ) > 0.
(2) x ' weakly invades x -population if j (x ',x ;x ) ³ 0,
(3) no x ' can weakly invade x -population if j (x ',x ;x ) < 0
(4) no x ' can strongly invade x -population. if j (x ',x ;x ) £ 0
9
10.
Symmetric uni-pop. games with continuous strategy spaces
Definition 1: Consider a uni-population game with p : S ´ S ® R .
(1) x * Î S is a Nash equilibrium strategy (NES) if for all x Î S
j (x ,x * ;x * ) º p (x ,x * ) - p (x * ,x * ) £ 0.
We call it a strict NES if strict inequality holds for all x Î S {x * } .
(2) x * Î S is a locally superior strategy if for all x ¹ x * sufficiently close to x * ,
j (x * ,x ;x ) º p (x * ,x ) - p (x,x ) > 0.
(3) x * Î S is a locally m-stable strategy if for all x ¹ x * sufficiently close to x * ,
p ((1 - l )x + l x * , x ) - p (x , x ) > 0 for small l > 0; and
p ((1 - l )x + l x * , x ) - p (x , x ) < 0 for small l < 0 as long as (1 - l )x + l x * Î S .
10
11.
Local superiority and m-stability
(1) Superiority requires that x * strictly minimize j (x * ,x ;x ) º p (x * ,x ) - p (x,x ) .
(2) Superiority is different from Pareto optimality, which requires that p (x * ,x * ) > p (x ,x )
for all x Î S {x * } .
(3) Superiority is weaker than dominance, which requires that p (x * ,z ) > p (x ,z ) for all
x Î S {x * } and z Î S .
(4) Superiority is weaker than the following condition: p (x * ,x ') > p (x ,x ') for all
(x ,x ') ¹ (x * ,x * ) .
(5) M-stability requires that x * strictly minimize:
l j ((1 - l )x + l x * ,x ;x ) º l (p ((1 - l )x + l x * ,x ) - p (x ,x ))
for small l Î R } .
{0
11
12.
Figure 1: Superiority and m-stability
superiority: yes superiority: no
m-stability: yes
x’
x’
x’=x x’=x
(-)
(+) (-) (+)
φ(x’,x;x)=0
x* (+) *
x
(-) (-)
φ(x’,x;x)=0
(+)
x x
m-stability: no x’ x’
x’=x
x’=x
(+) (-) (+)
φ(x’,x;x)=0 (-)
*
x *
x
(+) (-)
(-) (+)
φ(x’,x;x)=0
x
x
LINK Word.Document.8 "E:Woojin5evolutionThe concept of evolutionary stable
strategies.doc" OLE_LINK2 a r
Note: Consider the set of points (x ', x ) at which j (x ', x ;x ) = 0. Since j (x ', x ;x ) = 0 for
x ' = x , the 45 degree line is included in this set. For x * to be an equilibrium point, there
must be other curves in this set which cross the 45 degree line at x * . Here we suppose that
the set of such points, together with the 45 degree line, divides the space around (x * , x * ) into
four regions. (This may not be true in general.) For x * to be superior, the horizontal line at
1
13.
'
x * should be in the positive region. For x * to be m-stable, the sign of j (x , x ;x ) must be as
shown in the two diagrams of the first row.
2
14.
Figure 2: Various equilibrium concepts
Global m- Local m-
stability stability
If π is affine in the first argument
If π is convex in If π is concave in If π is convex in If π is concave in
the first argument the first argument the first argument the first argument
Global Local
superiority superiority
If π is affine in the first argument
If π is concave in
the first argument If π is strictly
quasi-convex in the
first argument
ESS-II
If π is continuous in
the second argument
If π is affine in the
second argument
ESS-I
Strict Nash (Univadabili
If π is strictly concave ty)
in the first argument
If π is strictly concave
If π is strictly concave in the first argument
in the first argument Nash
3
15.
Table 1: Sufficient conditions in symmetric uni-
population games
Concept Sufficient conditions for an interior solution
Local strictness Ñ 1p (x * ,x * ) = 0; and
Ñ 11p (x * ,x * ) is negative definite.
Local superiority Ñ 1p (x * ,x * ) = 0; and
Ñ 11p (x * , x * ) + 2 12p (x * , x * ) is negative definite.
Ñ
Local m-stability Ñ 1p (x * ,x * ) = 0; and
(Ñ 11p (x ,x ) + Ñ 12p (x ,x ) - l
2 Ñ 11p (x ,x )) x =x * is negative definite
for small l Î R {0 .
}
Local * *
Ñ 1p (x ,x ) = 0; and
uninvadability Ñ 11p (x * ,x * ) + 2 Ñ 12p (x * , x * ) is negative definite for sufficiently
e
small ε > 0.
Local ESNIS Ñ 1p (x * , x * ) = 0;
Ñ 11p (x * , x * ) is negative definite; and
Ñ 11p (x * ,x * ) + 2 12p (x * ,x * ) is negative definite.
Ñ
Local CSS Ñ 1p (x * , x * ) = 0;
Ñ 11p (x * , x * ) is negative definite; and
(Ñ 11p (x ,x ) + Ñ 12p (x ,x ) - l
2 Ñ 11p (x ,x )) x =x * is negative definite
for sufficiently small l Î R {0 .
}
4
16.
Evolutionary stabilities in uni-pop. games
Definition 2: Consider a uni-population game in which the payoff function is given by
p :S ´ S ® R .
(1) x * Î S is an uninvadable strategy (UIS) if it is a Nash equilibrium strategy for
which p (x , x * ) = p (x * , x * ) implies p(x * ,x ) > p (x ,x ) for all x ∈ S {x * } .
(2) An evolutionarily stable neighborhood invader strategy (ESNIS) is a strict
Nash equilibrium strategy that is locally superior.
(3) A continuously stable strategy (CSS) is a strict Nash equilibrium strategy that is
locally m-stable.
1
17.
Evolutionary stabilities in uni-pop. games
(1) An uninvadable strategy is what Maynard Smith and Price (1973) call an
evolutionarily stable strategy (ESS).
(2) An UIS is a stronger than a NES and weaker than a strict NES.
(3) An UIS is undominated; there is no x Î S {x * } that weakly dominates x * .
(4) An UIS is identical to the following definition:
ESS-I: x * Î S is an uninvadable strategy (UIS) if for all x Î S {x * } and sufficiently small
ε > 0 , (1 - e)p (x , x * ) + ep (x , x ) < (1 - e)p (x * , x * ) + ep (x * , x ) .
(5) An UIS is equivalent to the following definition only when p is affine in the second
argument.
ESS-II: x* Î S is an evolutionarily stable strategy if for all x Î S {x * } and sufficiently
small e > 0, p (x * , ex + (1 - e)x * ) > p (x, ex + (1 - e)x * ) .
(7) An interior strict Nash equilibrium is a CSS if the slope of the best reply curve at x *
is less than 1 and an ESNIS if it is less than 1/2.
2
18.
Example 1: Cournot duopoly game
Cournot firms, which face a linear inverse market demand function:
P (X ) = 2 - X , where X = x + y .
x
We assume a convex and increasing cost function c (x ) , where c '(x ) ³ 0, c ''(x ) ³ 0, and
0< c '(0 < 2 .
) x
The profit of each firm is p (x,y ) = x (2 - x - y ) - c (x ) and thus
x
¶p
= 2 - 2 - y - c '(x ) .
x x
¶x
The unique pure strategy Nash equilibrium of this game is (x * ,x * ) , where x * is defined
* ' * ¶ 2p ¶ 2p
by 2 - 3 = c (x ) . Because
x x = - 2- c (x ) < 0,
''
¶x¶y
= - 1 < 0, the interior Nash
¶x 2
equilibrium strategy is strict, uninvadable, superior, and m-stable.
1
Note that b '(y ) = - ; the slope of the best reply curve is less than 1/2 for all x .
2+ c ''(x )
3
19.
Example 2: Hotelling’s price competition model
Consider two stores separately located in the two end points of a unit interval.
There is a continuum of consumers distributed by a distribution function F in the unit
interval. The median of F will be denoted by m; F (m ) = 2 .
1
Each consumer has a unit demand and incurs travel cost t per unit of distance from
their location.
æ1
Then p (p,q) = (p - c )F ç +
ç ÷and ¶ p = F æ + q - p ö- (p - c ) f æ + q - p ö.
q - pö
÷ ç
ç2
1 ÷
÷ ç
ç2
1 ÷
÷
è
2 2t ø ¶p è 2t ø 2t è 2 ø
t
If the payoff function is strictly concave, the unique pure strategy Nash equilibrium
*
strategy is p = c + 2m
f (0.5) t , which is in the interior.
2
p * is a locally strict NES (or a locally uninvadable strategy) if mf '(0 ) < 2[ f (0 )] .
.5 .5
¶ 2p (p,q) ¶ 2p (p,q) f (0 )
.5
*
p is locally m-stable is because ( 2
+ ) =- < 0.
¶p ¶ p ¶ q (p ,q )=(p * ,p * ) 2t
p * is locally superior only when f '(0 ) > 0.
.5
4
20.
Example 3: Bryant’s coordination game
This is a game possessing infinitely many strict Nash equilibria (and thus infinitely
many uninvadable strategies) but none of them survive the test of superiority or m-
stability.
Consider a two player game in which each person chooses an effort level e Î [0e ]
,
simultaneously.
The payoff of an individual choosing e and facing an opponent choosing f is given by
p (e, f ) = a m e, f ]- be where a > b > 0.
in[
This game possesses a continuum of pure strategy Nash equilibria; any (e * ,e * ) where
e * Î [0 e ] is a Nash equilibrium.
,
Because p (e * ,e * ) = (a - b )e * > p (e,e * ) = a min[e,e * ]- be for all e ¹ e * , all of these Nash
equilibria are strict and thus uninvadable.
But none of them are superior, because p (e * ,e ) = a min[e * ,e ]- be * < p (e,e ) = (a - b )e for all
e ¹ e * . Also none of the Nash equilibria are m-stable.
5
21.
Bi-pop. games with continuous strategy spaces
-- There are two groups: group 1 and group 2.
-- When there are two populations, random matching can generally take two forms:
(i) a matching with a fellow player in the same group; and
(ii) a matching with a player in the other group.
'
-- Suppose p i (x i ;x i ,x j ) is the encompassing payoff of a player in group i = 1,2
choosing x i' Î S i when she faces a fellow player in the same group playing x i Î S i and
another player of group j ¹ i choosing x j Î S j .
-- The net payoff gain of a player in group i choosing strategy x i' compared with her
playing x i when the entire population is settled with y = (y 1,y 2) is
j i (x i',x i ;y ) º p i (x i';y ) - pi (x i ;y ) .
6
22.
Bi-pop. games with continuous strategy spaces
Let x = (x 1,x 2) and x ' = (x 1',x 2) . x ' is said to
'
(1) most strongly invade x -population if (j 1(x 1',x 1;x ),j 2(x 2,x 2;x )) ? (0 ) .
'
,0
(2) very strongly invade x -population if (j 1(x1',x 1;x ),j 2(x 2,x 2;x )) > (0 ) .
'
,0
(3) strongly invade x -population if j 1(x 1',x 1;x ) > 0 or j 2(x 2,x 2;x ) > 0
'
(4) weakly invade x -population if j 1(x 1',x 1;x ) ³ 0 or j 2(x 2,x 2;x ) ³ 0.
'
(5) If (j 1(x 1',x 1;x ),j 2(x 2,x 2;x )) £ (0 ) , we say no x ' can strongly invade x -population.
'
,0
(6) If (j 1(x1',x 1;x ),j 2(x 2,x 2;x )) = (0 ) , we say no x ' can weakly invade x -population.
'
,0
7
23.
Bi-pop. games with continuous strategy spaces
Definition 3:
(1) (x1* ,x 2) Î S 1 ´ S 2 is a Nash equilibrium of the bi-population game if
*
(j 1(x 1,x 1* ;x* ),j 2(x 2,x 2;x* )) £ (0 ) (i.e., (p1(x1 ;x* ), p 2(x 2;x* )) £ (p1(x1* ;x* ), p 2(x 2;x* )) ) for all
*
,0 *
(x 1,x 2) Î S 1 ´ S 2.
(2) (x 1* ,x 2) Î S 1 ´ S 2 is a locally superior equilibrium if there is no (x 1,x 2) ¹ (x 1* ,x 2)
* *
in the neighborhood of (x1* ,x 2) such that (j 1(x 1* ,x 1;x ),j 2(x 2,x 2;x )) £ (0 )
* *
,0
Û (p 1(x 1 ;x ), p 2(x 2;x )) £ (p 1(x 1;x ), p 2(x 2;x )) .
* *
(3) * *
(x 1 ,x 2) Î S 1 ´ S 2 is a locally m-stable equilibrium if
(i) for small (l 1,l 2) ? (0 ) , there is no (x 1,x 2) ¹ (x 1* ,x 2) in the neighborhood of (x 1* ,x 2)
,0 * *
such that (j 1((1 - l 1 )x1 + l 1x 1* ,x 1;x ),j 2((1 - l 2)x 2 + l 2x 2,x 2;x )) £ (0 ) ; and
*
,0
(ii) for small (l 1,l 2) = (0 ) , there is no (x 1,x 2) ¹ (x 1* ,x 2) in the neighborhood of
,0 * * *
(x 1 ,x 2)
such that (j 1((1 - l 1 )x 1 + l 1x 1* ,x 1;x ),j 2((1 - l 2)x 2 + l 2x 2,x 2;x )) ³ (0 ) for
*
,0 (1 - l i )x i + l i x i* Î S i .
8
24.
Superior equilibrium
(1) (x 1* ,x 2) Î S 1 ´ S 2 is a superior equilibrium if and only if either p1(x 1* ;x ) > p1(x 1;x ) or
*
p 2(x 2;x ) > p 2(x 2;x ) hold for all (x 1,x 2) ¹ (x 1 ,x 2) . Thus it requires that x * strongly invade x -
* * *
population. This is weaker than the following : p1(x 1* ;x ) > p1(x 1;x ) and p 2(x 2;x ) > p 2(x 2;x )
*
hold for all (x 1,x 2) ¹ (x 1* ,x 2) .
*
(2) Thus (x 1* ,x 2) Î S 1 ´ S 2 is a superior equilibrium if and only if
*
p i (x i ;x ) ³ p i (x i* ;x ) Þ p j (x j ;x ) < p j (x j* ;x ) ,
or equivalently, j i (x i , x i ;x ) £ 0 Þ j j (x j , x j ;x ) > 0. This condition resembles strong Pareto
* *
optimality, but is different from it. Strong Pareto optimality requires that we have:
p 2(x 2;x ) ³ p 2(x 2;x* ) Þ p 1(x 1;x ) < p 1(x 1 ;x* ) .
* *
(3) A sufficient condition for a locally superior equilibrium is that * *
(x 1 ,x 2) is a strict local
minimum of the following problem:
m j 1(x 1 ,x 1;x ) subject to j 2(x 2,x 2;x ) £ 0.
in * *
9
25.
(4) (x 1* ,x 2) Î S 1 ´ S 2 is an m-stable equilibrium if and only if for all (x 1,x 2) ¹ (x 1* ,x 2) in the
* *
neighborhood of (x 1* ,x 2) the following is true:
*
(i) for small (l 1,l 2) ? (0 ) , either j 1((1 - l 1 )x 1 + l 1x 1* ,x 1;x ) > 0 or
,0
j 2((1 - l 2)x 2 + l 2x 2,x 2;x ) > 0 hold; and
*
(ii) for small (l 1,l 2) = (0 ) , either j 1((1 - l 1 )x 1 + l 1x 1* ,x 1;x ) < 0 or
,0
j 2((1 - l 2)x 2 + l 2x 2,x 2;x ) < 0 hold as long as (1 - l i )x i + l i x i* Î S i .
*
(5) Thus * *
(x 1 ,x 2) Î S 1 ´ S 2 is an m-stable equilibrium if and only if
p i (x i ;x ) ³ p i ((1 - l i )x i + l i x i* ;x ) Þ p j (x j ;x ) < p j ((1 - l j )x j + l j x j* ;x ) fo (l i ,l j ) ? 0
r ,
and
p i (x i ;x ) £ p i ((1 - l i )x i + l i x i* ;x ) Þ p j (x j ;x ) > p j ((1 - l j )x j + l j x j* ;x ) fo (l i ,l j ) = 0
r .
(6) A sufficient condition for an m-stable equilibrium is that * *
(x 1 ,x 2) is a strict local
minimum of the following problem:
m j 1((1 - l 1 )x 1 + l 1x 1 ,x 1;x )
in *
s.t j 2((1 - l 2)x 2 + l 2x 2,x 2;x ) £ 0 for small (l 1,l 2) ? 0,
*
m j 1(x 1,(1 - l 1 )x 1 + l 1x 1 ;x )
in *
s.t j 2(x 2,(1 - l 2)x 2 + l 2x 2;x ) £ 0 for small (l 1,l 2) = 0.
*
10
26.
Table 3: Sufficient conditions in bi-pop. games
Concept Sufficient conditions for an interior solution
Local Ñ 1p 1(x 1 ;x* ) = Ñ 1p 2(x 2;x* ) = 0; and
* *
strictness Ñ 11p 1(x 1 ;x* ) and Ñ 11p 2(x 2;x* ) are negative definite.
* *
Local Ñ 1p 1(x 1 ;x* ) = Ñ 1p 2(x 2;x* ) = 0; and
* *
superiority é 11p 1(x 1 ;x * ) + 2 12p 1(x 1 ;x * ) Ñ 13p 1(x 1 ;x * ) + Ñ 12p 2(x 2;x * ) ù
* * * *
Ñ Ñ
ê ú
ê * ú
is negative definite.
êÑ 13p 1(x 1 ;x ) + Ñ 12p 2(x 2;x ) Ñ 11p 2(x 2;x ) + 2 13p 2(x 2;x )ú
* * * * * *
Ñ *
ë û
Local m- Ñ 1p 1(x 1 ;x ) = Ñ 1p 2(x 2;x ) = 0; and
* * * *
stability é 1((2- l 1 )Ñ 11p 1(x 1* ;x * ) + 2 12p 1(x 1* ;x * ))
l
ê Ñ l 1Ñ 13p1(x 1* ;x * ) + l 2Ñ 12p 2(x 2;x * ) ù
*
ú
ê * ú
is
ê l 1Ñ 13p 1(x 1 ;x ) + l 2Ñ 12p 2(x 2;x )
* * * *
l 2((2- l 2)Ñ 11p 2(x 2;x ) + 2 13p 2(x 2;x ))ú
* *
Ñ *
ë û
negative definite for small (l 1,l 2) ? 0 and positive definite for small (l 1,l 2) = 0.
Local Ñ 1p 1(x 1 ;x* ) = Ñ 1p 2(x 2;x* ) = 0; and
* *
uninvadability éÑ 11p 1(x 1 ;x * ) + 2 1Ñ 12p 1(x 1 ;x * ) e1Ñ 13p 1(x 1 ;x * ) + e2Ñ 12p 2(x 2;x * )ù
* * * *
ê e ú
ê * ú
is negative
ê 1Ñ 13p 1(x 1 ;x ) + e2Ñ 12p 2(x 2;x ) Ñ 11p 2(x 2;x ) + 2 2Ñ 13p 2(x 2;x )ú
e * * * * * *
e *
ë û
definite for small (e1, e2) ? 0.
Local ESNIE Ñ 1p 1(x 1 ;x* ) = Ñ 1p 2(x 2;x* ) = 0;
* *
Ñ 11p 1(x 1 ;x* ) and Ñ 11p 2(x 2;x* ) are negative definite; and
* *
é 11p 1(x 1 ;x * ) + 2 12p 1(x 1 ;x * ) Ñ 13p 1(x 1 ;x * ) + Ñ 12p 2(x 2;x * ) ù
Ñ *
Ñ * * *
ê ú
ê * ú
is negative definite.
êÑ 13p 1(x 1 ;x ) + Ñ 12p 2(x 2;x ) Ñ 11p 2(x 2;x ) + 2 13p 2(x 2;x )ú
* * * * * *
Ñ *
ë û
Local CSE Ñ 1p 1(x 1 ;x ) = Ñ 1p 2(x 2;x ) = 0;
* * * *
Ñ 11p 1(x 1 ;x* ) and Ñ 11p 2(x 2;x* ) are negative definite; and
* *
é l 1((2- l 1 )Ñ 11p 1(x 1* ;x * ) + Ñ 12p 1(x 1* ;x * ))
2 l 1Ñ 13p 1(x1* ;x * ) + l 2Ñ 12p 2(x 2;x * )
* ù
ê ú
ê * ú
is
ê l 1Ñ 13p 1(x 1 ;x ) + l 2Ñ 12p 2(x 2;x )
* * * *
2 2((2- l 2)Ñ 11p 2(x 2;x ) + Ñ 13p 2(x 2;x ))ú
l * * *
ë û
negative definite for small (l 1,l 2) ? 0 and positive definite for small (l 1,l 2) = 0.
1
27.
Definition 4: Consider a bi-population game with the encompassing payoff
function of p i : S i ´ S i ´ S j ® R for each i = 1,2.
(1) (x 1* ,x 2) Î S 1 ´ S 2 is an uninvadable equilibrium of the bi-population game if it
*
is a Nash equilibrium for which there is no (x 1,x 2) ¹ (x 1* ,x 2) such that
*
(j 1(x 1,x 1 ;x* ),j 2(x 2,x 2;x* )) = (0 ) and (j 1(x 1 ,x 1;x ),j 2(x 2,x 2;x )) £ (0 ) .
* *
,0 * *
,0
(2) An evolutionarily stable neighborhood invader equilibrium (ESNIE) is a
strict Nash equilibrium that is locally superior.
(3) A continuously stable equilibrium (CSE) is a strict Nash equilibrium that is
locally m-stable.
1
28.
Theorem 7: x * = (x 1* ,x 2) Î S 1 ´ S 2 is an uninvadable equilibrium of the bi-population
*
game if there is no x = (x 1,x 2) Î S 1 ´ S 2 {(x 1* ,x 2)} such that
*
æe1p 1(x 1 ;x ) + (1 - e1 )p 1(x 1 ;x * )ö
* *
÷ æe1p 1(x 1;x ) + (1 - e1 )p1(x 1;x* ) ö
÷
ç
ç ÷ ç
ç ÷
ç * ÷
÷ £ ç * ÷
÷
çe p (x ;x ) + (1 - e )p (x ;x )÷
ç2 2 2
* *
÷ çe p (x ;x ) + (1 - e )p (x ;x )÷
ç2 2 2 ÷
è 2 2 2 ø è 2 2 2 ø
for sufficiently small (e1, e2) ? (0 ) .
,0
Theorem 8: Suppose there are no within-group interactions. Then * *
(x 1 ,x 2) Î S 1 ´ S 2 is
an uninvadable equilibrium if and only if it is a strict Nash equilibrium.
2
29.
Example 5: Cournot duopoly game
Asymmetric Cournot equilibrium with convex cost functions is always m-stable and
uninvadable.
Superiority, however, depends on a specific form the cost function takes. The Cournot
equilibrium is not superior if the cost function is affine. If the cost function is quadratic,
on the other hand, the Cournot equilibrium is superior.
Example 6: Hotelling’s price competition model
Asymmetric Hotelling equilibrium with a uniform distribution function is m-stable and
uninvadable, but not superior.
Example 8: Bryant’s coordination game
All of the Nash equilibria are strict and thus uninvadable.
But none of these equilibria are superior; nor are they m-stable.
3
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