Evolutionary Stabilities in

Evolutionary Stabilities in

Games with Continuous Strategy Spaces:

Characterizations with Examples

Woojin Lee

Department of Economics

University of Massachusetts

Amherst, MA 01002

1

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.

w hat is common : the idea of stability of a strategy (or a strategy profile) against

small perturbations

what is different: all other details

2

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

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

Evolutionary stabilities

-- The first sense of evolutionary stability does not necessarily imply the second sense

of stability.

-- An -population being resistant to invasion in the first sense would be immune to

a mutant, but a shifted -population living near 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

5

of actual recovery from small perturbations and vice versa, but this is not true in

general non-linear games with continuous strategy spaces.

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.

6

Apaloo (1997)’s evolutionarily stable neighborhood invader strategy (ESNIS) is

an uninvadable strategy that is locally superior.

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.

7

-- 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.

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.

is the payoff of a player who chooses against an opponent who

chooses .

Strategies take values on a convex subset of the k-dimensional real Euclidean

space.

9


-- Suppose that all individuals are initially programmed to play a certain consensus

strategy (say, ), and then we inject into this -population a small population share

of individuals who are likewise programmed to play some other strategy (say, ).

-- Then is the net payoff gain of a player playing the minority

strategy relative to a player playing the consensus strategy . We say:

(1) strongly invades -population if .

(2) weakly invades -population if ,

(3) no can weakly invade -population if

(4) no can strongly invade -population. if

10


Definition 1: Consider a uni-population game with .

(1) is a Nash equilibrium strategy (NES) if for all

.

We call it a strict NES if strict inequality holds for all .

(2) is a locally superior strategy if for all sufficiently close to ,

.

(3) is a locally m-stable strategy if for all sufficiently close to ,

for small ; and

for small as long as .

11

Local superiority and m-stability

(1) Superiority requires that strictly minimize .

(2) Superiority is different from Pareto optimality, which requires that

for all .

(3) Superiority is weaker than dominance, which requires that for all

and .

(4) Superiority is weaker than the following condition: for all

.

(5) M-stability requires that strictly minimize:

for small .

12

Figure 1: Superiority and m-stability

superiority: yes superiority: nom-stability: yes

m-stability: no

LINK Word.Document.8 "E:\\Woojin5\\evolution\\The concept of evolutionary stable

strategies.doc" OLE_LINK2 \a \r

Note: Consider the set of points at which . Since for

, the 45 degree line is included in this set. For to be an equilibrium point, there

must be other curves in this set which cross the 45 degree line at . Here we suppose that

the set of such points, together with the 45 degree line, divides the space around into

four regions. (This may not be true in general.) For to be superior, the horizontal line at

x’=x

x’

x

x*

φ(x’,x;x)=0

(-)

(-)(+)

(+)

x’=x

x’

x

x*

φ(x’,x;x)=0

(+)

(+)(-)

(-)

x’=x

x’

x

x*φ(x’,x;x)=0

(-)

(+)

(+)

(-)

x’=x

x’

x

x*φ(x’,x;x)=0

(+)

(-)

(-)

(+)

1

should be in the positive region. For to be m-stable, the sign of must be as

shown in the two diagrams of the first row.

2

Figure 2: Various equilibrium concepts

Global m-stability

Local superiority

Global superiority

Local m-stability

ESS-II

ESS-I (Univadability)

Nash

If is affine in the first argument

If is affine in the first argument

If is concave in the first argument

If is convex in the first argument

If is concave in the first argument

If is convex in the first argument

If is concave in the first argument If is strictly

quasi-convex in the first argument

If is affine in the second argument

Strict Nash

If is strictly concave in the first argumentIf is strictly concave

in the first argument

If is continuous in the second argument

If is strictly concave in the first argument

3

Table 1: Sufficient conditions in symmetric uni-

population games

Concept Sufficient conditions for an interior solution

Local strictness ; and

is negative definite.

Local superiority ; and


Local m-stability ; and

is negative definite

for small .

Local

uninvadability

; and

is negative definite for sufficiently

small .

Local ESNIS ;

is negative definite; and


Local CSS ;

is negative definite; and

is negative definite

for sufficiently small .

4

Evolutionary stabilities in uni-pop. games

Definition 2: Consider a uni-population game in which the payoff function is given by

.

(1) is an uninvadable strategy (UIS) if it is a Nash equilibrium strategy for

which implies for all .

(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

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 that weakly dominates .

(4) An UIS is identical to the following definition:

ESS-I: is an uninvadable strategy (UIS) if for all and sufficiently small

, .

(5) An UIS is equivalent to the following definition only when is affine in the second

argument.

ESS-II: is an evolutionarily stable strategy if for all and sufficiently

small , .

(7) An interior strict Nash equilibrium is a CSS if the slope of the best reply curve at

is less than 1 and an ESNIS if it is less than 1/2.

2

Example 1: Cournot duopoly game

Cournot firms, which face a linear inverse market demand function:

, where .

We assume a convex and increasing cost function , where , , and

.

The profit of each firm is and thus

.

The unique pure strategy Nash equilibrium of this game is , where is defined

by . Because , , the interior Nash

equilibrium strategy is strict, uninvadable, superior, and m-stable.

Note that ; the slope of the best reply curve is less than 1/2 for all .

Example 2: Hotelling’s price competition model

3

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 will be denoted by m; .

Each consumer has a unit demand and incurs travel cost t per unit of distance from

their location.

Then and .

If the payoff function is strictly concave, the unique pure strategy Nash equilibrium

strategy is , which is in the interior.

is a locally strict NES (or a locally uninvadable strategy) if .

is locally m-stable is because .

is locally superior only when .

4

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

simultaneously.

The payoff of an individual choosing e and facing an opponent choosing f is given by

where .

This game possesses a continuum of pure strategy Nash equilibria; any where

is a Nash equilibrium.

Because for all , all of these Nash

equilibria are strict and thus uninvadable.

But none of them are superior, because for all

. Also none of the Nash equilibria are m-stable.

5

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 is the encompassing payoff of a player in group

choosing when she faces a fellow player in the same group playing and

another player of group choosing .

-- The net payoff gain of a player in group i choosing strategy compared with her

playing when the entire population is settled with is

.

6


Let and . is said to

(1) most strongly invade -population if .

(2) very strongly invade -population if .

(3) strongly invade -population if or

(4) weakly invade -population if or .

(5) If , we say no can strongly invade -population.

(6) If , we say no can weakly invade -population.

7


Definition 3:

(1) is a Nash equilibrium of the bi-population game if

(i.e., ) for all

.

(2) is a locally superior equilibrium if there is no

in the neighborhood of such that

.

(3) is a locally m-stable equilibrium if

(i) for small , there is no in the neighborhood of

such that ; and

(ii) for small , there is no in the neighborhood of

such that for .

8

Superior equilibrium

(1) is a superior equilibrium if and only if either or

hold for all . Thus it requires that strongly invade -

population. This is weaker than the following : and

hold for all .

(2) Thus is a superior equilibrium if and only if

,

or equivalently, . This condition resembles strong Pareto

optimality, but is different from it. Strong Pareto optimality requires that we have:

.

(3) A sufficient condition for a locally superior equilibrium is that is a strict local

minimum of the following problem:

9

subject to .

10

(4) is an m-stable equilibrium if and only if for all in the

neighborhood of the following is true:

(i) for small , either or

hold; and

(ii) for small , either or

hold as long as .

(5) Thus is an m-stable equilibrium if and only if

and

(6) A sufficient condition for an m-stable equilibrium is that is a strict local

minimum of the following problem:

s.t for small ,

s.t for small .

11

Table 3: Sufficient conditions in bi-pop. games

Concept Sufficient conditions for an interior solution

Local

strictness

; and

and are negative definite.

Local

superiority

; and


Local m-

stability

; and

is

negative definite for small and positive definite for small .

Local

uninvadability

; and

is negative

definite for small .

Local ESNIE ;

and are negative definite; and


Local CSE ;

and are negative definite; and

is

negative definite for small and positive definite for small .

1

Definition 4: Consider a bi-population game with the encompassing payoff

function of for each .

(1) is an uninvadable equilibrium of the bi-population game if it

is a Nash equilibrium for which there is no such that

and .

(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

Theorem 7: is an uninvadable equilibrium of the bi-population

game if there is no such that

for sufficiently small .

Theorem 8: Suppose there are no within-group interactions. Then is

an uninvadable equilibrium if and only if it is a strict Nash equilibrium.

2

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

Figure 4: Cournot duopoly

Case 1: Linear cost function

Case 2: Quadratic cost function

x2

x1

(-)

(-)(+)

(+)

(0)(0)

slope = c

x2

x1

(+)

(+)

(-)

(-)(0)

(0)

slope = c

x2

x1

(-)

(-)(+)

(+)

(0)(0)

slope =

x2

x1

(+)

(+)

(-)

(-)

(0)

(0)

slope =

1

Figure 5: Hotelling’s price competition model

p2

p1

(-)

(-)

(+)(+)

(0) (0)

slope =

(+)

p2

p1

(-)

(-)

(+)

(0)

(0)

slope =

2

Figure 7: Coordination game

(-)

e2

e1

(-)

(+)(+)

slope= 1/1

slope= (1-1)/1

(0)

(0) e2

e1

(-)

(-)

(+)

(+)

slope=2/2

slope=2/(2-2)

(0)

45 degree 45 degree(0)

3

Documents

Evolutionary Stabilities in