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arXiv:cond-mat/0209597v1 [cond-mat.stat-mech] 25 Sep 2002

On Spatial Asymmetric Games

E. Ahmed1,2, A. S. Hegazi1,2and A. S. Elgazzar3

1Mathematics Department, Faculty of Science

35516 Mansoura, Egypt

2Mathematics Department, Faculty of Science

Al-Ain PO Box 17551, UAE

3Mathematics Department, Faculty of Education

45111 El-Arish, Egypt

Abstract

The stability of some spatial asymmetric games is discussed. Both

linear and nonlinear asymptotic stability of asymmetric hawk-dove

and prisoner’s dilemma are studied. Telegraph reaction diffusion equa-

tions for the asymmetric spatial games are presented. Asymmetric

game of parental investment is studied in the presence of both ordi-

nary and cross diffusions.

1 Introduction

In asymmetric games [Hofbauer and Sigmund 1998], different players have

different strategies and different payoffs. In reality, most games are asym-

metric e.g. battle of the sexes and owners-intruders games. The differential

equations of the asymmetrical games are

dpi

dt= pi[(Aq)i− pAq]

dqi

dt= qi[(Bp)i− qBp], i = 1,2,...,n, (1)

where A(B) is the payoff matrix of the first (second) player, and pi(qi) is

the fraction of adopters of the strategy i in the first (second) population,

respectively.

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The problem of Turing instability (diffusion induced instability) [Okubo

1980] for symmetric games have been already studied [Cressman and Vickers

1997]. In this case, the standard replicator equation for the symmetric game

is given by

∂pi

∂t

= pi[(Ap)i− pAp] + D∇2pi

(2)

It is known [Okubo 1980] that including spatial effects may significantly

change the stability of equilibrium points. Also spatial effects are crucial

in many biological phenomena. Some authors [Boerlijst and Hogeweg 1995]

have studied general spatial hypercycle systems. Therefore studying spatial

asymmetric games is an important problem. Due to the difficulty in defin-

ing evolutionarily stable strategy (ESS) in asymmetric games [Hofbauer and

Sigmund 1998], we will concentrate on asymptotically stable strategies. The

equations of spatial asymmetric games are [Hofbauer et al 1997]

∂pi

∂t= pi[(Aq)i− pAq] + D1∇2pi,

∂qi

∂t= qi[(Bp)i− qBp] + D2∇2qi.

(3)

In this paper, we will attempt to answer the following questions:

1. Given an asymptotically stable solution to the system (1), does Turing

instability exist for the corresponding spatial game (3)?

2. Given an asymptotically unstable solution to the system (1), can dif-

fusion stabilize it?

3. Given an asymptotically linearly stable solution to the system (1), is it

nonlinearly stable?

Our typical examples will be the asymmetric hawk-dove (AHD) and the

asymmetric prisoner’s dilemma (APD) games.

The paper is organized as follows: In section 2, the asymmetric hawk-

dove game is studied. Conditions for Turing stability and nonlinear finite

amplitude instability are derived. In section 3, The asymmetric prisoner’s

dilemma is presented. Telegraph reaction diffusion equation is applied for

stability analysis of the asymmetric prisoner’s dilemma game in section 4. In

section 5, an asymmetric game of parental investment will be studied. Some

conclusions are summarized in section 6.

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2Asymmetric hawk-dove game

In this case the possible strategies are hawk (H) and dove (D), and the payoff

matrices A and B in Eq. (1) become

A =

?

1

2(v1− c1) v1

0

v1

2

?

, B =

?

1

2(v2− c2) v2

0

v2

2

?

, (4)

where ci> vi, i = 1,2. The corresponding partial differential equations for

the spatial AHD are

∂p

∂t= D1∇2p +1

∂q

∂t= D2∇2q +1

2p(1 − p)(v1− c1q),

2q(1 − q)(v2− c2p),

(5)

where p (q) is the fraction of hawks in the population of the first (second)

player. It is direct to see that the solution p = 1, q = 0 is linearly asymptot-

ically stable solution for the system (5), without diffusion (D1= D2= 0).

The first question is about Turing (diffusion induced) instability [Okubo

1980]. It occurs if the following system

∂p

∂t= D1

∂q

∂t= D2

∂2p

∂x2+ f1(p,q),

∂2q

∂x2+ f2(p,q),

(6)

has an equilibrium solution (pss,qss) which is stable if D1= D2= 0, and the

corresponding linearized system

p = pss+ ε(x,t), q = qss+ η(x,t),

∂ε

∂t= D1∂2ε

∂η

∂t= D2∂2η

∂x2+ a11ε + a12η,

∂x2+ a21ε + a22η,

(7)

satisfies the condition

H(k2) = D1D2k4− (D1a22+ D2a11)k2+ a11a22− a12a21< 0

In this case diffusion will destabilize the solution (pss,qss). This is Turing

instability.

Applying the above procedure to the spatial AHD one gets:

(8)

Proposition (1): The equilibrium solution p = 1, q = 0 of the AHD

game is Turing stable.

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The second question to be discussed is: Can diffusion stabilize an unstable

solution of the AHD game? Consider the internal solution p = v2/c2, q =

v1/c1. It is asymptotically unstable solution to the diffusionless case (D1=

D2= 0). Including diffusion and linearizing around p = v2/c2, q = v1/c1,

and assuming the following boundary conditions:

p =v2

c2+ ε(x,t), q =v1

ε(0,t) = ε(1,t) = 0, η(0,t) = η(1,t) = 0,

c1+ η(x,t),

(9)

then

Proposition (2): The interior solution p = v2/c2, q = v1/c1, with the

boundary conditions (9) is linearly asymptotically stable if

D1D2π4−v1v2

4

(1 −v1

c1)(1 −v2

c2) ≥ 0. (10)

Linear stability analysis is useful if the perturbations of equilibrium are

infinitesimally small. This is not always the case in biological systems. In

this case one has to study finite amplitude instability (FAI) [Stuart 1989] of

the equilibrium solution. In the following, we generalize the work of Stuart

to the two species case. Consider the following equation

∂θ

∂t=∂2θ

∂x2+ f(θ), θ(0,t) = θ(1,t) = 0. (11)

Linearizing around the solution θ = 0, i.e. let

θ(x,t) = v(x,t),

linearize in v, then

∂v

∂t=∂2v

σφ = φ′′+ f′(0)φ, φ(0) = φ(1) = 0.

∂x2+ vf′(0), Let v = φ(x) exp(σt) ⇒

(12)

Set φ =?

1,2,.... Studying the stability of the first bifurcation point l = 1 using

Matkowsky two-time nonlinear stability analysis, one defines λ = a − k,

set λ = π2+ λ0ε2, ε is a small parameter. Decompose the time into fast t′

and slow τ, then

lalsin(πlx), then bifurcation points are given by f′(0) = (πl)2, l =

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∂

∂t=

∂

∂t′+ ε2∂

∂τ.

Expand u in powers of ε2, then u ≃ εv1+ε3v3+..., (notice that f′′(0) = 0),

then substituting in Eq. (12), one gets

v1(x,t′,τ) =

?

l

al(τ)sin(πlx)exp(1 − l2)π2t′.

Substituting into the cubic term and setting the constant term in t′equal to

zero, one finally gets

da1

dτ= λ0a1+ b(a1)3,

b = 4π4f′(0)

?1

0sin4πx dx

6?1

0sin2πx dx.

Thus a nonlinear (finite amplitude) instability arises if

λ0< 0 and |a1(0)| >

?

|λ0|

b

. (13)

In the beginning of the section the condition f′′(0) = 0 was imposed, here

we will assume f′′(0) ?= 0. Thus we consider

∂u

∂t= D∇2u + f(u), f(0) = 0, u(0,t) = u(1,t) = 0.

The solution u = 0 is a steady state solution, so expanding near it we set

(14)

u =

?

m=1

εmvm(t′,t′′,x),

∂

∂t=

∂

∂t′+ ε∂

∂t′′, f′(0) = λ0+ ελ1.

Substituting in Eq. (14) and equating terms O(ε), one gets

v1(t′,t′′,x) =

?

l=1

al(t′′)sin(πlx)exp[(1 − l2)π2].

Let λ0= π, and consider the equation O(ε2), we set the secular term (inde-

pendent of t′) equal to zero, then

da1

dt′′= λ1a1+

?b′

2f′′(0)

?

a2

1, b′=

?1

0sin3πx dx

?1

0sin2πx dx.

(15)

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Thus the conditions for FAI are

f′′(0) > 0, λ1< 0 and a1(0) > −

2λ1

b′f′′(0).

For systems of two partial differential equations

∂u1

∂t= D1∇2u1+ f(u1,u2),

u1(0,t) = u1(1,t) = u2(0,t) = u2(1,t) = 0,

f(0,0) = g(0,0) = 0.

∂u2

∂t= D2∇2u2+ g(u1,u2),

(16)

Expanding near the steady state solutions u1= u2= 0, we get

u1=?

m=1εmvm(t′,t′′,x), u2=?

f1=

g1= µ11+ εµ12, g2= µ21+ εµ22.

m=1εmωm(t′,t′′,x),

∂

∂t=

∂

∂t′+ ε∂

∂t′′,

∂f

∂u1(0,0)= λ11+ ελ12, f2= λ21+ ελ22,

After some tedious calculations, we got the following conditions for FAI in

the system (16):

(i) (π2D1− λ11)(π2D2− µ21) − λ21µ11= 0.

(ii) λ12+ κλ22=

µ12

κ+ µ22< 0, where κ =π2D1−λ11

λ21

.

(iii)

f11

2+ κf12+κ2

2f22=g11

2κ+ g12+κ.

2g22> 0.

(iv) a1(0) > −

λ12+κλ22

f11

2+κf12+κ2

b′?

2f22

?,

where b′is defined in Eq. (15). Applying the condition (i) to the spatial

AHD, one gets

(π2D1+ v1)(π2D2+ v2) < 0,

which is not possible, thus we find:

Proposition (3): There is no finite amplitude instability for the solution

p = 1, q = 0 of the AHD game.

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3 Asymmetric prisoner’s dilemma game

In the prisoner’s dilemma game, the possible strategies are cooperate (C)

and defect (D). The payoff matrices A, B in Eq. (1) are given as follows:

A =

?

R1

T1

S1

P1

?

, B =

?

R2

T2

S2

P2

?

(17)

such that 2Ri > Ti+ Si, and Ti > Ri > Pi > Si, where i = 1, 2. The

dynamical equations for the spatial asymmetric prisoner’s dilemma game

(spatial APD) are:

∂u1

∂t= D1∇2u1+

u1(1 − u1)[−(P1− S1) + u2(P1− S1− T1+ R1)],

∂u2

∂t= D2∇2u2+

u2(1 − u2)[−(P2− S2) + u1(P2− S2− T2+ R2)],

where u1(u2) is the fraction of cooperators in the first (second) players pop-

ulation. The solution u1 = u2 = 0 is linearly asymptotically stable. It

represents the always defect strategy.

Two questions arise the first is: can diffusion stabilize the cooperation

solution u1 = u2 = 1? And does the always defect solution have (FAI)

nonlinear instability? Using the techniques of the previous section we get:

(18)

Proposition (4):

(i) If Diπ2> Ti− Ri, then the cooperation solution is linearly asymptoti-

cally stable.

(ii) The always defect solution does not have FAI.

4Telegraph reaction diffusion in spatial asym-

metric games

The standard spatial games depend on the familiar reaction-diffusion equa-

tion

∂u(x,t)

∂t

= D∂2u(x,t)

∂x2

+ f(u). (19)

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A basic weakness in this equation is that the flux j reacts simultaneously to

the gradient of u consequently an unbounded propagation speed is allowed.

This manifests itself in many solutions to Eq. (1) e.g. (if f = 0), then

u(x,t) =

1

√4πDte

−x2

4Dt, u(x,0) = δ(x) i.e u(x,t) > 0∀x, ∀t > 0.

This is unrealistic specially in biological and economical systems, where it is

known that propagation speeds are typically small. To rectify this weakness,

Fick’s law is replaced by

j + τ∂j

∂t= D∂u

∂x,

and the resulting telegraph diffusion equation becomes

τ∂2u

∂t2+∂u

∂t= D∂2u

∂x2.

The corresponding telegraph reaction diffusion (TRD) is

τ∂2u

∂t2+ (1 − τdf

du)∂u

∂t= D∇2u + f(u) (20)

The time constant τ can be related to the memory effect of the flux j as a

function of the distribution u as follows: Assume that [Compte and Metzle

1997]

j(x,t) = −

?t

0K(t − t′)∂u(x,t′)

∂x

dt′, (21)

hence

j + τ∂j

∂t= −τK(0)u(x,t) −

?t

0

?

τ∂K(t − t′)

∂t

+ K(t − t′)

?∂u

∂xdt′.

This equation is equivalent to the telegraph equation if

K(t) =D

τexp(−t

τ).

This lends further support that TRD is more suitable for economic and bi-

ological systems than the ordinary diffusion equation since e.g. it is known

that we take our decisions according to our previous experiences, so memory

effects are quite relevant. Further evidence comes from the work of Chopard

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and Droz [Chopard and Droz 1991], where they have shown that starting

from discrete time and space then the continuum limit does not give the

standard reaction diffusion but the telegraph one.

Since it is known that new technologies, habits etc... takes time to spread,

we believe that TRD equation is more relevant than the ordinary diffusion

equation in modelling economic and biological systems [Ahmed et al 2001].

Now we apply TRD to spatial APD game. The TRD for a system of

equations are [Hadeler 1998]

τ∂2ui

D∇2ui+ fi(u1,u2,...,un),

∂t2 +∂ui

∂t− τ?

j

∂uj

∂t

∂fi

∂uj=

(22)

hence applying it to the APD (18), we get

τ∂2u1

∂t2 +∂u1

u2(P1− S1− T1+ R1)] = D1∇2u1+

u1(1 − u1)[−(P1− S1) + u2(P1− S1− T1+ R1)],

τ∂2u2

∂t(1 − 2u2)[−(P2− S2)+

u1(P2− S2− T2+ R2)] = D2∇2u2+

u2(1 − u2)[−(P2− S2) + u1(P2− S2− T2+ R2)].

The following question arises: Can diffusion stabilize the cooperation so-

lution u1= u2= 1? Using the techniques of the second section, we get

∂t− τ∂u1

∂t(1 − 2u1)[−(P1− S1)+

∂t2 +∂u2

∂t− τ∂u2

(23)

Proposition (5): If the following conditions are satisfied

Diπ2> Ti− Ri, 1 > τ(Ti− Ri),

4τ(Diπ2− Ti+ Ri) ≤ [−1 + τ(Ti− Ri)]2, i = 1,2,

then the cooperation solution is linearly asymptotically stable.

(24)

Proof. Assume that

u1= 1 − ε1exp(σ1t)sin(πx), u2= 1 − ε2exp(σ2t)sin(πx).

Substituting one gets

σ1=

1

2τ[(−1 + τ(T1− R1))±

?

?

(−1 + τ(T1− R1))2− 4τ (R1− T1+ D1π2)],

σ2=

(−1 + τ(T2− R2))2− 4τ (R2− T2+ D2π2)].

1

2τ[(−1 + τ(T2− R2))±

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Stability requires that the real part of σi, i = 1,2 is negative. The first two

conditions in the proposition guarantee this requirement. Furthermore since

ui, i = 1,2 are real and nonnegative, then the term under the square root

should be nonnegative. The third condition of the proposition guarantees

ui≥ 0.

Thus the conditions for cooperation stability for TRD are more stringent

than those for ordinary diffusion (c.f. proposition (4)).

5Asymmetric game of parental investment

Parents are faced with the decision whether to care for the offsprings or to

desert. A model has been given for this asymmetric game [Krebs and Davies

2000]. Let p0, p1, p2, be the probabilities of survival of offsprings which are

not cared for, cared for by a single parent and cared for by both parents,

respectively, then p0< p1< p2. A deserting male has a chance q of mating

again while a caring (deserting) female has w1(w2) offsprings. The payoff

matrices for male (female) corresponding to the strategies C (care) or D

(desert) are denoted by A(B), and given by

A =

?

w1p2

w1p1(1 + q) w2p0(1 + q)

w2p1

?

, B =

?

w1p2 w1p1

w2p1 w2p0

?

. (25)

The spatial asymmetric equations for the above game are:

∂u

∂t= u[w1p2v + w2p1(1 − v) − w1p2uv − w2p1u(1 − v)−

w1p1(1 + q)v(1 − u) − w2p0(1 + q)(1 − u)(1 − v)] + D1∂2u

∂v

∂t= v[w1p2u + w1p2(1 − u) − w1p2uv − w2p1u(1 − v)−

w1p1v(1 − u) − w2p0(1 − u)(1 − v)] + D2∂2v

In this system, we introduced both ordinary and cross diffusion. Cross dif-

fusion is the diffusion of one type of species due to the presence of another

[Okubo 1980]. This phenomena is abundant in nature e.g. predator-prey sys-

tems where the predator diffuses towards the regions where the prey is more

abundant. On the other hand the prey tries to avoid predators by diffusing

away from it. Another area of application is in epidemics where susceptible

individuals try to avoid infected ones.

Here we will see that ordinary diffusion is unable of destabilizing the

diffusionless ESS:

∂x2+ D12∂2v

∂x2,

∂x2+ D21∂2u

∂x2.

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(i) ESS1 where both male and female desert i.e. (u = 0,v = 0). It requires

w2p0> w1p1and p0(1 + q) > p1.

(ii) ESS2 where male cares and female desert i.e. (u = 1,v = 0). It requires

w2p1> w1p2and p0(1 + q) < p1.

(iii) ESS3 where female cares and male desert i.e. (u = 0,v = 1). It requires

w1p1> w2p0and p1(1 + q) > p2.

(iv) ESS4 where both male and female care i.e. (u = 1,v = 1). It requires

w1p2> w2p1and p1(1 + q) < p2.

Following steps similar to the previous games, we get

Proposition (6): The solution ESSi, i = 1,2,3,4 is destabilized if the

following condition is satisfied:

D12D21π4> (D1π2− ai)(D2π2− bi) (26)

where

a1 = w2p1− w2p0(1 + q), b1= w1p1− w2p0,

a2 = −w2p1+ w2p0(1 + q), b1= w1p2− w2p1,

a3 = w1p2− w1p1(1 + q), b1= −w1p1+ w2p0,

a4 = −w1p2+ w1p1(1 + q), b1= w2p1− w1p2.

Notice that all ai, bi, i = 1,2,3,4 are negative hence ordinary diffusion

cannot destabilize the ESS.

Applying the above procedure to the battle of the sexes [Schuster and

Sigmund 1981] where the female has two strategies coy or fast while the

male can be either faithful or philanderer. The male (female) payoff matrix

is A(B)

?

−2 0

hence the spatial battle of the sexes equations are

A =

0−10

?

, B =

?

0 5

3 0

?

,

∂u

∂t= u(1 − u)(−10 + 12v) + D1∂2u

∂v

∂t= v(1 − v)(5 − 8u) + D2∂2v

∂x2,

∂x2.

(27)

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Proposition (7): Diffusion stabilizes the internal equilibrium of the sys-

tem (27).

Proof. There is a unique internal homogeneous equilibrium solution E =

(5/8,5/6), which (for the diffusionless case) is stable but not asymptotically

stable. Substituting with

u =5

8+ εexp(σt)sinπx, v =5

6+ ς exp(σt)sinπx, 1 ≥ x ≥ 0,

in Eq. (27), and linearizing in ε, ζ, one gets

(σ + D1π2)ε =45

16ζ, (σ + D2π2)ζ = −10

9ε.

Hence

[(σ + D1π2)(σ + D2π2) +25

8]ζ = 0,

i.e.

(σ + D1π2)(σ + D2π2) +25

8

= 0,

then

σ2+ σ(D1π2+ D2π2) + (25

8+ D1D2π4) = 0.

Therefore the real part of σ is negative, then the internal equilibrium (in the

presence of diffusion) is asymptotically stable. It is clear that if the diffusion

coefficients are set equal to zero (D1= D2= 0), then one regains the stability

but not asymptotic stability. This completes the proof.

6 Conclusions

Based on replicator equations, a mathematical approach for the analysis of

spatial asymmetric games is introduced. Some questions regarding spatial

stability for asymmetric hawk-dove and the asymmetric prisoner’s dilemma

(APD) games are answered. Telegraph reaction diffusion equation is applied

for stability analysis of the asymmetric prisoner’s dilemma game. Asymmet-

ric game of parental investment is studied in the presence of both ordinary

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and cross diffusions. Ordinary diffusion cannot destabilize the ESS for this

game. Conditions for destabilizing the ESS are given in the case of cross

diffusion.

Acknowledgments

We thank the referees for their helpful comments.

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