ON SPATIAL ASYMMETRIC GAMES
ABSTRACT The stability of some spatial asymmetric games is discussed. Both linear and nonlinear asymptotic stability of asymmetric hawkdove and prisoner's dilemma are studied. Telegraph reaction diffusion equations for the asymmetric spatial games are presented. Asymmetric games of parental investment is studied in the presence of both ordinary and cross diffusions.

Article: The Diversity in the Decision Facilitates Cooperation in the Sequential Prisoner's Dilemma Game.
[Show abstract] [Hide abstract]
ABSTRACT: The condition of cooperation in social conflicts of interest has been an interesting topic. On the one hand people usually desire to make their own profit. On the other hand, they mutually cooperate. This fact has motivated many researchers. Some solutions for this question have been proposed, and particular studies indicate that the diversity in decisionmaking or relationships promotes cooperation. In this research, we achieve the diversity by utilizing the novel method that refers to the mechanism of correction regarding each probability that every strategy comes to the representative by decisionmaking of group. This mechanism works when difference between the probability of the first and others becomes quite large. If once every group adopts this corrected decision, he/she achieves mutual cooperation of high level in the sequential prisoner's dilemma game in case the number of strategies (= players) is within the definite range. We also note that this game can effectively describe the property of evolution of strategy only with a small number of players. When each group has many players, in contrast to previous research, the decision with correction also has an effect on the suppression of prevalence of defection. In addition, we also show that the decision of this model is analogous to the system of redistribution of revenue, which provides balance of strength between several teams in professional sports.Advances in Complex Systems 11/2011; 14:377401. · 0.79 Impact Factor  SourceAvailable from: E. Ahmed[Show abstract] [Hide abstract]
ABSTRACT: The concept of cross diffusion is applied to some biological systems. The conditions for persistence and Turing instability in the presence of cross diffusion are derived. Many examples including: predatorprey, epidemics (with and without delay), hawk–dove–retaliate and prisoner's dilemma games are given.Advances in Complex Systems 11/2011; 07(01). · 0.79 Impact Factor
Page 1
arXiv:condmat/0209597v1 [condmat.statmech] 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
AlAin PO Box 17551, UAE
3Mathematics Department, Faculty of Education
45111 ElArish, Egypt
Abstract
The stability of some spatial asymmetric games is discussed. Both
linear and nonlinear asymptotic stability of asymmetric hawkdove
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 ownersintruders 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.
1
Page 2
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 hawkdove (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.
2
Page 3
2Asymmetric hawkdove 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.
3
Page 4
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 twotime 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 =
4
Page 5
∂
∂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)
5
Page 6
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.
6
Page 7
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 reactiondiffusion equa
tion
∂u(x,t)
∂t
= D∂2u(x,t)
∂x2
+ f(u). (19)
7
Page 8
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
8
Page 9
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))±
9
Page 10
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. predatorprey 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.
10
Page 11
(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)
11
Page 12
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 hawkdove 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
12
Page 13
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.
References
Ahmed E., Abdusalam H. A. and Fahmy I. (2001), Telegraph reaction
diffusion equations. Int. J. Mod. Phys. C 12, 717726.
Boerlijst M. C. and Hogeweg P. (1995), Attractors and spatial patterns
in hypercycles with negative interactions. J. Theor. Biol. 176, 199210.
Chopard B. and Droz M. (1991), Cellular automata model for the dif
fusion equation, J. Stat. Phys. 64, 859892.
Compte A. and Metzle R. (1997), The generalized Cattaneo equation
for the description of anomalous transport processes. J. Phys. A 30,
72777289.
Cressman R. and Vickers G. T. (1997), Spatial and density effects in
evolutionary game theory. J. Theor. Biol. 184, 359369.
Hadeler K. P. (1998). in O. Diekman et al (eds), Mathematics inspired
by Biology. Springer, Berlin.
Hofbauer J., Hutson V. and Vickers G. T. (1997), Travelling waves for
games in economics and biology. Nonlinear Analysis, Theory. Methods
and Applications 30, 12351244.
Hofbauer J. and Sigmund K. (1998), Evolutionary games and popula
tion dynamics. Cambridge University Press, Cambridge.
Krebs J. R. and Davies N. B. (2000), An introduction to behavioral
ecology. Blakwell scientific pub., Oxford.
Okubo A. (1980), Diffusion and ecological problems. Springer, Berlin.
13
Page 14
Schuster P. and Sigmund K. (1981), Coyness philandering and stable
strategies. Anim. Behav. 29, 186192.
Stuart A. (1989), Nonlinear instability in dissipative finite difference
scheme. Siam Rev. 31, 191220.
14