- Email: [email protected]
A note on the Replicator Equations of dynamical game theory
Appl. Math. Lett. Vol. 6, No. 4, pp. 31-33, 1993 Printed in Great Britain. All rights reserved
Copyright@
0893-9659193 $6.00 + 0.00 1993 Pergamon Press Ltd
A NOTE ON THE REPLICATOR EQUATIONS OF DYNAMICAL GAME THEORY C.
CHRISTOPHER
Department
Aberystwyth, (Received
J. DEVLIN
AND
of Mathematics,
University of Wales
Dyfed SY23 3BZ, Wales
and accepted February
1993)
Abstract-The rteplicator Equations introduced by Maynard Smith and Price [l] are examined in the continuous form given by Taylor and Jonker (21. A simple, explicit classification of the stable classes for three competing strategies is given in terms of the system parameters.
We follow the paper of Zeeman [3], but giving a slightly different derivation of the equations. Given a population in which each member can play one of n+l strategies (i) (i = 0, . . . , n), denote the number playing (i) by Ni. Suppose that an encounter between an individual playing (i) with one playing (j) results in a net pay-off of aij to the first. We hypothesise that the population growth, with a suitably chosen time scale, is given by
that is, the per capita growth rate of Ni is proportional to the average pay-off of strategy (i). Writing Nil cr, Nk = Xi, we have
dt
-
cc;=,m2
’
which, substituting from (l), gives
These are the so-called Replicator Equations. The region of physical significance is the n-simplex 1}, w hich can easily be shown to be invariant with respect to the flow. Our {x]xi>O,Cixi= interest is in the case of three competing strategies, for which Zeeman [3] conjectured that there can be no limit cycles. Hofbauer [4] confirmed this by showing that equations (2) are equivalent to the n-dimensional Lotka-Volterra equations:
dyi dt
=
Yi
(GO - aO0) +
g(%j
-
aOj)Yj
(3)
*
j=l
Since the Lotka-Volterra equations in the plane cannot support a limit cycle, neither can the bstrategy Replicator Equations. (We note that this result can be proved more directly by projecting the flow of (2) onto the xcxr-plane and taking x&r~(l - x0 - ~r)(‘+~+~) as a Dulac ‘bp=t 31
by h@-W
32
C. CHRISTOPHER,J. DEVLIN Table Critical Point
1.
I
(O,O,1)
PO2 +
P12
0)
PO1 +
P21
-?.zl-)
P20 po-PlzPzl
p12 Plz+Pzl
( iGY%?
POlP21
>
*
Plop20 -m
P12+P21
’ P12+P21 )O,
PO2Pl2
PlO +
_ ( *
Determinant
(O,l, (LO,O)
(0,
Trace
P20fP02 _
-w -m
2
0
PO1 +Plo
iGkik?O>
909142 (qo+q1+92)~
Table 2
Sign Class
-.. .
samesign
90, 91. Q all
not same sign
(10, q,q2
A qi >O
‘Ik’O Sj
A qi Co
qkO A
--
A
A
A /
>\
A
A
A
A
a
X X
X A A
d&b
A
6#0 :(Pij,PjklPki)*(Pji*Pkj*Pik)>
0 A
:(Pij+PiklPjk) v(Pji,PkivPkj)>
?i >O 6#0 c(Pij,Pji*Pki)*(Pik*Pjk*Pkj)> qco
X
.
Replicator
equations
33
function, where C and m are suitably chosen polynomials in the coefficients aij.) A consequence of this is that the nineteen stable classes listed by Zeeman are the only ones. The aim of this paper is to give a clear, explicit classification of these stable classes. For while Zeeman lists the possible phase portraits and gives a concrete example of an equation in each class, he does not give conditions on the system parameters whereby it can be determined into which class a given equation falls. We note that Bomze [5] has listed all possible classes (including degenerate classes), but again, giving only one example of each. Two systems (2) are considered equivalent if there exists a homeomorphism of the simplex onto itself, sending vertices to vertices, such that orbits of one system are mapped onto orbits of the other. Clearly, the change of parameters aij --+ aij + b, leaves the system (3) unaffected for any b; the key to our classification is the choice of parameters pi, = aij - ajj. We define the following quantities: 9% = PijPjk
+PikPkj
-p~kPk~,
for {i,j,
k} = (0,1,2};
6 = PO2P21PlO + P2OPOlPl2~
Table 1 gives information on the seven possible critical points on the simplex. Table 2 gives the classification; open circles represent saddles, closed circles antisaddles (foci or nodes), and lines separatrices. The rest of the phase portrait, up to time reversal, can be deduced from elementary topological considerations. (i, j, k) is any permutation of (0,1,2). Following Zeeman, we say that a system is in sign class (A, B) if all elements of A are of one sign and all elements of B of the other. REFERENCES 1. J. Maynard Smith and G.R. Price, The logic of animal conflicts, Nature 246, 15-18 (1973). 2. P.D. Taylor and L.B. Jonker, Evolutionarily stable strategies and game dynamics, Math. Bias. 40, 145-156 (1978). 3. E.C. Zeeman, Population dynamics from game theory, Lecture Notes zn hfathematzcs 819, 472-497 (1979). 4. J. Hofbauer, On the occurrence of limit cycles in the Volterra-Lotka differential equation, J. Nonlznear Anal. 5, 1003-1007 (1981). 5. I.M. Bomze, Lotka-Volterra eqrratiou and Replicator dynamics: A two-dimensional classification. Bzol. Cybern. 48, 201-211 (1983).
Copyright@
0893-9659193 $6.00 + 0.00 1993 Pergamon Press Ltd
A NOTE ON THE REPLICATOR EQUATIONS OF DYNAMICAL GAME THEORY C.
CHRISTOPHER
Department
Aberystwyth, (Received
J. DEVLIN
AND
of Mathematics,
University of Wales
Dyfed SY23 3BZ, Wales
and accepted February
1993)
Abstract-The rteplicator Equations introduced by Maynard Smith and Price [l] are examined in the continuous form given by Taylor and Jonker (21. A simple, explicit classification of the stable classes for three competing strategies is given in terms of the system parameters.
We follow the paper of Zeeman [3], but giving a slightly different derivation of the equations. Given a population in which each member can play one of n+l strategies (i) (i = 0, . . . , n), denote the number playing (i) by Ni. Suppose that an encounter between an individual playing (i) with one playing (j) results in a net pay-off of aij to the first. We hypothesise that the population growth, with a suitably chosen time scale, is given by
that is, the per capita growth rate of Ni is proportional to the average pay-off of strategy (i). Writing Nil cr, Nk = Xi, we have
dt
-
cc;=,m2
’
which, substituting from (l), gives
These are the so-called Replicator Equations. The region of physical significance is the n-simplex 1}, w hich can easily be shown to be invariant with respect to the flow. Our {x]xi>O,Cixi= interest is in the case of three competing strategies, for which Zeeman [3] conjectured that there can be no limit cycles. Hofbauer [4] confirmed this by showing that equations (2) are equivalent to the n-dimensional Lotka-Volterra equations:
dyi dt
=
Yi
(GO - aO0) +
g(%j
-
aOj)Yj
(3)
*
j=l
Since the Lotka-Volterra equations in the plane cannot support a limit cycle, neither can the bstrategy Replicator Equations. (We note that this result can be proved more directly by projecting the flow of (2) onto the xcxr-plane and taking x&r~(l - x0 - ~r)(‘+~+~) as a Dulac ‘bp=t 31
by h@-W
32
C. CHRISTOPHER,J. DEVLIN Table Critical Point
1.
I
(O,O,1)
PO2 +
P12
0)
PO1 +
P21
-?.zl-)
P20 po-PlzPzl
p12 Plz+Pzl
( iGY%?
POlP21
>
*
Plop20 -m
P12+P21
’ P12+P21 )O,
PO2Pl2
PlO +
_ ( *
Determinant
(O,l, (LO,O)
(0,
Trace
P20fP02 _
-w -m
2
0
PO1 +Plo
iGkik?O>
909142 (qo+q1+92)~
Table 2
Sign Class
-.. .
samesign
90, 91. Q all
not same sign
(10, q,q2
A qi >O
‘Ik’O Sj
A qi Co
qk
--
A
A
A /
>\
A
A
A
A
a
X X
X A A
d&b
A
6#0 :(Pij,PjklPki)*(Pji*Pkj*Pik)>
0 A
:(Pij+PiklPjk) v(Pji,PkivPkj)>
?i >O 6#0 c(Pij,Pji*Pki)*(Pik*Pjk*Pkj)> qco
X
.
Replicator
equations
33
function, where C and m are suitably chosen polynomials in the coefficients aij.) A consequence of this is that the nineteen stable classes listed by Zeeman are the only ones. The aim of this paper is to give a clear, explicit classification of these stable classes. For while Zeeman lists the possible phase portraits and gives a concrete example of an equation in each class, he does not give conditions on the system parameters whereby it can be determined into which class a given equation falls. We note that Bomze [5] has listed all possible classes (including degenerate classes), but again, giving only one example of each. Two systems (2) are considered equivalent if there exists a homeomorphism of the simplex onto itself, sending vertices to vertices, such that orbits of one system are mapped onto orbits of the other. Clearly, the change of parameters aij --+ aij + b, leaves the system (3) unaffected for any b; the key to our classification is the choice of parameters pi, = aij - ajj. We define the following quantities: 9% = PijPjk
+PikPkj
-p~kPk~,
for {i,j,
k} = (0,1,2};
6 = PO2P21PlO + P2OPOlPl2~
Table 1 gives information on the seven possible critical points on the simplex. Table 2 gives the classification; open circles represent saddles, closed circles antisaddles (foci or nodes), and lines separatrices. The rest of the phase portrait, up to time reversal, can be deduced from elementary topological considerations. (i, j, k) is any permutation of (0,1,2). Following Zeeman, we say that a system is in sign class (A, B) if all elements of A are of one sign and all elements of B of the other. REFERENCES 1. J. Maynard Smith and G.R. Price, The logic of animal conflicts, Nature 246, 15-18 (1973). 2. P.D. Taylor and L.B. Jonker, Evolutionarily stable strategies and game dynamics, Math. Bias. 40, 145-156 (1978). 3. E.C. Zeeman, Population dynamics from game theory, Lecture Notes zn hfathematzcs 819, 472-497 (1979). 4. J. Hofbauer, On the occurrence of limit cycles in the Volterra-Lotka differential equation, J. Nonlznear Anal. 5, 1003-1007 (1981). 5. I.M. Bomze, Lotka-Volterra eqrratiou and Replicator dynamics: A two-dimensional classification. Bzol. Cybern. 48, 201-211 (1983).