Stable Limit Cycles, Multiple Steady States and
Complex Attractors in Logit Dynamics
Marius-Ionut Ocheay
University of Amsterdam
15th January 2008
Abstract
We investigate, via standard Rock-Scissors-Paper and Coordination games, the
occurrence of periodic and complex behaviour in a smoothed version of the Best
Response Dynamics, the Logit Dynamics. The …ndings are that, …rst, in contrast
with the Replicator Dynamics, generic Hopf bifurcation and thus, stable limit cycles,
do occur under the Logit, even for three strategy games. Second, the Logit displays
another phenomenon which does not occur under the Replicator: multiple, interior
steady states created via a sequence of fold bifurcations. The basins of attraction
of these interior point-attractors vary non-monotonically both with the payo¤ and
the behavioural parameters; they allow the computations of the long-run aggregate
welfare which is maximized when agents are only moderately ’rational’. Last, we
…nd, in a four strategy game, a period-doubling route to chaotic dynamics under a
’weighted’version of the Logit dynamics.
I am particularly grateful to my supervisor, Prof. Cars Hommes, for his valuable comments and kind
support during writing this paper.
y
PhD student, Center for Nonlinear Dynamics in Economics and Finance(CeNDEF), University of
Amsterdam, the Netherlands; e-mail: m.i.ochea@uva.nl; phone: + 31 20 525 7356
1
1
Introduction
1.1
Motivation
A large part of the research on evolutionary game dynamics focused on identifying conditions, both in the class of dynamics (Samuelson and Zhang (1992), Hofbauer and Weibull
(1996), Hofbauer (2000), Cressman (1997), Sandholm (2005)) and underlying games (Milgrom and Roberts (1990), Nachbar (1990), Hofbauer and Sandholm (2002)), for uniqueness
of and convergence towards point-attractors such as Nash Equilibrium and Evolutionary
Stable Strategy (ESS). Nevertheless, there are some examples of periodic and chaotic behaviour in the literature, mostly under a particular kind of evolutionary dynamic, the
Replicator Dynamics. Motivated by the idea of adding an explicit dynamical process to
the static concept of Evolutionary Stable Strategy, Taylor and Jonker (1978) introduced
the Replicator Dynamics; it found soon applications in biological, genetic or chemical systems, those domains where organisms, genes or molecules evolve over time via replication.
The common features of these systems are that they can be well approximated by an
in…nite population game with random pairwise matching, giving rise to a replicator-like
evolutionary dynamics given by a low-dimensional dynamical system.
In the realm of non-convergence literature two issues are particularly important: the
existence of stable periodic and complicated solutions and their robustness(to slight perturbations in the payo¤s matrix). Hofbauer et al. (1980) and Zeeman (1980) investigate
the phase portraits resulting from three-strategy games under the replicator dynamics
and conclude that only ’simple’ behaviour - sinks, sources, centers, saddles - can occur.
In general, an evolutionary dynamics together with a n strategy game de…ne a proper
1 dynamical system on the n
n
1 simplex. An important result is Zeeman (1980)
conjecture that there are no generic Hopf bifurcations on the 2-simplex: "When n = 3
all Hopf bifurcations are degenerate"1 . Bomze (1983,1995) provides a thorough classi1
Zeeman(1980), pg. 493
2
…cation of the planar phase portraits for all 3-strategy games according to the number,
location(interior or boundary of the simplex) and stability properties of the Replicator Dynamics …xed points and identi…es 49 such di¤erent phase portraits; again, only non-robust
cycles are created usually via a degenerate Hopf bifurcation. Hofbauer (1981) proves that
in a 4-strategy game stable limit cycles are possible under Replicator Dynamics; the proof
consists in …nding a suitable Lyapunov function whose time derivative vanishes on the
! limit set of a periodic orbit. Stable limit cycles are also reported in Akin (1982) in a
genetic model where gene ’replicates’ via the two allele-two locus selection; this is not
surprising as the dynamical system modeling gametic frequencies is three-dimensional, the
dynamics is of Replicator type and the Hofbauer (1981) proof applies here as well. Furthermore, Maynard Smith and Hofbauer (1987) prove the existence of a stable limit cycle for
an asymmetric, Battle of Sexes-type genetic model where the allelic frequencies evolution
de…nes again a 3-D system. Their proof hinges on normal form reduction together with
averaging and elliptic integrals techniques for computing the phase and angular velocity of
the periodic orbit. Stadler and Schuster (1990) perform an impressive systematic search for
both generic(…xed points exchanging stability) and degenerate(stable and unstable …xed
points colliding into a one or two-dimensional manifold at the critical parameter value)
transitions between phase portraits of the replicator equation on 3 x 3 normal form game.
Chaotic behaviour is found by Schuster et al. (1991) in Replicator Dynamics for a
4-strategy game derived via Hofbauer transform from an autocatalytic reaction network.
They report the standard Feigenbaum route to chaos: a cascade of period-doubling bifurcations intermingles with several interior crisis and collapses to a chaotic attractor.
Numerical evidence for strange attractors is provided by Skyrms (1992, 1999) for a Replicator Dynamics ‡ow on two examples of a four-strategy game.
Although periodic and chaotic behaviour is substantially documented in the literature for the Replicator Dynamics, there is much less evidence for such complicated behaviour in classes of evolutionary dynamics that are more appropriate for humans inter-
3
action(…ctitious play, best response dynamics, adaptive dynamics, etc.). While Shapley
(1964) constructs an example of a non-zero sum game with a limit cycle under …ctitious
play (as reported by Skyrms (1993) ) and Berger and Hofbauer (2005) …nd stable periodic
behaviour - two limit cycles bounding an asymptotically stable annulus - for a di¤erent
dynamic - the Brown-von Newmann Nash (BNN) - a systematic characterization of (non)
generic bifurcations of phase portraits is still missing for the Best Response dynamics.
In this paper we take a …rst step in this direction and use a smoothed version of the
Best Response dynamics - the Logit Dynamics - to study evolutionary dynamics in simple
three and four strategies games from the existing literature. The qualitative behaviour of
the resulting ’evolutionary’games is investigated with respect to changes in the payo¤ and
behavioural parameters, using analytical and numerical tools from non-linear dynamical
system theory.
1.2
’Replicative’vs. ’rationalistic’dynamics
Most of the earlier discussed evolutionary examples came from the biological (animal
contests (Zeeman (1980), Bomze (1983,1995)), genetics (Maynard Smith and Hofbauer
(1987))) or chemical (catalytic networks (Schuster et al. (1990), Stadler and Schuster
(1990)) approach and not from economics. From the perspective of strategic interaction
the main criticism of the ’biological’ game-theoretic models is targeted at the intensive
use of preprogrammed, simple imitative play with no role for optimization and innovation.
Speci…cally, in the transition from animal contests and biology to humans interactions and
economics the Replicator Dynamics seems no longer adequate to model the rationalistic
and ’competent’forms of behaviour (Sandholm (2006)). Best Response Dynamics would
be more applicable to human interaction as it assumes that agents are able to optimally
compute and play a (myopic) ’best response’to the current state of the population. But,
while the Replicator Dynamics appeared to impose an unnecessarily loose rationality assumption the Best Response dynamics moves to the other extreme: it is too stringent in
4
terms of rationality. Another drawback is that, technically, the best reply is not necessarily
unique and this leads to a di¤erential inclusions instead of an ordinary di¤erential equation. One way of solving these problems was to stochastically perturb the matrix payo¤s
and derive, via the discrete choice theory, a "noisy" Best Response Dynamics, called the
Logit Dynamics. Mathematically it is a ’smoothed’, well-behaved dynamics while from
the strategic interaction point of view it models a boundedly rational player/agent. Third,
from nonlinear dynamical systems perspective the Replicator Dynamic is non-generic in
dimension two and only degenerate Hopf bifurcations can arise on the 2-simplex. In sum,
apart from its conjectured generic properties, the Logit is recommended by the need for
modelling players with di¤erent degrees of rationality and for smoothing the Best Response
correspondence. Thus, the main part of this paper investigates the qualitative behaviour
of simple evolutionary games under the Logit dynamics when the level of noise and/or
underlying normal form game payo¤s matrix is varied with the goal of …nding attractors(periodic or perhaps more complicated) which are not …xed points or Nash Equilibria.
The paper is organized as follows: Section 2 overviews the Logit Dynamics while Section
3 gives a brief introduction into the Hopf bifurcation theory. In Sections 4 the Logit
Dynamics is implemented on various versions of Rock-Scissors-Paper and Coordination
games while Section 5 discusses an example of chaotic dynamics under a modi…ed version
of the Logit Dynamics. Section 6 contains concluding remarks.
2
The Logit Dynamics
2.1
Evolutionary dynamics
Evolutionary game theory deals with games played within a (large) population over long
time horizon(evolution scale). Its main ingredients are the underlying normal form game with payo¤ matrix A[n n] - and the evolutionary dynamic class which de…nes a dynamical
5
system on the state of the population. In a symmetric framework, the strategic interaction
takes the form of random matching with each of the two players choosing from a …nite set of
available strategies E = fE1 ; E2; :::En g: For every time t; x(t) denotes the n-dimensional
vector of frequencies for each strategy/type Ei and belongs to the n 1 dimensional simplex
n
P
n 1
= fx 2 Rn :
xi = 1g: The assumption of random interactions proves crucial for
i=1
the linearity of strategy Ei payo¤: this would be simply determined by averaging the
payo¤s from each strategic interaction with weights given by the state of the population
x . Denoting with f (x) the payo¤ vector, its components - individual payo¤ or …tness of
strategy i in biological terms - are:
(1)
fi (x) = (Ax)i
Sandholm (2006) rigorously de…nes an evolutionary dynamics as a map assigning to
each population game a di¤erential equation x_ = V(x) on the simplex
n 1
: In order
to derive such an ’aggregate’ level vector …eld from individual choices he introduces a
revision protocol
ij (f (x); x)
indicating, for each pair (i; j) , the rate of switching(
ij )
from the currently played strategy i to strategy j: The mean vector …eld is obtained as:
x_ i = Vi (x) = in‡ow into strategy i - out‡ow from strategy i
n
n
X
X
=
xj ji (f (x); x) xi
ij (f (x); x)
j=1
(2)
j=1
Based on the computational requirements/quality of the revision protocol
the set of
evolutionary dynamics splits into two large classes: imitative dynamics and pairwise comparison (’competent’ play). The …rst class is represented by the most famous dynamics,
the Replicator Dynamic (Taylor and Jonker (1978) ) which can be easily derived by substituting into (2) the pairwise proportional revision protocol:
ij (f (x); x)
= xj [fj (x) fi (x)]+
(player i switches to strategy j at a rate proportional with the probability of meeting an
6
j-strategist(xj ) and with the excess payo¤ of opponent j-[fj (x) fi (x)]- i¤ positive):
x_ i = xi [fi (x) f (x)] = xi [(Ax)i xAx]
(3)
where f (x) = xAx is the average population payo¤. Although widely applicable to biological/chemical models, the Replicator Dynamics lacks the proper individual choice, microfoundations which would make it attractive for modelling humans interactions. The alternative - Best Response dynamic - already introduced by Gilboa and Matsui (1991)
requires extra computational abilities from agents, beyond merely sampling randomly a
player and observing the di¤erence in payo¤: speci…cally being able to compute a best
reply strategy to the current population state:
x_ i = BR(x)
xi
where,
BR(x) = arg max yf (x)
y
2.2
Discrete choice models-the Logit choice rule
Apart from the highly unrealistic assumptions regarding agents capacity to compute a
perfect best reply to a given population state there is also the drawback that (??) de…nes
a di¤erential inclusion, i.e. a set-valued function. The best responses may not be unique
and multiple trajectory paths can emerge from the same initial conditions. A ’smoothed’
approximation of the Best Reply dynamics - the Logit dynamics - was introduced by
Fudenberg and Levine (1998); it was obtained by stochastically perturbing the payo¤
vector f (x) and deriving the Logit revision protocol:
exp[ 1 Ax)i ]
exp[ 1 fj (x)]
P
P
(f
(x);
x)
=
=
ij
1 f (x)]
1 Ax) ]
k
k exp[
k exp[
k
7
(4)
where
> 0 is the noise level. Here
ij
represents the probability of player i switching to
strategy j when provided with a revision opportunity. For high levels of noise the choice is
fully random(no optimization) while for
close to zero the switching probability is almost
one. This revision protocol can be explicitly derived from random utility model or discrete
choice theory (Mc Fadden (1981), Anderson, de Palma, and Thisse (1992)) by adding
to the payo¤ vector f (x) a noise vector " with a particular distribution, i.e. "i are i.i.d
1
following the extreme value distribution G("i ) = exp( exp(
"i
));
= 0:5772(Euler
constant). The density of this Weibull type distribution is
g("i ) = G0 ("i ) =
1
1
exp(
"i
) exp( exp(
1
"i
))
With noisy payo¤s, the probability that strategy Ei is a best response can be computed
as follows:
i
h
P (i = arg max (Ax)j + "j ) = P [(Ax)i + "i > (Ax)j + "j ]; 8j 6= i
j
= P ["j < (Ax)i + "i
=
Z1
(Ax)j ]j6=i =
Z1
g("i )
exp(
1
"i
) exp( exp(
G((Ax)i + "i
(Ax)j )d"i
j6=i
1
1
Y
1
"i
))
Y
exp( exp(
1
[(Ax)i + "i
j6=i
1
which simpli…es to our logit probability of revision:
ij
exp[ 1 Ax)i ]
P
=
1 Ax) ]
k exp[
k
An alternative way to obtain (4) is to deterministically perturb the set-valued best
reply correspondence (??) with a strictly concave function V (y) (Hofbauer (2000)):
BR (x) = arg max
[y (Ax) + V (y)]
n 1
y2
8
(Ax)j ]
)
For a particular choice of the perturbation function V (y) =
P
i=1;n
yi log yi ; y 2
n 1
the
resulting objective function is single-valued and smooth; the …rst order condition yields
the unique logit choice rule:
exp[ 1 Ax)i ]
BR (x)i = P
1 Ax) ]
k exp[
k
Plugging the Logit revision protocol (4) back into the general form of the mean …eld
dynamic (2) and making the substitution
=
1
we obtain a well-behaved system of
o.d.e.’s, the Logit dynamics as a function of the intensity of choice (Brock and Hommes
(1997) ) parameter :
When
exp[ Ax)i ]
x_ i = P
k exp[ Ax)k ]
xi
(5)
! 1 the probability of switching to the discrete ’best response’ j is close
to one while for a very low intensity of choice ( ! 0) the switching rate is independent
of the actual performance of the alternative strategies (equal probability mass is put on
each of them). The Logit displays some properties characteristic to the logistic growth
function, namely high growth rates(x_ i ) for small values of xi and growth ’levelling o¤’
when close to the upper bound. This means that a speci…c frequency xi grows faster when
it is already large in Replicator Dynamics relative to the Logit dynamics.
3
Hopf and degenerate Hopf bifurcations
As the main focus of the thesis is the detection of stable cyclic behaviour this section
will shortly review the main bifurcation route towards periodicity, the Hopf bifurcation. In a one-parameter family of continuous-time systems, the only generic bifurcation
through which a limit cycle is created or disappears is the non-degenerate Hopf bifurcation. The planar case will be discussed …rst and then, brie‡y, the methods to reduce
higher-dimensional systems to the two-dimensional one. The main mathematical result
9
(see, for example, Kuznetsov (1995)) is:
Assume we are given a parameter-dependent, two dimensional system:
x = f (x; ); x 2 R2 ;
2 R; f smooth and analytic
(6)
with the Jacobian matrix evaluated at the …xed point x = 0 having a pair of purely
imaginary, complex conjugates eigenvalues:
1;2
and ( ) > 0;
= ( )
i!( ); ( ) < 0;
< 0; (0) = 0
> 0:
If , in addition the following genericity2 conditions are satis…ed:
h
i
(i) @ @( )
6= 0 - transversality condition
=0
(ii) l1 (0) 6= 0, where l1 (0) is the …rst Lyapunov coe¢ cient3 - nondegeneracy condition
then the system (6) undergoes an Andronov- Hopf bifurcation at
= 0: Basically, as
is varied from negative to positive the Hopf bifurcation describes two ways in which the
stable focus equilibrium (single pair of complex eigenvalues with negative real part, ( ) <
0 for
< 0) x is losing stability, depending on the sign of the …rst Lyapunov coe¢ cient
l1 (0) :
(a) l1 (0) < 0 then the stable focus x becomes unstable for
> 0 and is surrounded by
an isolated, stable closed orbit(limit cycle).
(b) l1 (0) > 0 then, for
< 0 the basin of attraction of the stable focus x is surrounded
by an unstable cycle which gets smaller and disappears as
crosses the critical value
o
=0
while the system diverges quickly from the neighbourhood of x .
2
Kuznetsov(1995): certain nonzero functions of partial derivatives of f (x; ) with respect to x give the
nondegeneracy condition(i.e. the singularity x is typical/nondegenerate for a class of singularities satisfying certain bifurcation conditions) while the nonzero functions of partial derivatives wrt the parameter
are called the transversality conditions(the parameter unfolds the singularity in a generic way);
3
This is the coe¢ cient of the third order term in the normal form of the Hopf bifurcation
10
In the …rst case the (stable) cycle is created immediately after
reaches the critical
value and thus the Hopf bifurcation is called supercritical, while in the latter the (unstable)
cycle already exists before the critical value, i.e. subcritical Hopf bifurcation (Kuznetsov
(1995)). The supercritical Hopf is also known as a soft or non-catastrophic bifurcation
because , even when the system becomes unstable it still lingers within a small neighbourhood of the equilibrium bounded by the limit cycle, while the subcritical case is a
sharp/catastrophic one as the system now moves quickly away of the unstable equilibrium.
If l1 (0) = 0 then there is a degeneracy in the third order terms of normal form and ,
if other, higher order nondegeneracy conditions hold(i.e. non-vanishing second Lyapunov
coe¢ cient) then the bifurcation is called Bautin or generalized Hopf bifurcation. This
happens when the …rst Lyapunov coe¢ cient vanishes at the given equilibrium x but the
following higher-order genericity4 conditions hold:
(i) l2 (0) 6= 0, where l2 (0) is the second Lyapunov coe¢ cient - nondegeneracy condition
(ii) the map
value
! ( ( ); l1 ( )) is regular(Jacobian matrix is nonsingular) at the critical
= 0 - transversality condition.
At a Bautin point, depending on the sign of l2 (0) the system may display a limit cycle
bifurcating into two or more cycles , coexistence of stable and unstable cycles which collide
and disappear, together with cycle blow-up.
Computation of the …rst Lyapunov coe¢ cient
For the planar case, l1 (0) can be computed without explicitly deriving the normal
form, from the Taylor coe¢ cients of a transformed version of the original vector …eld.
The computation hinges on the Center Manifold Theorem by which the orbit structure
of our original system near (x ;
0)
is fully determined by its restriction to the center
manifold(the manifold spanned by the eigenvectors corresponding to the eigenvalues with
4
technically, the genericity conditions ensure that there are smooth invertible coordinate transformations depending smoothly on parameters together with parameter changes and (possible) time reparametrizations such that (6) can be reduced to a ’simplest’form, the normal form.
11
zero real part). On the center manifold (6) takes the form (Wiggins (2003)):
0
1
0
B x_ C B Re ( )
@ A=@
y_
Im ( )
where
10
1
0
1
1
Im ( ) C B x C B f (x; y; ) C
A@ A + @
A
Re ( )
y
f 2 (x; y; )
(7)
( ) is an eigenvalue of the linearized vector …eld around the steady state and
f1 (x; y; ); f2 (x; y; ) are nonlinear functions of order O(jxj2 ) to be obtained from the original vector …eld. Wiggins (2003) also provides a procedure for transforming (6) into
(7). Speci…cally, for any vector …eld x_ = F (x); x 2 R2 let DF (x ) denote the Jacobian
evaluated at the …xed point x .Then x_ = F (x) is equivalent to:
x_ = Jx + T
1
(8)
F (T x)
with J stands for the real Jordan canonical form of DF (x ); T is the matrix transforming
DF (x ) into the Jordan form, and F (x) = F (x)
0;
1;2
=
l1 ( ) =
DF (x )x: At the Hopf bifurcation point
i! and the …rst Lyapunov coe¢ cient is (Wiggins (2003)):
1 1
1
[f xxx +f 1xyy +f 2xxy +f 2yyy ]+
[f 1xy (f 1xx +f 1yy )
16
16!
f 2xy (f 2xx +f 2yy )
2
2
f 1xx fxx
+f 1yy fyy
]
(9)
4
Three strategy games
In this section we …rst perform a local bifurcation analysis for a classical example of threestrategy games, the Rock-Scissors-Paper. Two types of evolutionary dynamics - Replicator
and Logit Dynamics - is considered while the qualitative change in their orbital structure
is studied with respect to the behavioural ( ) and payo¤ ("; ) parameters. Second, we
provide numerical evidence for a fold bifurcation in the Coordination game under the Logit
Dynamics and depict the fold curves in the parameter space.
12
4.1
Rock-Scissors-Paper
The …rst 3-strategy example we look at is a generalization of the classical game of cyclical
competition, Rock-Scissor-Paper as discussed in Hofbauer and Sigmund(2003) with ; " >
0 . In general, any game can be transformed into a ’simplest ’form -the normal form - by
substracting a constant from each column such that all the elements of the main diagonal
are zeros (Zeeman (1980)):
0
B 0
B
A=B
B "
@
0
"
1
" C
C
C
C
A
0
(10)
Letting x(t) = (x(t); y(t); z(t)) denote the population state at time instance t de…ne a
point from the 2-dimensional simplex, the payo¤ vector [Ax] is obtained via (1):
0
z"
B y
B
[Ax] = B
B x" + z
@
x
y"
1
C
C
C
C
A
Next, the average …tness of the population is:
xAx = x (y
z") + y ( x" + z ) + z (x
y")
With these preliminary computations two types of (evolutionary dynamics) are discussed focusing on the existence of stable cycling within the unit simplex.
4.1.1
Replicator Dynamics
While Hofbauer and Sigmund (2003) employ the Poincare-Bendixson theorem together
with the Dulac criterion to prove that limit cycles cannot occur in games with 3 strategies
under the replicator we will derive this negative result using tools from dynamical systems,
13
in particular the Hopf bifurcation and ’normal form’theory. This toolkit is applied next
to the logit dynamic and a positive result - stable limit cycles do occur - is derived.
The replicator equation (3) with the game matrix (10) induce on the 2-simplex the
following vector …eld:
2
z"
6 x_ = x[y
6
6 y_ = y[ x" + z
6
4
z_ = z[x
y"
(x (y
z") + y ( x" + z ) + z (x
(x (y
(x (y
z") + y ( x" + z ) + z (x
z") + y ( x" + z ) + z (x
3
y"))] 7
7
y"))] 7
7
5
y"))]
(11)
As we are interested in limit cycles within the simplex we consider only interior …xed points
of this system(any replicator dynamic has the simplex vertices as steady states, too). For
the parameter range "; > 0 the barycentrum x = [x = 1=3; y = 1=3; z = 1=3] is always
an interior …xed point of (11). In order to analyze its stability properties we obtain …rst,
by substituting z = 1
x
y into (11), a proper 2 dimensional dynamical system of the
form:
2
3
2
2
2
6 x_ 7 6 x" + 2xy" x + 2x " + x
4 5=4
y
2xy
2y 2 + y 2 " + y 3
y_
3
3
x " + xy
2
2
+x y
y 3 " + xy 2 + x2 y
2
xy "
xy 2 "
2
3
x y" 7
5 (12)
2
x y"
We can detect the Hopf bifurcation threshold at the point where the trace of the Jacobian
matrix
2 of (12) is equal3to zero. The Jacobian evaluated at the barycentrical steady state
p p
"
+" 7
6
3
1
x is 4
,
with
eigenvalues:
(";
)
=
(" + )2 and trace
("
)
i
5
1;2
2
2
"
"
: For < "; x is an unstable focus, at = " a pair of imaginary eigenvalues crosses
p
the imaginary axis( 1;2 = i 3 ), while for > " x becomes a stable focus(see Figure
1 below). This is consistent with Theorem 6 in Zeeman (1980) which states that the
determinant of the matrix A determines the stability properties of the interior …xed point.
In example (10) DetA = "3
3
= 0 (i.e." = ) and, by the same theorem, the vector …eld
(11) has a center in the 2-simplex and a continuum of cycles. Our local bifurcation analysis
14
suggests that a Hopf bifurcation occurs when " =
and in order to ascertain its features -
sub/supercritical or degenerate - we have to further investigate the nonlinear vector …eld
near the (x ; " = ) point. At the Hopf bifurcation necessary condition Re
1;2 (";
) = 0;
vector …eld (12) takes a simpler form:
2
3
2
6 x_ 7 6 x + 2xy + x
4 5=4
y_
y
2xy
y2
2
3
7
5
Using formula (7) we can back out the nonlinear functions f 1 ; f 2 needed for the computation of the …rst Lyapunov coe¢ cient:
2
3
p
x + 2xy + x2 7
6 f1 (x; y) = y 3
4
5
p
2xy
y2
f2 (x; y) = x 3 + y
Lemma 1 All Hopf bifurcations are degenerate in the cyclical Rock-Scissor-Paper game
under Replicator Dynamics.
Proof. From the nonlinear functions f1 (x; y); f2 (x; y) derived above we can compute
the …rst Lyapunov coe¢ cient (9) as l1 ("Hopf =
Hopf
) = 0 which means that there is a
…rst degeneracy in the third order terms from the Taylor expansion of the normal form.
The detected bifurcation is a Bautin or degenerate Hopf bifurcation( assuming away other
higher order degeneracies: technically, the second Lyapunov coe¢ cient should not vanish).
Although, in general, the orbital structure at a degenerate Hopf bifurcation may be
extremely complicated (see Section 3.1), for our particular vector …eld induced by the
Replicator it can be shown by Lyapunov function techniques (Zeeman (1980), Hofbauer
and Sigmund(2003)) that a continuum of cycles is born exactly at the critical parameter
value5 .
5
The absence of a generic Hopf bifurcation does not su¢ ce to conclude that the vector …eld admits no
15
1
1
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.9
0.8
0.7
0.6
y
y
y
1
0.9
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0
0
(a )
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
= 0:6, u n sta b le fo c u s
1
0.1
0
0
0
(b )
0.1
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
= 1, D e g e n e ra te H o p f
0
0.1
(d )
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
= 2, sta b le fo c u s
Figure 1. Replicator Dynamics and RSP game for …xed " = 1 and di¤erent
(a) unstable focus, (b) degenerate Hopf bifurcation for
= 1, (d) stable focus
It would be worth pointing out the connection between our local (in)stability results
and the static concept of Evolutionary Stable Strategy (ESS). As already noted in the
literature (Zeeman (1980), Hofbauer (2000)) ESS implies (global) asymptotic stability
under a wide class of evolutionary dynamics - Replicator Dynamics, Best Response and
Smooth Best Response Dynamics, BNN, etc. The reverse implication does not hold in
general, i.e. the local stability analysis does not su¢ ce to qualify an attractor as an ESS.
However, for
< " we have shown that x is an unstable focus and we can conclude that,
for this class of RSP games, the barycentrum is not an ESS. Indeed, the case
so called bad RSP game which is known not to have an ESS. For
< " is the
> " we are in the good
RSP game and it does have an interior ESS which coincides with the asymptotically stable
rest point x of the Replicator Dynamics. Last, if
= " (i.e. standard RSP) then x is a
neutrally stable strategy/state and we proved that in this zero-sum game the Replicator
undergoes a degenerate Hopf bifurcation.
isolated periodic orbits and ’global’results are required(the Bendixson-Dulac method, positive divergence
of the vector …eld on the simplex)
16
4.1.2
Logit Dynamics [ =
1
]
The logit evolutionary dynamics (5) applied to our normal form game (10) leads to the
following vector …eld:
2
6 x_ =
6
6 y_ =
6
4
z_ =
By substituting z = 1
x
exp( (y
exp( (y z"))
z"))+exp( ( x"+z ))+exp( (x
y"))
exp( (y
exp( ( x"+z ))
z"))+exp( ( x"+z ))+exp( (x
y"))
exp( (y
exp( (x y"))
z"))+exp( ( x"+z ))+exp( (x
y"))
3
x 7
7
y 7
7
5
z
(13)
y into (13) we can reduce initial system to a 2-dimensional
system of equations and solve for its …xed points in the interior of the simplex, numerically,
for di¤erent parameters values ( ; "; ):
2
6
4
The 2
exp( (x
exp( (y "( x y+1)))
y"))+exp( ( x"+ ( x y+1)))+exp( (y
"( x y+1)))
exp( (x
exp( ( x"+ ( x y+1)))
y"))+exp( ( x"+ ( x y+1)))+exp( (y
"( x y+1)))
3
x=0 7
5
y=0
(14)
dim simplex barycentrum [x = 1=3; y = 1=3; z = 1=3] remains a …xed point
irrespective
of the value of 3 . The Jacobian of (14) evaluated at this steady state is:
2
1
1
q
+ 13 " 7
6 3 " 1
3
1
1
1
1 i 2 13 ( + "):The
4
5 ; with eigenvalues: 1;2 = 6 " 6
1
1
1
"
1
3
3
3
Hopf bifurcation (necessary) condition Re(
Hopf
1;2 )
=
= 0 leads to:
6
"
(15)
Lemma 2 The Logit Dynamics (13) on the cyclic Rock-Scissors-Paper game exhibits a
generic Hopf bifurcation and, therefore, has limit cycles. Moreover, all such Hopf bifurcations are supercritical, i.e. the limit cycles are born stable.
Proof. Condition (15) gives the necessary for Hopf bifurcation to occur ; in order to
show that it is non-degenerate we have to compute, according to (9) the …rst Lyapunov
17
coe¢ cient l1 (
Hopf
; "; ) and check whether it is non-zero. The analytical form of this
coe¢ cient takes a complicated expression of exponential terms(see Appendix 1) which,
after some tedious computations, boils down to:
l1 (
Hopf
; "; ) =
864( " + 2 + "2 )
< 0; (8) " >
3(3" 3 )2
> 0:
Computer simulations of this route to cycling are shown in Figure 2 below. We notice
that as
moves up from 10 to 35 (i.e. the noise level is decreasing) the interior stable
steady state looses stability via a supercritical Hopf bifurcation and a small, stable limit
cycle emerges around the now unstable steady state. Unlike Replicator Dynamics, stable
cyclic behavior does occur under the Logit dynamics even for three-strategy games.
1
1
0.5
0.9
0.9
0.45
0.8
0.8
0.7
0.7
0.5
0.4
0.4
0.3
0.3
0.2
0.2
y
0.6
0.5
y
y
0.4
0.6
0.35
0.3
0.25
0.1
0.1
0
0
0.1
(a )
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
= 1 0 , sta b le fo c u s
0.9
1
0
0
0.1
(c )
0.2
0.3
0.4
0.5
x
0.6
0.7
0.8
0.9
1
0.2
0.2
= 3 0 , g e n e ric H o p f
Figure 2. Logit Dynamics and RSP. …xed " = 1;
0.25
(c )
0.3
0.35
x
0.4
0.45
0.5
= 3 5 , lim it c y c le
= 0:8, free
(a) stable focus, (b) generic Hopf bifurcation, (c)-stable limit cycle
Similar periodic behaviour can be detected in the payo¤ parameter space as in Figure
18
3 below where the noise level is kept constant.
(a)
=1
(b)
= 0:3999
Figure 3. Logit Dynamics and RSP. …xed " = 1;
(c)
= 0:1
= 10; free :
(a), stable focus, (b) supercritical Hopf bifurcation, (c)-stable limit cycle
Figure 4 depicts depicts curves of Hopf bifurcations in the ( ; ") and ( ; ) parameter
space(…xing the third parameter to a speci…c level). As we cross these Hopf curves from
below the stable interior …xed point loses stability and a stable periodic attractor surrounds
it.
30
25
stable
limit cycle
beta
20
stable
limit cycle
15
10
stable
focus
5
0
0
0.2
0.4
0.6
0.8
1
1.2
delta
2
4
6
8
10
eps
F ig u re 4 . L o g it D y n a m ic s a n d R S P : S u p e rc ritic a l H o p f c u rve s in (
19
; , ; ")
sp a c e
4.2
Coordination game
Using topological arguments, Zeeman (1980) shows that games have at most one interior
isolated …xed point under Replicator Dynamics(Theorem 3 in Zeeman(1980)). In particular, fold catastrophe (two isolated …xed points which collide and disappear when some
parameter is varied) cannot occur within the simplex. In this section we provide - by means
of the classical coordination game - numerical evidence for the occurrence of multiple, isolated interior steady-states under the Logit Dynamics and show that fold catastrophe is
possible when we alter the intensity of choice . The coordination game we consider is
given by the following payo¤ matrix:
0
0
B 1 " 0
B
A=B
1
0
B 0
@
0
0 1+"
4.2.1
1
C
C
C ; " 2 (0; 1)
C
A
Replicator Dynamics
Vector …eld: x_ = x[Ax
xAx]
2
2
2
2
6 x_ = x[x ( " + 1) (y + z (" + 1) + x ( " + 1))]
6
6
y_ = y[y (y 2 + z 2 (" + 1) + x2 ( " + 1))]
6
4
z_ = z[z (" + 1) (y 2 + z 2 (" + 1) + x2 ( " + 1))]
This systems has the following …xed points in simplex coordinates:
(a) simplex vertices(stable nodes): (1; 0; 0); (0; 1; 0); (0; 0; 1)
(b) interior(source): O[x =
"+1
;y
3 "2
=
1 "2
;z
3 "2
(b) on the boundary(saddles):
M ( 2 1 " ; 12 "" ; 0); N ( 2 1 " ; 0; 21 "" ); P (0; 2 1 " ; 12 "" )
20
=
1 "
]
3 "2
3
7
7
7
7
5
(16)
Local stability analysis:
A [x = 0; y = 1; z = 0] : eigenvalues:
1
B[x = 1; y = 0; z = 0] : eigenvalues: "
C [x = 0; y = 0; z = 1]: eigenvalues:
O[x =
"+1
;y
3 "2
=
1 "2
;z
3 "2
=
1 "
]
3 "2
M ( 2 1 " ; 12 "" ; 0): eigenvalues:
1
"
1
: eigenvalues:
"2 1
"2 3
> 0; for " 2 (0; 1)
1 "
" 2
N ( 2 1 " ; 0; 12 "" ): eigenvalues:
1
4"+"2 +4
(2" + "2
P (0; 2 1 " ; 12 "" ): eigenvalues:
1
4"+"2 +4
(" + "2
"3
"3
2)
2)
This …xed points structure is consistent with Zeeman(1980) result that no fold catastrophes(multiple isolated interior steady states) can occur under the Replicator Dynamics.
The three saddles together with the interior source de…ne the basins of attractions for the
simplex vertices. The boundaries of the simplex are invariant under Replicator Dynamics(proof in Hofbauer and Sigmund(2003)) and it su¢ ces to show that the segment lines
[OM ]; [ON ], and [OP ](Figure 5) are also invariant under this dynamics.
A
A (1,0,0)
B (0,1,0)
C (0,0,1)
N
M
O
B
P
F ig u re 5 . R e p lic a to r: inva ria nt m a n ifo ld s
21
C
Lemma 3 The segment lines [OM ]; [ON ], and [OP ] are invariant under Replicator Dynamics and they form, along with the simplex edges, the basins of attraction boundaries
for the three stable steady states.
Proof. Using the standard substitution z=1-x-y system (16) becomes:
2
2
3
2
2
2
3
2
3
x
xy
x " + x " x ( x y + 1)
x" ( x y + 1) 7
6 x_ = x
4
5
y_ = y 2 y 3 x2 y + x2 y" y ( x y + 1)2 y" ( x y + 1)2
Segment [M O] is de…ned, in simplex coordinates, by y = x(1
").
(17)
Along this line we
have:
y_
x_
=
x ( " + 1) x ( " + 1)
[M O]:y=x(1 ")
= 1
x2 ( " + 1)
x x ( " + 1)
";
x2 ( " + 1)2
x2 ( " + 1)
x2 ( " + 1)2
(" + 1) ( x
(" + 1) ( x
y + 1)2
which is exactly the slope of [M O]. Similarly, invariance results are obtained for segments
[ON ] and [OP ] de…ned by x(1
") = z(1 + ") and y = z(1 + "):
Analytically, the basins of attraction are determined by the areas of the polygons delineated by the invariant manifolds [OM ]; [ON ], and [OP ] and the 2-dim simplex boundaries,
as follows:
2
6
6
A(1; 0; 0) = A[AM ON ] = 6
6 +
4
B(0; 1; 0) = A[BM OP ] =
C(0; 0; 1) = A[CN OP ] =
1
4
1
4
p
p
p
p
p
p
3 63 2""2 14 2 3" 63 2""2
p
p
p "+1
3
1
3 2"
1
1
3 (" + 1) (4 2")(3
+
3 3 "2
4
"2 )
4 4 2"
8
p
p
p
1 3 3 "
"+1
+ 14 3 63 2""2 + 14 3 (4 2")(3
2
" )
4 2 " 6 2"2
p
1 3
82 "
3 (4
1
4
3 34 2"
2"
"+1
2")(3 "2 )
3("+1)(1 ") (1+")
(2 ")(3 "2 )
22
1
8
3 3"+1
"2
p
1 3 3 "
4 2 " 6 2"2
1
4
p
+ 14
p
1 3
82 "
+
+
3 3"+1
"2
1
4
p
1
4
3 (" + 1) (2
p
3 (" + 1) (4
1 "
")(3 "2 )
3 2"
2")(3 "2 )
y + 1)2
3
7
7
7
7
5
The basins’sizes vary with the payo¤ perturbation parameter " :
A(1; 0; 0) B(0; 1; 0) C(0; 0; 1) Long-Run Average Fitness/Welfare
"nA
0
33:33%
33:33%
33:33%
1
0:1
29:9%
33:2%
36:7%
1: 0048
0:5
12:92%
33:76%
53:32%
1: 1986
0:9
0:7%
34:21%
65:03%
1: 6427
1
0%
33:33%
66:66%
1:666
When the payo¤ asymmetries are small, the simplex of initial conditions is divided
equally among the three …xed points/equilibria(Figure 8, panel (a)). As the payo¤ discrepancies increase most of the initial conditions are attracted by the welfare-maximizing
equilibrium (0; 0; 1).(Figure 6, panel (b).)
Coordination Game, Replicator Dynamics [eps=0.1]
Coordination Game, Replicator Dynamics [eps=0.8]
0.9
0.9
(1,0,0)--29%
(0,1,0)--33.31%
(0,0,1)--37.69%
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
0.8
(1,0,0)--3%
(0,1,0)--34.4%
(0,0,1)--62.6%
0.8
0
1
0
(a) A" [" = 0:1]
0.2
0.4
0.6
0.8
1
(b) A" [" = 0:8]
Figure 6. Coordination Game and Replicator Dynamics-basins of attraction
4.2.2
Logit Dynamics
For a particular payo¤ perturbation [" = 0:1] the Logit Dynamics together this payo¤
matrix de…ne the following vector …eld on the simplex of frequencies [x; y; z]:
23
2
6 x_ =
6
6 y_ =
6
4
z_ =
exp(0:9 x)
exp(0:9 x)+exp( y)+exp(1:1 z)
exp( y)
exp(0:9 x)+exp( y)+exp(1:1 z)
exp(1:1 z)
exp(0:9 x)+exp( y)+exp(1:1 z)
3
x 7
7
y 7
7
5
z
In order to ascertain the number of (asymptotically) stable …xed points on the twodimensional simplex we run simulations for increasing values of
and for di¤erent initial
conditions within the simplex:
Case I.
0: One interior steady state (initial conditions, time series and attracting
point in Figure 7, panel (a)). The barycentrum is globally attracting, i.e. irrespective of
the initial proportions, the population will settle down to a state with equalized fractions.
Case II.
of the simplex as
10: Three stable steady states asymptotically approaching the vertices
increases (Figure 7, panels (b)-(d)) plus other unstable steady states.
The size of their basins of attraction is determined both by the strategy relative payo¤
advantage and by the value of the intensity of choice: This is to say that, whichever of the
three strategy types - E1 ; E2 ; E3
happens to outnumber the other two, it will eventually
24
win the evolutionary competition.
(a)
(c)
=0, any (x,y,z)–>(1/3,1/3,1/3)
=10, (0.25,0.4,0.35)–> (0,1,0)
(b)
(d)
=10, (0.4,0.3,0.3) –> (1,0,0)
=10, (0.33,0.33,0.34)–> (0,0,1)
Figure 7. Coordination game with extreme levels of noise— Time series: x/blue, y/red, z/green
(a) -unique interior stable steady state, (b)-(d)-three stable steady states
25
(a)
(b)
=3, (0.2,0.6,0.2)–> (0.1,0.8,0.1)
(c)
=2.6, (0.8,0.1,0.1)–>
=3, (0.5,0.3,0.2)–> (0.06,0.06,0.88)
(d)
(0.11,0.11,0.78)
=2.6, (0.4,0.5,0.1)–>
(0.11,0.11,0.78)
Figure 8. Coordination game with moderate levels of noise— x/blue, y/red, z/green
(a)-(b)- two interior, isolated stable steady states; (c)-(d)-one stable steady state
The most interesting situations occur in transition from high to low values of the
behavioral parameter : we conjecture that the Logit Dynamics undertakes a sequence of
fold bifurcations by which each of the three stable …xed points collide with other unstable
steady states from within the simplex and disappear. First, for
= 3 the previously stable
(1; 0; 0) steady state disappears as all orbits are attracted by two stable …xed points close
26
to the other vertices: if we start in a region where type E2 is relatively more frequent
then we converge to the …xed point close to the (0; 1; 0) vertex ( Figure 8, panel (b)). All
other orbits (including the ones originating in a region with relatively higher frequency of
type E1 !)end up in a steady state close to (0; 0; 1) ( Figure 8, panel (a)).Next, if we further
decrease
a second fold catastrophe occurs and we are left with only one stable …xed point.
Panels (c), (d) in Figure 8 above show trajectories starting from initial conditions where
type E2 and E1 largely outnumber E3 strategists but, they are nonetheless driven out by
the evolutionary dynamics as the unique, interior point attractor lies in the neighbourhood
of the (0; 0; 1) vertex. Interestingly, with moderate levels of rationality the population
ends up coordinating close to the Pareto-optimal equilibrium, irrespective of the initial
distribution of frequencies.
Proposition 4 Unlike Replicator, the Logit Dynamics displays multiple, interior isolated
steady states created via a fold bifurcation. In the particular case of a 3-strategy Coordination game, three interior stable stady states emerge through a sequence of two saddle-node
bifurcations.
Figure 9 depicts the fold bifurcations scenario by which the multiple, interior …xed
points appear when the intensity of choice(Panel (a)) or the payo¤ parameter(Panel (b))
changes. For small values of
the unique, interior stable steady state is the barycentrum
(1=3; 1=3; 1=3). A …rst fold bifurcation occurs at
appear, one stable and one unstable. If we increase
= 2:77 and two new …xed points
even further (
3:26) a second
fold bifurcation takes place and two additional equilibria emerge. Last, two new …xed
points arise at
= 4:31 via a saddle-source bifurcation. A similar pattern is visible in
the payo¤ parameter space (Panel (b)) where a family of fold bifurcations is obtained for
27
di¤erent values of the switching intensity.
1
1
0.9
0.9
0.8
0.8
LP
0.7
LP
0.7
H
0.6
H
H
H
H
0.5
x
x
0.6
LP
H
0.4
H
0.4
0.3
2.7 3.2
4.3
LP
H
LP
H
2
0.2
H
H
0.1
0
0
0.3
H
LP
0.2
beta
6
LP
0.5
8
10
(a) (x, ) "=0.1
0.1
0
0
0.1
0.2
0.3
0.4
0.5
eps
(b) (x,")
0.6
0.7
0.8
0.9
1
=5
Figure 9. Equilibria curves as function of model parameters
The continuation of the fold curves in the ( ; ") parameter space allows the detection of
codimension II bifurcations(cusp points-CP). The cusp points in Figure 10 below organize
the entire bifurcating scenario which consists in the emergence of multiple, interior pointattractors in Coordination Game. For low vales of the intensity of choice
there is only
one (logit) equilibrium and, as we cross each of the successive fold curves from below, two
28
additional equilibria arise.
9
7
8
3.6
7
7
3.4
6
ZH
5
5
beta
beta
ZH
5
3.2
4
3
3
1
2.8
CP
CP
2
1
0
-0.4
ZH
ZH
3
3
CP
2.6
-0.3
-0.2
-0.1
0 eps 0.1
0.2
0.3
0.4
(a) three curves of fold bifurcations
-0.05
-0.03
-0.01
CP
eps 0.01
1
0.03
0.05
(b) cusp points and multiple equilibria
Figure 10. Continuation of the three fold points in the parameter space ( ,")
The numerical computation of the basins of attraction for di¤erent equilibria reveals
interesting properties of the Logit dynamics from an social welfare perspective. While
for extreme values of the switching parameters the basins of attraction are similar in size
with the Replicator Dynamics (Panels (a), (c), (d) in Figures 11, 12), for moderate levels
of rationality the population manages to coordinate close to the Pareto optimal Nash
29
equilibria (Panel (b) in Figures 11, 12 ).
(1,0,0)
(1,0,0)
(1/3,1/3,1/3)--100%
(0,1,0)
(0,0,1)
(a )
(0.13,0.14,0.73)--100%
(0,1,0)
=1
(1,0,0)
(0,0,1)
(b )
= 10=4
(0.05,0.89,0.06)--76.9%
(0.03,0.03,0.94)--23.1%
(1,0,0)
(1,0,0) - 29%
(0,0,1) - 37.5%
(0,1,0) - 33.5%
(0,0,1)
(0,1,0)
(c )
(0,1,0)
= 10=3
(0,0,1)
(d )
= 15
Figure 11. Basins of Attraction: Coordination Game["=0.1] and Logit Dynamics
30
(1,0,0)
(1,0,0)
( 1 /3 , 1 /3 , 1 /3 ) - - 1 0 0 %
(0,1,0)
(0,0,1)
(0.06,0.06,0.88)--100%
(0,1,0)
= 0:5
(a )
(1,0,0)
(0,0,1)
(b )
=2
(1,0,0)
(0.04,0.91,0.05)-25%
(1,0,0)-5%
(0,0,1) - 60%
(0,1,0) - 35%
(0.001,0,0.999)-75%
(0,1,0)
(0,0,1)
(c )
0,1,0)
=4
(0,0,1)
(d )
= 15
Figure 12. Basins of Attraction: Coordination Game["=0.6] and Logit Dynamics
We construct an long-run aggregate welfare index W as an weighted average of steady
states welfare with weights given by the set of initial conditions converging to the respective
steady state(i.e. its basin of attraction). W evolves non-monotonically with respect to the
behavioural parameter
(Figure 13, (a), (b)): …rst increases as the fully mixed equilibrium
slides slowly towards the (0; 0; 1) vertex, attains a maximum before the fold bifurcation
occurs
LP1
welfare in
LP1
= 2:77 and then decreases approaching the Replicator Dynamics average
! 1 limit. Intuitively, there are two e¤ects driving the welfare peak before
value is hit: on the one side the welfare at the steady state which is higher the
closer the steady state is to the Pareto optimal equilibrium (0; 0; 1) and, on the other side,
the relative size of the basins of attractions of each stable steady state (for
31
<
LP1
the
entire simplex of initial conditions is attracted by the unique steady state lying close to the
aforementioned optimal equilibrium). However, when
is large enough, rational agents are
locked into rational (given initial mixtures of the population), albeit payo¤ sub-optimal
choices. In the limiting case
! 1; the …xed points of the Logit Dynamics (i.e. the
logit equilibria) coincide with the Nash equilibria of the underlying game which, for this
Coordination Game, are exactly the …xed points of the Replicator Dynamics. Thus the
analysis - stable …xed points, basins’of attraction sizes - for the ’unbounded’rationality
case is identical to the one pertaining to the Replicator Dynamics in Coordination game(see
subsection 4.2.1).
Coordination Game-Welfare
1.9
0.1
0.3
0.5
0.6
0.7
0.9
1.8
1.7
1.5
2
1.4
welfare
welfare
1.6
1.3
0.1
1.5
0.3
1.2
1
15
1.1
1
0
2.77
5
(a ) We lfa re -
beta
10
p lo t fo r d i¤e re nt
0.5
0.6
0.7
10
4.31 2.77
15
beta
"
(b ) su rfa c e p lo t [
0.9
eps
0
W, ; "]
Figure 13. Coordination Game-Long-Run Welfare
5
Weighted Logit Dynamics(iLogit)
In this last section we run computer simulations for the three-strategy Rock-Scissor-Paper
game and Schuster et al. (1991) example- from the perspective of a di¤erent type of
evolutionary dynamic closely related to the Logit dynamic: Weighted Logit[ =
xi exp[ 1 Ax)i ]
x_ i = P
1 Ax) ]
k xi exp[
k
32
xi
1
]
(18)
This evolutionary dynamic has the property that when
approaches 0 it converges to
the Replicator Dynamics (with adjustment speed scaled down by a factor ) and when
the intensity of choice is very large it approaches the Best Response dynamic (Hofbauer
and Weibull (1996)).
A) Rock-Scissors-Paper and iLogit Dynamics.
2
6 x_ =
6
6 y_ =
6
4
z_ =
x exp[z " y
x exp[z " y
]+y exp[x " z
]
]+z exp[y " x
]
x exp[z " y
y exp[x " z
]+y exp[x " z
]
]+z exp[y " x
]
x exp[z " y
z exp[y " x
]+y exp[x " z
]
]+z exp[y " x
]
E&F Chaos simulations:(x
x 7
7
y 7
7
5
z
y subspace, Runtime 1500-2500):
33
3
(a)
(c)
= 0:1; " = 1
= 30; " = 1
(b)
= 2; " = 1
(d)
= 50; " = 1
Figure 14. RSP and Weighted Logit for …xed " and free
(a)-(b) stable limit cycles, (c) stable steady state, (d) Best response triangle
B) Schuster et al.(1991) and iLogit Dynamics.
Payo¤ matrix:
0
B
B
B
B
A=B
B
B
@
0
0:5
1:1
0
0:5
1:7 +
1
1
34
1
0:1 0:1 C
C
0:6 0 C
C
C
0
0 C
C
A
0:2 0
(19)
The weighted logit with the underlying game (19) generate the following vector …eld
on the 4-simplex:
2
6 x_ =
6
6 y_ =
6
6
6 z_ =
6
4
t_ =
x exp[ (0:1t+0:5y 0:1z)]
x exp[ (0:1t+0:5y 0:1z)]+y exp[ (1: 1x 0:6z)]+z exp[ ( 0:5x+y)]+t exp[ ( 0:2z+y(
1)+x( +1: 7))]
y exp[ (1: 1x 0:6z)]
x exp[ (0:1t+0:5y 0:1z)]+y exp[ (1: 1x 0:6z)]+z exp[ ( 0:5x+y)]+t exp[ ( 0:2z+y(
1)+x( +1: 7))]
z exp[ ( 0:5x+y)]
x exp[ (0:1t+0:5y 0:1z)]+y exp[ (1: 1x 0:6z)]+z exp[ ( 0:5x+y)]+t exp[ ( 0:2z+y(
1)+x( +1: 7))]
t exp[ ( 0:2z+y(
1)+x( +1: 7))]
x exp[ (0:1t+0:5y 0:1z)]+y exp[ (1: 1x 0:6z)]+z exp[ ( 0:5x+y)]+t exp[ ( 0:2z+y(
1)+x( +1: 7))]
Schuster et al.(1991) show that, within a speci…c payo¤ parameter region [
<
3
x 7
7
y 7
7
7
z 7
7
5
t
0:2 <
0:105]; a Feigenbaum sequence of period doubling bifurcations unfolds under the
Replicator Dynamics and, eventually, chaos sets in. Here we consider the question whether
the weighted version of the Logit Dynamics displays similar patterns, for a given payo¤
perturbation
; when the intensity of choice
is varied. First we …x
to
0:2(the
parameter value for which the Replicator generates ’only’ periodic behaviour) and run
computer simulations for di¤erent :Cycles of increasing period multiplicity are reported
in Figure 15 below.
35
(a)
(c)
= 0:085 period 1 cycle
(b)
= 1 period 2 cycle
= 2:1 period 4 cycle
(d)
= 2:2 period 8 cycle
Figure 15. Road to chaos. Cascade of period-doubling bifurcations, projections onto the x-y state subspace
For
= 2:5 the iLogit Dynamics already enters the chaotic regime on a strange at-
tractor(Figure 16).
36
(a) x-y
(b) x-z
(c) x-t
(d) y-z
Figure 16. Schuster et al.(1991) and Weighted Logit dynamics[
=
0:2;
= 2:5]
(a)-(d)-projections of the strange attractor onto two-dimensional state subspaces
6
Conclusions
The main goal of this paper was to show that, even for ’simple’ three-strategy games,
periodic attractors do occur under a rationalistic way of modelling evolution in games,
the Logit dynamics. The resulting dynamical systems were investigated with respect to
changes in both the payo¤ and behavioral parameters. Then, identifying stable cyclic
37
behaviour in such a system translates into proving that the Hopf bifurcations are nondegenerate. Consequently, by means of normal form computations, we showed …rst that
this is not the case for Replicator Dynamics, when the number of strategies is three for
games like Rock-Scissors-Paper. In these games, under the Replicator Dynamics, only a
degenerate Hopf bifurcation can occur. However, the Logit dynamics is generic even for
three strategy case and stable cycles are created, via a supercritical Hopf bifurcation. Another …nding is that the periodic attractors can be generated either by varying the payo¤
parameters ("; ) or the intensity of choice ( ). Moreover, via computer simulations on
the Coordination game, we showed that the Logit may display multiple, isolated, interior
steady states together with the fold catastrophe, a bifurcation which is known not to occur under the Replicator. Last, in case of the Weighted Logit dynamics we were able to
detect strange attractors and a period-doubling route to chaos in the intensity of choice
parameter space.
7
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Appendix 1 - Rock-Scissor-Paper game with Logit Dynamics: Non-linear
functions for the computation of the …rst Lyapunov coe¢ cient
exp
p "+
f1 (x; y) = y 3
"
f2 (x; y) =
x+
exp
p "+
x 3
"
6(x
y")
+"
exp
6(x
y")
+"
"( x y+1))
+"
6( x"+ ( x y+1))
+"
+ exp
exp
y+
6(y
+ exp
+ exp
6(y
"( x y+1))
+"
6( x"+ ( x y+1))
+"
6( x"+ ( x y+1))
+"
+ exp
6(y
"( x y+1))
+"
Next, applying formula (9) for the computation of the …rst Lyapunov coe¢ cient, we
obtain after some further simpli…cations:
l1 ("; ) =
1728 "
4320 2 4320"2 4320 " + 1728 2 + 1728"2
19 2 38 " + 19"2 16 " + 8 2 + 8"2
2592 " 2592 2 2592"2
=
27 2 54 " + 27"2
" + 2 + "2
= 2592
3(3" 3 )2
41