Coexistence equilibria of evolutionary games on graphs under deterministic imitation dynamics

Abstract

Cooperative behaviour is often accompanied by the incentives to defect, i.e., to reap the benefits of others' efforts without own contribution. We provide evidence that cooperation and defection can coexist under very broad conditions in the framework of evolutionary games on graphs under deterministic imitation dynamics. Namely, we show that for all graphs there exist coexistence equilibria for certain game-theoretical parameters. Similarly, for all relevant game-theoretical parameters there exists a graph yielding coexistence equilibria. Our proofs are constructive and robust with respect to various utility functions which can be considered. Finally, we briefly discuss bounds for the number of coexistence equilibria.

Full text

DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2016.21.803
DYNAMICAL SYSTEMS SERIES B
Volume 21, Number 3, May 2016 pp. 803–813
COEXISTENCE EQUILIBRIA OF EVOLUTIONARY GAMES ON
GRAPHS UNDER DETERMINISTIC IMITATION DYNAMICS
Jeremias Epperlein and Stefan Siegmund∗
Center for Dynamics & Institute for Analysis
Dept. of Mathematics
Technische Universit¨at Dresden, 01062
Dresden, Germany
Petr Stehl
´
ık and Vladim
´
ır ˇ
Sv
´
ıgler
Dept. of Mathematics and NTIS
Faculty of Applied Sciences
University of West Bohemia
Univerzitni 8, 30614 Pilsen
Pilsen, Czech Republic
(Communicated by Peter E. Kloeden)
Abstract. Cooperative behaviour is often accompanied by the incentives to
defect, i.e., to reap the benefits of others’ efforts without own contribution.
We provide evidence that cooperation and defection can coexist under very
broad conditions in the framework of evolutionary games on graphs under de-
terministic imitation dynamics. Namely, we show that for all graphs there
exist coexistence equilibria for certain game-theoretical parameters. Similarly,
for all relevant game-theoretical parameters there exists a graph yielding co-
existence equilibria. Our proofs are constructive and robust with respect to
various utility functions which can be considered. Finally, we briefly discuss
bounds for the number of coexistence equilibria.
1. Introduction. Cooperative behaviour in complex systems and natural networks
is an exciting phenomenon occurring in groups of cells, animals [4,6,11] and,
most importantly, human societies [1], social organizations and related networks
[10]. Increased levels of cooperation can lead to advanced organizational structures.
It has been suggested that cooperation is the third fundamental driving force of
evolution besides mutation and natural selection [12]. Naturally, in most cases,
cooperative actions are accompanied by the presence of defective ones (i.e., free-
riding behaviour in which individuals collect the benefits of cooperation of others
without contributing themselves) and both coexist in various forms [7,9]. The
goal of this paper is to formally show that in a simple framework of evolutionary
games on graphs one can easily observe omnipresence of configurations in which
cooperation and defection coexist.
In standard evolutionary game theory [8,9], infinite homogeneous populations
are considered and cooperation and defection coexist in the case of Stag hunt and
2010 Mathematics Subject Classification. 05C90, 37N25, 37N40, 91A22.
Key words and phrases. Evolutionary games on graphs, game theory, coexistence, equilibrium,
cooperation.
∗Corresponding author.
803

804 EPPERLEIN, SIEGMUND, STEHL´
IK AND ˇ
SV´
IGLER
Hawk and dove games (whereby the coexistence equilibrium is unstable in the for-
mer and stable in the latter case). In recent years, numerous studies of finite and
heterogenous populations (modelled by evolutionary games on graphs) revealed that
the introduction of spatial structure could extend the areas of coexistence of coop-
eration and defection to other social-dilemma games, especially prisoner’s dilemma
[5,7,13,14,15,16]. We contribute to this line of research and provide construc-
tive proofs showing that, under deterministic imitation dynamics, for every social
dilemma parameter cooperation and defection can coexist. Similarly, we show that
for each graph/network there exist game-theoretical parameters such that cooper-
ation and defection can coexist (we find these parameters in SH and FC parameter
regions). Finally, in order to quantify the ubiquity of the states in which cooper-
ation and defection coexist, we construct a specific class of graphs such that the
number of coexistence equilibria grows exponentially with the number of vertices of
underlying graphs.
The paper is organized as follows. Firstly, in Section 2we introduce our formal
model of an evolutionary game on a graph as well as the concept of coexistence
equilibria. Next, in Section 3, we study the stationary boundaries of clusters of
cooperators and defectors. Based on this knowledge, we are able to show in Section
4that coexistence equilibria (or fixed points) exist for all social-dilemma game
parameters as well as for all graphs. Finally, we estimate the possible number of
coexistence equilibria on graphs in Section 5. In Section 6we provide a robustness
analysis for our constructions to show that they could be used also if another utility
function is considered. We conclude with final remarks and open problems in Section
7.
2. Evolutionary games on graphs. We consider undirected graphs whose ver-
tices represent players and the edges/links represent the interaction between the
players. Each player can either cooperate Cor defect D, the set of possible states
for each vertex is thus S={D, C }={0,1}. We use the following notation for
neighbourhoods on graphs. N1(i) denotes all vertices with distance 1 from vertex
iand N≤1(i) includes all vertices whose distance is at most one (similarly N2(i)
denotes all vertices with distance 2 etc.).
In each time step each vertex (player) determines its utility ufrom interactions
with its neighbours. This utility is given by the underlying game-theoretical pa-
rameters:
C D
Ca b
Dc d
Based on the values of its utility and the utilities of its neighbours it then chooses
its next state (following a dynamical rule ϕ).
Putting those ideas together, we can formulate formally an evolutionary game
on a graph as a dynamical system in the following way (see [5] for more details):
Definition 2.1. An evolutionary game on a graph is a quintuple (G, p, u, T, ϕ),
where
(a) G= (V, E ) is a connected graph,
(b) p= (a, b, c, d) are game-theoretical parameters,
(c) u:SV→RVis a utility function,
(d) T:N0→2Vis an update order,
(e) ϕ: (N0)2
≥×SV→SVis a (generally nonautonomous) dynamical system.

COEXISTENCE EQUILIBRIA OF EVOLUTIONARY GAMES 805
FC
HD
SH
PD
-1.0 -0.5 0.0 0.5 1.0
0.0
0.5
1.0
1.5
2.0
b
c
Figure 1. Game-theoretical parameters b,dand underlying
social-dillemas (for a= 1, d= 0). Stability diagrams show the
standard replicator dynamics on the interval [0,1] for spatially ho-
mogenenous and well-mixed populations. Coexistence equilibria
exist only in Stag-hunt (unstable, white) and Hawk and dove (sta-
ble, black).
Remark 1. (a) We assume that the game-theoretical parameters p= (a, b, c, d)
satisfy min{a, c}>max{b, d}, i.e. we consider the so-called social-dilemmas
which include Prisoner’s dilemma (PD, c>a>d>b), Stag hunt (SH, a >
c>d>b), Hawk and dove (HD, c>a>b>d) and Full cooperation (FC,
a>c>b>d), see Figure 1. Without loss of generality one could assume that
a= 1 and d= 0, see [5, Remark 9] for details.
(b) There are two most common choices of utility functions, either the aggregate
utility
uA
i(x) =aX
j∈N1(i)
xixj+bX
j∈N1(i)
xi(1 −xj) + cX
j∈N1(i)
(1 −xi)xj
+dX
j∈N1(i)
(1 −xi)(1 −xj),(1)
for x∈SVor the mean utility
uM
i(x) = 1
|N1(i)|uA
i(x).(2)
In the case of regular graphs, the dynamics is the same, but it differs for irregular
graphs, see [5].
(c) Two major examples of update orders (for others see [5, Section 5]) are syn-
chronous (T(t) = Vfor each t∈N0), and sequential (vertices can be ordered

806 EPPERLEIN, SIEGMUND, STEHL´
IK AND ˇ
SV´
IGLER
so that T(t) = {(t+ 1)(mod n)}). However, in this paper we deal with fixed
points and our results apply to any update order.
(d) In this paper we use the (deterministic) imitation dynamics ϕID , in which a
vertex follows the strategy in its 1-neighbourhood which currently yields the
highest utility. For other dynamics (birth-death, death-birth), see [5, Remark
4].
Mathematically, we define ϕID via its components ϕI D
i:= proji◦ϕ: (N0)2
≥×
SV→Sby
ϕID
i(t+ 1, t, x) = (xmax if i∈ T (t),|Ai(x)|= 1 and Ai(x) = {xmax},
xiotherwise,(3)
where Ai(x) is the set of strategies in the neighbourhood of xwhich yield the
highest utility and is given by
Ai(x) = {xk:k∈argmax {uj(x) : j∈N≤1(i)}} .(4)
The cardinality of Ai(x) is used to ensure that all vertices with the highest
utility have the same state. If that is not the case, the vertex preserves its
current state (in order to keep the dynamics deterministic).
In this paper we study states in which cooperation and defection coexist and
which remain unchanged by the dynamics ϕ.
Definition 2.2. We say that a state x∈SVis a coexistence equilibrium (coexis-
tence fixed point) of the evolutionary game on a graph (G, p, u, T, ϕ) if
(a) it is a fixed point, i.e., ϕ(t+ 1, t, x) = xfor all t∈N0,
(b) it is a coexistence state, i.e., 0 <Pi∈Vxi<|V|.
The following observation enables us to easily consider all update orders Tat
once.
Lemma 2.3. Let T:N0→2Vbe the synchronous update order, i.e., T(t) = Vfor
all t∈N0. If a state x∈SVis a coexistence equilibrium of the evolutionary game
on a graph (G, p, u, T, ϕ)then it is a coexistence equilibrium of any evolutionary
game (G, p, u, ˜
T, ϕ), where ˜
Tis an arbitrary update order.
Proof. Indeed, if ϕ(t+1, t, x) = xin the synchronous case, all vertices i∈V=T(t)
preserve their strategy xi. Thus, if ˜
T(t)⊂V(only a subset of vertices is being
updated) we have ϕ(t+ 1, t, x) = xas well.
Remark 2. (i) Consequently, we consider the synchronous update order through-
out the paper and shorten the nonautonomous notation ϕ(t+ 1, t, x) and write
autonomously ϕ(x) instead. However, note that the dynamical properties like
the stability of fixed points (which we do not study in this paper) need not be
preserved when we move from the synchronous (autonomous) to the general
(nonautonomous) update order, see examples in [5].
(ii) Following the idea of the proof, we could replace synchronous update order
by any fair update order in Lemma 2.3. An update order T:N0→2Vis fair
if for each vertex v∈Vand each time t0∈N0there exists t > t0such that
v∈ T (t) (i.e., vis updated at time t). See [5, Definition 15] for more details.

COEXISTENCE EQUILIBRIA OF EVOLUTIONARY GAMES 807
3. Cluster boundaries. In a deterministic imitation dynamics ϕI D which we in-
troduced above, the vertices which only have neighbours with the same state never
change their strategy. Consequently, we introduce clusters of cooperators and de-
fectors and study their boundaries on which the change of strategies could occur,
see Figure 2for the illustration and note that the importance of clusters has been
already pointed out [13]. For a given state x∈SVwe introduce the sets of inner
cooperators (abbreviated by IC, cooperators with only cooperative neighbours) and
inner defectors (ID, defectors with only defective neighbours)
VIC (x) := {i∈V:xi= 1,and xj= 1 for all j∈N1(i)},
VID (x) := {i∈V:xi= 0,and xj= 0 for all j∈N1(i)}.
In contrast, if a cooperator has at least one defective neighbour, we call it a
boundary cooperator (BC). Similarly, boundary defectors (BD) have at least one
cooperative neighbour. For a given state x∈SVwe define the set of boundary
cooperators by
VBC (x) := {i∈V:xi= 1,and there exists j∈N1(i) with xj= 0},
and the set of boundary defectors by
VBD (x) := {i∈V:xi= 0,and there exists j∈N1(i) with xj= 1}.
Obviously, for all x∈SVwe have V=VIC (x)∪VBC (x)∪VI D (x)∪VBD (x).
This definition of cluster boundaries enables us to prove the following simple
sufficient condition for a state xto be a coexistence equilibrium. This statement is
the cornerstone of our later constructions.
Lemma 3.1. Let (G, p, u, T, ϕID )be an evolutionary game on a graph and let
x∈SVbe a coexistence state. If for each i∈VBD(x)and each j∈VBC (x)∩N1(i)
there exists v∈VIC (x)∩N1(j)such that
uv> ui> uj,(5)
then xis a coexistence equilibrium of (G, p, u, T, ϕID ).
Proof. We need to prove that neither boundary cooperators nor boundary defectors
change states. Indeed, the fact that for all i∈VBD (x) and each j∈VBC (x)∩N1(i)
we have ui> ujimplies that (see (3)-(4))
Ai(x) = {0}and consequently ϕID
i(x) = 0 (= xi).
Similarly, each j∈VBC (x) has a cooperative neighbour v∈VI C (x)∩N1(j) such
that for all i∈VBD (x)∩N1(j) inequalities (5) hold, which implies that
Aj(x) = {1}and consequently ϕID
j(x) = 1 (= xj).
Remark 3. The inequalities (5) are not necessary (only sufficient) for xto be a fixed
point. For example, the boundary defector icould have a cooperating neighbour
j∈VBC (x) with a higher utility as long as it has a defective neighbour k∈VBD(x)
such that
uk> uj> ui.
4. Construction of coexistence equilibria. First, we show that for any given
game-theoretical parameters p= (a, b, c, d) there exists a graph such that the evo-
lutionary game has a coexistence equilibrium.

808 EPPERLEIN, SIEGMUND, STEHL´
IK AND ˇ
SV´
IGLER
IC BC BD ID
Figure 2. Construction of a graph with a coexistence equilibrium
for any given parameters, see Theorem 4.1. Black vertices represent
cooperators, white vertices defect.
Theorem 4.1. For each p= (a, b, c, d)and any update order Tthere exists a
connected graph Gsuch that the evolutionary game on a graph (G, p, uM,T, ϕID )
has a coexistence equilibrium.
Proof. We construct a graph Gand a state configuration x∈SVof the coexistence
equilibrium at once. We have four vertex types (see Figure 2). In our construction,
inner cooperators (IC) and inner defectors (ID) are always vertices of degree 1 (i.e.,
leaf vertices). On the other hand, mboundary cooperators (BC) and mboundary
defectors (BD) form a complete bipartite graph Km,m so that each boundary co-
operator is connected to all mboundary defectors and vice versa. Moreover, each
boundary defector has exactly `neighbouring inner defectors and each boundary co-
operator has exactly nneighbouring inner cooperators (see Figure 2for illustration
of this construction). For given parameters p= (a, b, c, d) the respective utilities
are
uM
IC =a,
uM
BC =na +mb
n+m,
uM
BD =`d +mc
`+m,
uM
ID =d.
The inequalities (5) hold if
a > `d +mc
`+m>na +mb
n+m.(6)
Since a > b we have that a > na+mb
n+mfor all m > 0. Moreover, the fact that a > d
and b<cimply that we can find `and msuch that (6) hold.

COEXISTENCE EQUILIBRIA OF EVOLUTIONARY GAMES 809
ID
BD
BC
IC
v
Figure 3. Construction of a coexistence equilibrium on an arbi-
trary graph, see Theorem 4.2.
So far, we showed that for any p= (a, b, c, d) there exists a graph such that a
coexistence equilibrium exists. Next, we prove that for any given graph we can find
a suitable parameter vector such that the evolutionary game on this graph yields a
coexistence equilibrium.
Theorem 4.2. For each connected graph Gand any update order Tthere ex-
ists a parameter vector p= (a, b, c, d)such that the evolutionary game on a graph
(G, p, uM,T, ϕID )has a coexistence equilibrium.
Proof. Firstly, we construct a coexistence equilibrium for noncomplete graphs, then
for complete graphs.
1. If the graph is not complete, we can pick any vertex vwhich is not connected
to all the other vertices. We define the states of vertices x∈SVby
xi=(1i∈N≤1(v),
0i /∈N≤1(v),
i.e., we define a cluster of cooperators (vertices which are at most at distance
one from v) and a set of defectors (vertices whose distance from vis at least
2, this set is not necessarily a connected subgraph of G). In other words, we
have (see Figure 3):
VIC (x) := {v},
VBC (x) := {j∈V:j∈N1(v)},
VBD (x) := {i∈V:i∈N2(v)}.
If we denote by kmax the maximal degree of the graph G, the utilities of
the relevant vertices satisfy
uM
v=a,
uM
j≤(kmax −1)a+b
kmax
,
uM
i≤c,

810 EPPERLEIN, SIEGMUND, STEHL´
IK AND ˇ
SV´
IGLER
uM
i≥c+ (kmax −1)d
kmax
,
for all j∈VBC (x) and i∈VBD (x).
Consequently, the former inequality in (5) holds if we choose
a > c,
and the latter if we choose
c > (kmax −1)(a−d) + b.
2. If the graph is complete, i.e. G=Knand we pick arbitrary mvertices as
cooperators and the remaining n−mas defectors, we have the following
utilities for cooperators and defectors (note that VI D (x) = VIC (x) = ∅in this
case)
uM
C= (m−1)a+ (n−m)b,
uM
D=mc + (n−m−1)d.
If uC=uD, we have that ϕi(x) = xi. This situation occurs if
(m−1)a+ (n−m)b=mc + (n−m−1)d.
Remark 4. Note that, with the exception of the complete graph, the coexistence
equilibria are stable with respect to small perturbations of parameters (since the
construction is based on inequalities rather than equalities as in the case of complete
graphs).
5. Number of coexistence equilibria. In this section, we construct a special
narrow class of graphs which shows that there can be an exponential number of
coexistence equilibria.
Theorem 5.1. For each n≥6, there exists a connected graph Gwith nvertices
such that the evolutionary game on a graph (G, p, uM,T, ϕID)has at least 2bn/3c
coexistence equilibria for some parameter vector p= (a, b, c, d)and any update order
T.
Proof. Consider the undirected cycle G=Cnof length nwith vertices numbered
0, . . . , n −1. Let Xbe the set of states xon Cnsuch that each cluster of defectors
and cooperators has at least size 3, that is
X={x∈ {0,1}n|W(x)∩ {(1,0,1),(1,0,0,1),(0,1,0),(0,1,1,0)}=∅}
where we write
W(x) = {(xjmod n, xj+1 mod n, . . . , xj+k−1 mod n)|j, k ∈ {0, . . . , n −1}}.
We calculate the following mean utilities
uM
j=a+b
2for all j∈VBC (x),
uM
i=c+d
2for all i∈VBD (x).
Thus for a > c+d
2>a+b
2the inequalities (5) hold and we get a coexistence equi-
librium. It remains to estimate the cardinality of X. Since for n= 3kwe have
{(1,1,1),(0,0,0)}k⊆X,|X|>2n
3holds. Consequently, for general n, we have
|X|>2bn/3c.

COEXISTENCE EQUILIBRIA OF EVOLUTIONARY GAMES 811
Remark 5. The parameter range that we used in the proof is a subset of SH, HD
and PD. We can get a parameter range intersecting all 4 scenarios by considering
the lexicographic product L=Cn[G] where Gis a kregular graph with `vertices
(i.e., k < `). Thus Lhas vertex set {1, . . . , n} × V(G) and ((i1, j1),(i2, j2)) ∈E(L)
if (i1, i2)∈E(Cn) or i1=i2,(j1, j2)∈E(G). Then Lis a k+ 2`regular graph.
Each state on xcan be mapped to a state f(x) with f(x)(i,j):= xi. Define T=
{x∈ {0,1}n|W(x)∩ {(0,0),(0,1,0),(0,1,1,0)}=∅}. Then for a state yin f(T)
we have
uM
j=(k+`)a+`b
k+ 2`for all j∈VBC (x),
uM
i=2`c +kd
k+ 2`for all i∈VBD (x).
Hence yis a fixed point, if
(k+ 2`)a > 2`c +kd > (k+`)a+`b. (7)
For a= 1 and d= 0 this is 1 + k
2`> c > k+`
2`+b
2. Again for fixed kand `the
number of elements in f(T) is exponential in n, but for every admissible (a, b, c, d)
with 1+b
2< c < 3/2 we can find kand `such that the inequalities in (7) are fulfilled.
Remark 6. Considering equivalence classes of states under automorphisms of the
graph can heavily reduce the number of states. In this sense, it is worth noting
that our constructions which yield the exponential number of coexistence equilibria
are based on symmetric graphs. Notice however that the number of coexistence
equilibria which differ under automorphisms in our example is at least 2bn/3c
2n(the
automorphism group of the cycle has size 2n), so we still get an exponential lower
bound. Also some of these equilibria cannot be attained from other states (the
influence of graph automorphisms and the problem of attainabilty has been studied
in the case of stochastic evolutionary dynamics on graphs in [2,3]).
6. Aggregate utility function. All above results can relatively simply be for-
mulated for the aggregate utility function uAgiven by (1). First, for any given
parameters there exists a graph such that the evolutionary game has a coexistence
equilibrium (counterpart of Theorem 4.1).
Theorem 6.1. For each p= (a, b, c, d)and any update order Tthere exists a
connected graph Gsuch that the evolutionary game on a graph (G, p, uA,T, ϕID )
has a coexistence equilibrium.
Proof. We only have to modify the construction in the proof of Theorem 4.1 slightly
by adding a set of kcooperators to every inner cooperator (see Figure 2). In that
case, the required inequalities (6) turn into
(k+ 1)a > `d +mc > na +mb.
The former inequality will always be satisfied if kis large enough (i.e., if we add
sufficiently many cooperating neighbours to inner cooperators in Figure 2. The
latter inequality will be satisfied if mis large enough (since b<c), i.e., if each
boundary cooperator will have enough boundary cooperators.
Also in the case of the aggregate utility function, for each graph we can find
admissible parameters for which the evolutionary game on a graph has a coexistence
equilibrium.

812 EPPERLEIN, SIEGMUND, STEHL´
IK AND ˇ
SV´
IGLER
Theorem 6.2. For each connected graph Gand any update order Tthere ex-
ists a parameter vector p= (a, b, c, d)such that the evolutionary game on a graph
(G, p, uA,T, ϕID )has a coexistence equilibrium.
Proof. In this case, the construction is the same as in the proof of Theorem 4.2,
we only assume that max{b, d} ≤ 0 and min{a, c}>0. Considering the worst case
scenarios we observe that a vertex j∈VBC (x) can be connected to every vertex from
the cooperating cluster and must be connected to at least one boundary defector.
A vertex i∈VBD(x) can be connected to all vertices from the cooperating cluster
except the vertex v(yielding the highest utility). On the other hand, the vertex i
could be connected to exactly one cooperator and to every other defecting vertex
in the graph (yielding the lowest utility). Consequently,
uA
v≥a,
uA
j≤(kmax −1)a+b,
uA
i≤kmaxc,
uA
i≥c+ (kmax −1)d.
These estimates imply that (5) hold if a > kmaxcand c+(kmax −1)d > (kmax −1)a+b.
Thus, if we set a>kmax cand bdboth inequalities are satisfied.
Finally, in Theorem 5.1 we showed that there is an exponential growth of the
number of coexistence equilibria with respect to the number of vertices of the under-
lying graph. Since the construction in the proof was based on 2-regular graphs (on
which the mean and aggregate utility functions uMand uAsatisfy uA= 2uM), we
can straightforwardly claim the same statement for the aggregate utility function.
Theorem 6.3. For each n≥6, there exists a connected graph Gwith nvertices
such that the evolutionary game on a graph (G, p, uA,T, ϕID)has at least 2bn/3c
coexistence equilibria for some parameter vector p= (a, b, c, d)and any update order
T.
7. Final remarks. Evolutionary games on graphs have been studied in very com-
plex settings under more general assumptions (large random nonconstant graphs,
random updating, etc.). In this paper, we showed analytically that even in the de-
terministic settings the theory offers rich behaviour and yields coexistence equilibria
for all graphs and for all game theoretical parameters. Consequently, we answered
problems (A) and (D) which we posed in [5, Section 9]. Besides the other ques-
tions listed there we mention other issues related to coexistence equilibria worth
investigation.
(A) Cluster dynamics: Our results are based on a simple observation, Lemma
3.1. This result could indicate that deeper analysis of dynamics on cluster
boundaries could provide finer insight into the behaviour of evolutionary games
on graphs. Specifically, since Lemma 3.1 is a sufficient condition for xbeing a
coexistence equilibrium can we, e.g., obtain a helpful necessary condition?
(B) Stability and attractivity: A natural and surprisingly nontrivial question
concerns the stability of equilibria of evolutionary games on graphs. In [5,
Definition 7] we defined stability via perturbation of a single vertex in a state
xand provided simple results for complete and k-regular graphs. However, it
seems that this concept is very difficult to study on general graphs. Is there a
sophisticated way to analyze stability of equilibria in this sense? Alternatively,

COEXISTENCE EQUILIBRIA OF EVOLUTIONARY GAMES 813
is there a better concept of stability/attractivity for evolutionary games on
graphs?
(C) Nonexistence: Describe conditions under which there is no coexistence equi-
librium for an evolutionary game on a graph.
(D) Periodic coexistence: In this paper we studied coexistence equilibria, i.e.,
fixed points. Note that we could ask similar questions related to coexistence
cycles, in which we could observe periodic dynamics. Most importantly, given
game-theoretical parameters (a, b, c, d) (see Figure 2), can we find a graph G
such that the evolutionary game on Gyields a cycle (or even a cycle of given
length)?
Acknowledgments. The first and second author were partly supported by the
German Research Foundation (DFG) through the Cluster of Excellence (EXC 1056),
Center for Advancing Electronics Dresden (cfaed). The third author acknowledges
the support by the Czech Science Foundation 15-00735S. We thank the anonymous
referee for valuable comments.
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Received July 2015; revised November 2015.
E-mail address:jeremias.epperlein@tu-dresden.de
E-mail address:stefan.siegmund@tu-dresden.de
E-mail address:pstehlik@kma.zcu.cz
E-mail address:sviglerv@kma.zcu.cz