Journal of Game Theory 2017, 6(2): 21-37
DOI: 10.5923/j.jgt.20170602.01
An Evolutionary Game Analysis of Balancing and
Bandwagoning in Unipolar Systems
Serdar Ş. Güner
Department of International Relations, Bilkent University, Ankara, Turkey
Abstract This article proposes a simple evolutionary game to analyze the stability of states’ balancing and bandwagoning
behavior towards the sole superpower called the unipole. The trajectories leading to evolutionarily stable strategies (ESSs)
demonstrate possible alignment paths given environmental constraints of unipolar systems and game rules. States are found
to bandwagon with or balance against the unipole. They can also become divided adopting opposite alignment behaviors. The
evolutionarily stable strategies imply alternative views of socialization and competition processes for structural realism,
liberalism, and constructivism.
Keywords
Unipole, Balancer, Bandwagoner, Evolutionary Stable Strategy (ESS), Unipolar system, Unipolar
environment
1. Introduction
A unipolar system contains a sole superpower called the
unipole outranking all other states in terms of resource
endowment and global operations ability. In essence, states
in the second-tier are mostly concerned with the behavior of
the unipole; their alignments constitute reactions to the
superpower. This article develops an evolutionary game to
investigate whether second-tier states choose to balance the
unipole by forming a common front against it, or, inversely,
align with the unipole corresponding to bandwagoning in a
unipolar system. The game equilibria reveal whether states
balance against or bandwagon with the unipole depending on
the environment of the system.
IR theory contains a rich vocabulary of evolutionary terms
and conjectures but few evolutionary games [1]. The
evolutionary game we propose models on-going interactions
among states and differs from conventional games by being
dynamic, that is, by explicitly dealing with how strategies
change over time and by not requiring states to be rational
[2-5].1 It pictures states as having a unique motive: increase
resources. States co-exist in an environment where some
policies they adopt toward the unipole enhance their
resources more than others. They can commit mistakes but
* Corresponding author:
sguner@bilkent.edu.tr (Serdar Ş. Güner)
Published online at http://journal.sapub.org/jgt
Copyright © 2017 Scientific & Academic Publishing. All Rights Reserved
1
Bennett (1995) indicates that game models are mostly criticized for their
static nature and their requirement of rationality assumption: Bennett, Peter G.
(1995). “Modelling Decisions in International Relations: Game Theory and
Beyond,” Mershon International Studies Review 39 (1): 19-52.
learn to select better strategies progressively. The policies
evolve as states discover that some are more rewarding than
others in increasing resource levels defined as fitness. They
may find that balancing or bandwagoning is a better
instrument as those states become fitter that balance or
bandwagon with unipole.
States are assumed to get fitter with increments of own
resources [6]. The assumption implies that unipolar
structures do not shape and shove states’ actions in a unique
way with all becoming balancers as structural realism
predicts. An overall opposition against the unipole does not
arise automatically. Thus, balancing is not a one-way street
in unipolar systems. The game implies that states’ actions
can get similar while they err in imitating successful
alignment modes. States can become divided as balancers
and bandwagoners depending upon the unipolar
environment as well or a single mutant bandwagoner can
become successful and be imitated even if all states are
balancers.2 The unipole then becomes the hegemon facing
no opposition but enjoying supreme power in the system.
We find also that states can bandwagon with a malign
unipole or balance against a benign unipole. If they are
distinct with respect to their individual traits, for example,
their particular history with the unipole, then they can decide
to become a bandwagoner or a balancer assuming alternative
roles. States sharing the same individual traits are found not
to adopt opposite alignment options. They all become either
bandwagoners or balancers.
The evolutionary trajectories leading to different game
equilibria constitute socialization and competition processes.
They indicate progressive similarity in states’ alignment
2
We use bandwagoners as the synonym of states that bandwagon with the
unipole from now on.
22
Serdar Ş. Güner: An Evolutionary Game Analysis of Balancing and Bandwagoning in Unipolar Systems
policies toward the unipole and changes in fitness. The
directional change of the processes varies depending on the
environment of unipolar systems and on initial partitions of
states adopting opposite alignment behaviors in some cases.
Both processes pinpoint connections between structural
realism, constructivism, and liberalism as the game
equilibria are conventions, that is, outcomes of continuing
practices.
The remainder of the article is divided into six parts. The
first briefly reviews theoretical and empirical analyses and
conjectures about states’ behavior towards the unipole. The
second motivates the stage game by stylizing alignment
interactions in unipolar systems. The third presents the game
and the equilibria different game variants imply. The fourth
discusses theoretical implications. The fifth concludes. The
appendix contains the proofs of the game equilibria.
2. Literature Review
Theoretical Views
Balance of power theory implies that states react against
overwhelming power aiming to survive, therefore they are
expected to balance against the unipole [7, 8].3 “If states
wished to maximize power, they would join the stronger side,
and we would see not balances forming but world hegemony
forged. This does not happen because balancing, not
bandwagoning, is the behavior induced by the system.”
States strive to maintain their positions in the system first;
they are not after power maximization. [9] States can align
with the unipole to benefit from increased resources but they
are able to assess that a resource improvement through
bandwagoning is not accompanied with higher safety.
Consequently, the most successful policy is balancing;
bandwagoning policies are progressively eliminated and the
superpower sooner or later faces counterbalancing efforts.
Balance-of-threat theory implies that states adopt
balancing if the unipole possesses a high amount of
aggregate power, it is aggressive, it is endowed with
offensive capabilities, and it is geographically proximate to
others [10]. It is impossible for a unipole to be the neighbor
of all remaining states; hence geographic factor in threat
definition applies in well-defined neighborhoods [11, 12]. If
the unipole harbors no aggressive intentions or it is
geographically distant and endowed with defensive
capabilities, then it can attract allies.
States can also bandwagon with the unipole for protection
against regional threats and the degree of order and public
goods the unipole generates [13-15]. If the unipole is
benevolent, that is, it uses its power with restraint, it engages
merely in defensive measures and it adopts cooperative
policies that benefit others, then there is no reason to oppose
it. If states perceive the unipole as a benign actor engaged in
3
It is impossible to review all theories explaining balancing and bandwagoning
in few pages. See, for example, Jervis (1978) and Schweller (1994) for the role
of offense-defense balance and states’ motives in states’ alignment decisions.
such global strategies, then balancing against it becomes
almost meaningless [16, 17]. Thus, states can bandwagon
with a non-aggressive unipole that protects the status quo
and opts for self-restraint [18].
Empirical Views
Empirical literature identifies the United States as the
unipole and focuses mostly on the U.S. behavior. The United
States can adopt isolationism, engage selectively, pursue
collective security, or aim at keeping its preponderance in the
system [19]. Each strategy would imply different constraints
for states affecting their actions. For example, states already
believe, or can be led to believe, that the United States is a
benign hegemon while it aims at preventing the rise of a peer
competitor. The United States is therefore expected to meet
with global support rather than opposition; “other states will
not balance against the United States” [20]. Similarly, states
will not balance against a non-aggressive United States they
perceive as highly selective in its aggressiveness [21]. If the
United States takes bellicose acts, states would react against
what they perceive as efforts of global social engineering
[22].
States are also argued to be engaged in soft balancing, that
is, they can use “nonmilitary tools to delay, frustrate, and
undermine aggressive unilateral U.S. military policies” [23].
To illustrate, China, France, Germany, Japan, and Russia
prefer not to challenge the United States directly by pooling
military resources but take soft balancing measures.
Nevertheless, soft balancing could later evolve into hard
balancing as the response to an expansionist unipole is the
formation of a global counter alliance [24]. Thus, soft
balancing argument is similar to that of balance-of-threat but
the measures are less conspicuous than formal alliances. The
argument is indeed criticized as soft-balancing measures are
nothing but “routine diplomatic friction” and “merely
bargaining moves not directly aimed at reducing American
hegemony” [25, 26].
The cursory presentation of theoretical and empirical
views on states’ alignment with respect to the unipole
demonstrates that we need a framework simple enough to
generate explanations and to discover connections between
opposing arguments. The framework should be transparent
in the sense that its assumptions are clearly stated so that one
can follow how explanations are constructed. One may not
assess how useful an assumption is before it is actually made
[27]. “In science, it is more important that the conclusions be
right than that the assumptions sound reasonable. The
assumption of gravitational force seems totally unreasonable
on the face of it, yet leads to correct conclusions” [28]. In our
case, we can only speculate whether the conclusions are
meaningful; not whether they are false or correct.
Evolutionary processes take time; one cannot immediately
assess empirical refutation or verification of conjectures the
game implies.
Journal of Game Theory 2017, 6(2): 21-37
3. Evolutionary Framework
States
States are the players of the game. We assume that more
resources a state controls, higher becomes its fitness and
therefore its survival chances in a unipolar system. Higher
resources are associated with higher likelihoods of
independent existence, that is, success. Resourceful states
are fitter and therefore successful.
The assumption is more primitive than prominent classical
realist views, because it is not associated either with
metaphysical or religious characterization of man as evil and
states’ aim to dominate others [29] or to increase power, that
is, to improve “man’s control over the minds and actions of
other men” [30]. 4 Survival comes before such motives.
States must first subsist and possess resources.
The assumption is in line with states’ aim to guarantee
their survival: “I assume that states seek to ensure their
survival…the aims of states may be endlessly varied; they
may range from the ambition to conquer the world to the
desire to be left alone. Survival is a prerequisite to achieving
any goals that states may have, other than the goal of
promoting their own disappearance as political entities” [31].
Hence, the assumption fixes states’ preferences and parallels
structural realism with one difference, however: it does not
ascribe multiple motives ranging from world domination to
isolation.5
Actions
Theoretical and empirical views concede that
bandwagoning constitutes an alignment option available to
states in a unipolar system. As to balancing, [32-34] argue
that collective action problems hinder the formation of a
joint front against the unipole. The resource gap between the
unipole and prospective allies would complicate states’
coordination efforts. It is difficult for the allies to agree on
how to share alliance costs and how much to contribute to
their external balancing efforts. States that do not contribute
would enjoy security benefits; therefore there are strong
incentives to free ride in the system. States would avoid
cooperation if their partners obtain higher benefits, as one’s
increased resources can be later used to harm others hinting
at the problem of whether states maximize absolute or
relative gains in unipolar systems [35]. However, the debate
over absolute and relative gains mistakenly takes effects for
causes [36, 37]. One cannot simply assume that states
maximize relative or absolute gains regardless situations
states find themselves in: states’ concerns for absolute and
4
Thayer (2000) connects evolutionary theory with classical realism to justify
states’ dominance motive as resulting from human evolution and selfish gene:
Thayer, Bradley A. (2000) “Bringing in Darwin: Evolutionary Theory, Realism,
and International Politics.” International Security 25 (2): 124-151. Masters
(1983) searches for general links between biology and political theory: Masters,
Roger D. (1983). “The Biological Nature of the State.” World Politics 35 (2):
161-193.
5
Legro and Moravcsik (1999: 14) state that Waltz’s assumption becomes is
vague and elastic: Legro, Jeffrey W. and Andrew Moravcsik (1999). “Is
Anybody Still a Realist?” International Security 24 (2): 5-55.
23
relative gains vary depending upon systemic changes of
strategic environments. Unipolar systems are structurally
different from bipolar and multipolar systems, and, as we
will later argue, not all unipolar systems possess equivalent
environments.
A more fundamental question is whether it is possible to
form a global counter alliance against the unipole. If a state is
the unipole and its resources are so large that they cannot be
matched even if all states in the second-tier combine their
resources, the global front does not constitute any
counterpoise. Hence, external balancing in unipolar systems
is almost impossible. The question of “who will gain more?”
then loses its meaning; relative gain concerns dissipate with
no prospect of cooperation against the unipole--revived
perhaps in regional interactions. Consequently, states can
choose either bandwagoning or internal balancing.6 States
can improve their resource levels by domestic efforts such as
allocations of more resources to defense, changes in military
doctrines in targeting the unipole even though the unipole’s
level of resources is out of reach [38].
Unipolar Environments
We assume that two parameters make up the environment
of a unipolar system: the behavior of the unipole and the
distribution of resources across states. Bandwagoning and
internal balancing generate different magnitudes of fitness
enhancements depending upon the environment.
Bandwagoners essentially obtain unipole’s protection and
technology transfers. The unipolar actor’s backing and help
in local security problems and improvement of defense
levels by direct military and economic aid bolster
bandwagoners’ domestic resources in a complex fashion.
Yet bandwagoning is costly for two reasons. First,
bandwagoners lose their freedom of action. An alignment
with a major power provides a large increase in security at
the cost of limited range of foreign policy options [39, 40].
The interests of states and the unipole would not be
concordant in an issue. Second, bandwagoners face the
prospect of sharing the costs of unipolar actor’s interventions
around the globe [41]. Thus, bandwagoners’ resource
improvements and costs depend upon the unipole’s type.
The unipole is assumed to be either benign, so that it
voluntarily constrains its power, or malign, that is, it pursues
its own ideals about the international system [42-44]. 7 A
benign unipole acts in such a way that it does not threaten
states: it delimits own actions in managing international
politics, it does not take advantage of its superior position,
and promotes actions and institutions for joint gains. In
contrast, a malign unipole intervenes around the globe and
these operations threaten states. It takes advantage of its
Levy (2003: 134-135) argues that balancing behavior “comes in degrees”
similar to “nonbalancing” behavior covering bandwagoning, buck-passing,
chainganging.
7
Nowadays, the United States does not face global balancing efforts as states
perceive it as “benign”: Kupchan (1998). Lieber and Alexander (2005: 113)
similarly argue that states do not balance against the United States because it is
selectively aggressive and not broadly threatening after 9/11 attacks.
6
24
Serdar Ş. Güner: An Evolutionary Game Analysis of Balancing and Bandwagoning in Unipolar Systems
superior position by putting no limits on own actions. It does
not necessarily promote actions and institutions for joint
gains. The unipole’s global management therefore
determines the magnitude of bandwagoners’ fitness.
The distribution of resources across states in turn
determines balancers’ resource improvement: Balancers
draw upon own resources. Higher states’ levels of
development, technology, and capabilities, higher become
their resource gains through internal balancing efforts.
Internal balancing is however less resource enriching than
bandwagoning: no state can attain the same quantity and
quality of resources through domestic efforts as those
resources and assistance the unipole can provide. The
unipole’s backing and help are assumed to be more effective
in bolstering resources. Balancers enjoy although one
advantage: they are not subject to the cost of bandwagoning
by definition.
Thus, bandwagoners’ and balancers’ resource
improvements can be high or low, the cost of bandwagoning
can also be high, low, or even zero. Each combination
implies a different class of unipolar environment. There can
be a unipolar system where the unipole is benign and
therefore helps states in resolution of local conflicts and
improvement of defense levels through its direct military and
economic aid. Bandwagoners then benefit from high
enhancements in resource levels. They would also suffer
from negligible costs provided that the benign unipole does
not conduct global interventions asking for states’ assistance
and it does not delimit bandwagoners’ range of foreign
policy choices. Suppose also that states benefit from high
levels of development, technology, and capabilities in the
system. Such an environment is in sharp contrast with the
one where a malign unipole forces states to act in its favor
and to pursue policies against their national interests, and
states have a low level of development, technology, and
capabilities. Therefore, unipolar systems are structurally
identical but can be environmentally different. Unipolar
environments produce constraints altering states’ resource
improvements through balancing and bandwagoning.
Foreign Policy Model
States are assumed to emulate those strategies that
improve fitness. Balancing and bandwagoning are opposed
behaviors but they are motivated by the same goal: achieve a
higher amount of resources and get fitter. States are not
“programmed” to bandwagon with the unipole or internally
balance against it either. They gradually learn to adapt to the
environment [45]. Adherents of alignments that are poor
responses to unipolar environments are eventually
overwhelmed by adherents of those that bolster fitness. We
therefore have to explain states’ particular actions. In other
words, the evolutionary game necessitates a model of foreign
policy making.
We start by assuming that states are not black boxes. The
leaders, elites, and organizations related to foreign policy
make up the state, and, they do not have enough information
about the environment [46, 47]. Leaders and organizations
make mistakes due to routines, limited capacities of
processing information, and misperceptions which generate
inconsistencies in reaching collective decisions [48, 49].
Elites can misperceive the unipole’s intentions and behavior
and each other’s preferences about whether balancing or
bandwagoning is a better tool to improve fitness. Moreover,
organizations’ standard operating procedures create routines
which are not optimal in assessment and comparisons of
resources. As a result, states do not adopt the most successful
action instantly when they compare resource levels.
Nevertheless, they find that either balancing or
bandwagoning
performs
better
through
lengthy
trial-and-error processes by adapting to successful alignment
trends in the system.
To illustrate, suppose that the leaders and bureaucrats of
various organizations of a state discuss and negotiate among
themselves what behavior to adopt with respect to the
unipole. In doing so, they compare the action and the
resource levels of other states with theirs. They discuss and
decide through internal politicking and controversies
whether bandwagoning or internal balancing is a better tool
to raise the level of fitness. They can however commit a
mistake by adopting internal balancing while a sensible
majority of states opt for bandwagoning and increase their
resource levels, or adopting to bandwagon with the unipole
while a sizable portion of states improve survival chances
through internal balancing. They might also assess that one
policy is better than the other yet they may repeat the mistake
without changing state’s foreign policy. Nevertheless, unfit
actions become sooner or later replaced by fitter ones so that
the state survives better in the system [50]. Adaptation to the
environment takes time.
Equilibria
A strategy becomes evolutionarily stable if all states
ultimately adopt it and no different strategy can replace it.
An evolutionarily stable strategy (ESS) is the state of
strategy distribution that cannot be “invaded” by alternative
actions. If, for example, all states are bandwagoners as
bandwagoning is the ESS, then if some states adopt
balancing, they obtain a lower amount of fitness and revert to
bandwagoning. In general terms, mutants, that is, small
minorities of states employing deviant actions eventually
disappear and finally conform to the ESS. The disappearance
of mutants does not correspond to states vanishing from the
international scene but to the conversion of their actions into
successful ones.
States’ behavior towards the unipole can approach some
stability over time, as fitter strategies progressively become
prevalent and other strategies become extinct. 8 If
bandwagoning provides a higher resource level depending
upon the prevalent bandwagoning-balancing configuration
in the population, a greater number of states would align with
unipole. Nevertheless, if the rest of the population chooses
8
Axelrod (1984) finds that TIT-FOR-TAT is the most successful strategy in
tournaments of infinitely repeated Prisoner’s Dilemma. The ESS of the game
does not confirm this finding (Osborne, 2004: 440-441; Samuelson, 1998: 20).
Journal of Game Theory 2017, 6(2): 21-37
bandwagoning, a state may be able to reap benefits if it
chooses internal balancing. Similarly, if the rest of the
population balance against the unipole, then a state may find
that to bandwagon with the unipole is the fitter strategy.
Hence, it does not mean that a prevailing strategy is the ESS
and the one that produces higher fitness. States’ actions
become adapted to the unipolar environment over the long
haul. States are after augmenting resources but they are
boundedly rational, as elites err in resource assessments and
whether balancing or bandwagoning is a better policy to
improve fitness.
Consequently, the evolution of states’ behavior towards
the unipole depends upon the dispersal of policies that
improve fitness while the environment does not change. If
bandwagoning with the unipole brings more resources, then
more states adopt it; otherwise they become balancers.
Successful bandwagoning breeds bandwagoning and
successful balancing breeds balancing.
4. The Game
The game assumes that states repeatedly compare
respective levels of fitness. We first assume that the
comparisons of fitness and actions are assumed to take place
randomly: the probability that a state’s matching with a
bandwagoner or a balancer equals the proportion of
bandwagoners and balancers, respectively. The assumption
points out to some indifference of elites and organizations
involved in foreign policy making who randomly pick a state,
say state X, and compare their own alignment efforts and
resulting resource improvements with those of state X.
Nevertheless, the decision makers might not be that
indifferent towards various states. They might rather be
interested in how a specific state is performing in the system.
Therefore randomness assumption can be relaxed. We
accordingly assume that a state can also be matched at a
higher probability with those that share its alignment action.
This amounts to assortative matching [51]. Some states, for
example, would be less inclined to bandwagon with the
unipole for some historical, internal political or some other
reasons. Once a state cannot easily become a bandwagoner,
then the assessment of how those allies of the unipole fare in
the system becomes useless. Thus, it is more likely that a
bandwagoner (balancer) compares its fitness with a fellow
bandwagoner (balancer). The assumption then implies that
balancers and bandwagoners can be quite differentiated on
the basis of various unipolar environments. For example, if
balancers’ resource and technology levels are low, then they
are disfavored compared to bandwagoners during the
evolution of actions towards the unipole.
We also differentiate between two types of populations.
First, we assume that the population of states is
homogeneous, that is, states’ roles in the evolutionary
process remain the same: they are not strategically different
from each other. Second, we assume that the population of
states is not homogeneous. States can condition their
25
alignment choices on their individual traits, for example,
their histories of relationship with the unipole, regime types,
or any other feature. States then become strategically distinct
when they decide to become a bandwagoner or a balancer
assuming alternative roles.
We first assume that the population of states is either
homogeneous or non-homogeneous and study evolutionary
dynamics under random matching in the variant 1
(homogeneous population-random matching) and the variant
2 (non-homogeneous population-random matching). We
later extend the analysis by assuming a homogeneous
population and assortative matching in the variant 3.
Let V, v, a respectively denote resources available to
bandwagoners and balancers and the cost of bandwagoning
in a unipolar environment. They are assumed to satisfy
conditions of V > 0, v > 0, V > v, and a ≥ 0, V ≥ a. The
parameters can change across environments. If V drops, it
still exceeds v, the cost of bandwagoning varies between V at
its maximum and 0 at its minimum, and, v never attains 0,
that is, internal balancing always yields some amount of
fitness.
States in the second-tier occupy identical positions with
respect to each other and the unipole. Unipolar systems
produce structural symmetry. Therefore, if two states adopt
bandwagoning, their resource enhancements will be equal:
v
V a
. Similarly, two balancers obtain
. If they adopt
2
2
opposite alignment policies, their fitness improvements
become asymmetric. The state that aligns with unipole
obtains V − a, and the balancer obtains the whole resource v.
Payoffs in the game below indicate states’ fitness depending
on their alignment policies toward the unipole.
Table 1. Stage Game
Bandwagon (BAN)
Balance (BAL)
Bandwagon (BAN)
Balance (BAL)
,
V− a, v
V a
2
V a
v, V− a
2
v
2
,
v
2
Variant 1
An ESS is a refinement of Nash equilibrium. If U(BAN,
BAN) = U(BAL, BAN), so that (BAN, BAN) is a non-strict
Nash equilibrium, then, by definition, BAN is evolutionarily
V a
v , then
stable if U(BAN, BAL) > U(BAL, BAL). If
2
v
V a . Therefore, if bandwagoners and balancers do
2
equally good among bandwagoners, then bandwagoners
must be more successful than balancers among balancers for
bandwagoning to be the ESS. If the Nash equilibrium is strict,
that is, U(BAN, BAN) > U(BAL, BAN), then the condition
U(BAN, BAL) > U(BAL, BAL) is satisfied automatically.
Bandwagoners obtain a strictly higher amount of resources
than balancers among bandwagoners and bandwagoning
Serdar Ş. Güner: An Evolutionary Game Analysis of Balancing and Bandwagoning in Unipolar Systems
26
becomes established as an evolutionarily stable strategy
among states. As a result, BAN is the ESS provided that
V a
v.
2
Similarly, if U(BAL, BAL) = U(BAN, BAL) and U(BAL,
BAN) > U(BAN, BAN), or, simply, if U(BAL, BAL) > U(BAN,
BAL), then BAL is the ESS. The ESS become all states’
balancing against the unipolar actor if balancers and
bandwagoners perform equally well among balancers but
balancers fare better than bandwagoners among
bandwagoners or if balancers are more successful than
bandwagoners among balancers. Thus, we need either
v
v
V a
V a and v
V a , for
, or, simply,
2
2
2
v
V a ,
BAL to be an evolutionarily stable strategy. If
2
V a
then v
. Consequently, BAL is the ESS provided
2
v
that V a .
2
v
V a 2v . The unique symmetric Nash
2
2(a V ) v
V a 2v
equilibrium arises in mixed strategies with x
and 1 x
, where x is the fraction of
a V v
a V v
bandwagoners and 1 x is the fraction of balancers in the population. 9 The division of states as bandwagoners and
balancers is called a polymorphic equilibrium.
To see how bandwagoning and balancing evolve over time, we have to consider the expected resource improvements from
each strategy given that some states are bandwagoners and some others are balancers. Let f(BAN) denote the fitness of
bandwagoners and f(BAL) the fitness of balancers, x denote the fraction of bandwagoners and 1 x the fraction of balancers
in the population. From Table 1, it follows that:
No symmetric Nash equilibrium and therefore no pure ESS exist if
V a
f ( BAN )
x (V a)(1 x)
2
(1)
v
f ( BAL) vx (1 x)
2
(2)
The first equation means that, under random selection, the fitness of a bandwagoner equals to the sum of resources it
obtains among bandwagoners and among balancers, and, similarly, that the fitness of a balancer equals to the sum of
resources it obtains among bandwagoners and balancers both weighted by the number of bandwagoners and balancers
respectively. It is sufficient that one strategy brings a higher fitness than the other to be imitated. If a strategy yields a higher
resource return than the other, then the percentage of states adopting it expands in the population. Otherwise, the number of
its adherents shrinks over time. If strategies generate equal fitness, the fraction of states using them remains constant.10
The evolutionary dynamics generate there cases:
i)
2(a V ) v
V a
v , so that
1 , all states become bandwagoners, that is, x 1 ;
a V v
2
x=0
All BAL
x=1
All BAN
Figure 1. Homogeneous Population Monomorphic Equilibrium
ii) V a v / 2 , so that
9
10
2(a V ) v
0 , all states become balancers, that is, x = 0;
a V v
Remark that x increases when V increases and decreases when either a or v increases.
Differential equations and formal deductions are placed in the appendix. The equations set up fitness conditions as replicator dynamics.
Journal of Game Theory 2017, 6(2): 21-37
27
x=0
All BAL
x=1
All BAN
Figure 2. Homogeneous Population Monomorphic Equilibrium
v / 2 V a 2v , so that 0
iii)
2(a V ) v
1 , states become divided as bandwagoners and balancers.
a V v
k
x=0
All BAL
x=1
All BAN
Figure 3. Homogeneous Population Polymorphic Equilibrium
V a
f 2 ( BAN )
x (V a)(1 x)
2
Variant 2
The above results hold for a homogenous population of
states that cannot condition their choices upon their peculiar
traits. Asymmetry can be studied by focusing on players who
possess different strategies. There are different species in
biological applications; in economics there are players like
buyers and sellers. Revisionist states can be modeled to
either ask for concessions or not while status quo states give
in to demands or not in international politics. Different
strategies however require further payoff assumptions
together with their justifications producing a less tractable
model.
We instead do not change the strategic structure of the
game so that all states have identical payoffs but assume that
they condition their choice of balancing and bandwagoning
upon an exogenous feature to the game like the history of
relations with the unipole or domestic political regime
[52-54]. We suppose that there are two types of states called
as type-1 and type-2. The types do not have different
strategies or payoffs but condition their strategies on the
information of whether they are type 1 or type 2. The
strategies take the form of “opt for balancing if type 1 but for
bandwagoning if type 2” or “opt for bandwagoning if type 1
but for balancing if type 2.”
Now let x denote the population of type-1 states and y the
population of type-2 states. The fitness functions of type-1
states become:
V a
f 1 ( BAN )
y (V a)(1 y )
2
v
f 1 ( BAL) vy (1 y )
2
Similarly, fitness functions of type-2 states are:
(3)
(4)
v
f 2 ( BAL) vx (1 x)
2
(5)
(6)
The figures 4 and 5 are the phase portraits of the evolution
of bandwagoning and balancing under the conditions of
V a
2
v and
v
2
V a , respectively. The figure 6 shows
v
V a 2v .
2
In the figure 4, all types finally jump in the bandwagon,
thus all trajectories end up in the state of x y 1
whatever is the initial composition of types of bandwagoners
and balancers the population. Both type of states bandwagon
with the unipole ultimately. Similarly, all trajectories end up
in the state of x y 0 in the figure 5 regardless of initial
conditions; all types choose balancing.
In the figure 6, the direction of evolutionary processes
starting from initial points is toward x 1, y 0 if x y
but toward x 0, y 1 if x y . If type-1 bandwagoners
are more numerous than type-2 bandwagoners at the start, all
type-1 states finally bandwagon with but all type-2 states
balance against the unipole. No type-1 balancers or type-2
bandwagoners survive. The equilibrium is reversed when
type-2 bandwagoners are more numerous than type-1
bandwagoners. The evolution of actions is stabilized with all
type-2 states but no type-1 state becoming bandwagoners.
Similarly, type-2 balancers and type-1 bandwagoners get
extinct. Hence, either all type-1 states or type-2 states choose
bandwagoning ultimately; it is impossible that all states
bandwagon with or balance against the unipole regardless
their type.
the evolution of strategies when
Serdar Ş. Güner: An Evolutionary Game Analysis of Balancing and Bandwagoning in Unipolar Systems
28
V a
v and initially type-1 bandwagoners are more
2
numerous than type-2 bandwagoners, the evolution of
actions is stabilized with all type-1 states becoming
bandwagoners but type-2 states becoming divided into
factions of balancers and bandwagoners. Otherwise, if type-2
bandwagoners are initially more numerous than type-1
bandwagoners, then all type-2 states switch to
bandwagoning and the type-1 states become divided as
balancers and bandwagoners.
If
x=y=1
All BAN
y
x
Figure 4. Non-homogeneous Population, Monomorphic Equilibrium
y
x=y=0
All BAL
x
Figure 5. Non-homogeneous Population, Monomorphic Equilibrium
Journal of Game Theory 2017, 6(2): 21-37
29
(x = 0; y =1)
Type 1: BAL
Type 2: BAN
k
y
(x = 1; y = 0)
Type 1: BAN
Type 2: BAL
x
Figure 6. Non-homogeneous Population Polymorphic Equilibrium
y
x
Figure 7. Non-homogeneous Population Hybrid Equilibrium
y
x
Figure 8. Non-homogeneous Population Hybrid Equilibrium
Serdar Ş. Güner: An Evolutionary Game Analysis of Balancing and Bandwagoning in Unipolar Systems
30
v
V a and type-1 bandwagoners are more
2
numerous than type-2 bandwagoners, all type-2 states
become balancers and type-1 states becomes divided. Finally,
if type-2 bandwagoners are more numerous than type-1
bandwagoners, all type-1 states become balancers and the
type-2 states in turn become divided.
The bold lines in the figures 7 and 8 indicate that states in
one type of population all adopt the same action towards the
unipole while states of the other type become divided. For
example, in the figure above, if the number of Type-1
bandwagoners is greater than Type-2 bandwagoners, no
Type-2 bandwagoner survives but Type-1 states become
partitioned as balancers and bandwagoners. We qualify such
apportionments as hybrid equilibria.
Variant 3
If matching is assortative, the probability that a
bandwagoner compares its resources with those of a
bandwagoner is greater than the proportion x of
bandwagoners and the probability that a balancer compares
its resources with those of a balancer is greater than the
proportion (1 − x) of balancers. We accordingly assume that
a bandwagoner (balancer) is matched with another
bandwagoner (balancer) by probability z, whereas it is
matched with a random member of the population by
probability 1− z. The game then implies the following fitness
equations:
If
V a
V a
f ( BAN ) z
(1 z ) x
(1 x)(V a)
2
2
(7)
v
v
f ( BAL) z (1 z ) xv (1 x)
2
2
(8)
The equations (7) and (8) imply the following cases:
i)
If
(V a)(2 z ) v
1 ,
(V a v)(1 z )
all
states
become
(V a)(2 z ) v
0 , all states become
(V a v)(1 z )
bandwagoners;
ii) If
(V a)(2 z ) v
1 , states become divided
(V a v)(1 z )
balancers;
iii) If 0
as bandwagoners and balancers.
The cases imply conditions on the probability z. If
(V a)(2 z ) v
1 , all states become bandwagoners
(V a v)(1 z )
V a
v for all z such that 0<z<1. This
provided that
2
(V a)(2 z ) v
1 , all states become
(V a v)(1 z )
V a
bandwagoners provided that if V a v
and
2
2v a V
2v a V
z
1 . Similarly, if
where 0
v
v
v
(V a)(2 z ) v
V a , all states become
0 and
2
(V a v)(1 z )
result corresponds to the case under random matching. In
addition, if
balancers for all z, 0<z<1, as in the case of random
matching. It is also possible that all states become
v
2(V a ) v
and z
where
balancers if v V a
2
V a
2(V a) v
0
1.
V a
(V a)(2 z ) v
Finally,
if
and
1
0
(V a v)(1 z )
2(V a ) v a V 2v
a
polymorphic
z min{
,
}
V a
v
equilibrium is obtained. The minimum threshold is
v
2(V a ) v
a V 2v
V a v , and, it is
if
if
2
V a
v
V a
v V a . Thus, states become partitioned into
2
balancers and bandwagoners under assortative matching if
either v V a v or V a v V a . The environments
2
2
v
differ from V a 2v , the one where a polymorphic
2
equilibrium results under random matching. In general,
compared to random matching, the assumption of assortative
matching reveals the importance of additional unipolar
environments allowing additional interpretations and
implications. The table below summarizes all findings.
Journal of Game Theory 2017, 6(2): 21-37
31
Table 2
Environment
V a
2
V a
2
ESS
v
All states become bandwagoners regardless matching rules and population type.
v
v / 2 V a
v / 2 V a
2v V a
V av
v
2
V a
2
All states become bandwagoners under random matching if the population is
homogeneous; otherwise, if the population is non-homogeneous, one type of states
either becomes all bandwagoners or balancers while the other type of states become
divided adopting opposite alignments.
All states become balancers regardless matching rules and population type.
All states become balancers under random matching if the population is
homogeneous; otherwise, if the population is non-homogeneous, one type of states
either becomes all balancers or bandwagoners while the other type of states become
divided adopting opposite alignments.
States become partitioned as balancers and bandwagoners if the population is
homogeneous; otherwise, if the population is non-homogeneous, either type-1 states
become all balancers while type-2 states become all bandwagoners or vice versa.
a V 2v
a V 2v
If matching is assortative, then either all states become bandwagoners provided that
z
v
; otherwise, if z
v
, they become partitioned into
balancers and bandwagoners.
v V a
2(V a ) v
2(V a ) v
If matching is assortative, then either all states become balancers provided that
v
2
z
V a
; otherwise, if z
V a
, they become partitioned into
balancers and bandwagoners.
5. Implications
What does the game imply for unipolar systems? How
does the model help us in generating explanations of
opposite forms of alignment toward the unipole? We answer
these questions by classifying the equilibria with respect to
outcomes and processes. Some outcomes are conspicuous as
no balancers or bandwagoners survive through evolutionary
processes indicating an evolving similarity in actions. All
states’ alignment with the unipole corresponds to the
unipole’s absolute dominance and therefore the unipole’s
hegemony in the system; otherwise, a global resistance
through overall internal balancing emerges against the
unipole. The progressive similarities in actions and possible
behavioral dynamic bifurcations such as the division of
states as balancers and bandwagoners generate alternative
interpretations of socialization and competition processes
which are of interest to structural realism, liberalism, and
constructivism.11
Hegemony Versus Unipolarity
Unipolar and hegemonic systems are similar as they all
include one globally dominant state but boundaries between
them remain fuzzy. “Hegemony is a concept that is widely
used, but it is rarely defined with any degree of precision”
[55]. A distinction can be made by proposing that hegemony
11
It is impossible to discuss each aspect of these theories in terms of
evolutionary processes. Therefore, we limit our discussion with theoretical
implications for competition and socialization processes.
refers to power relations unlike unipolarity. The hegemon is
“the state with control over raw materials, control over
resources of capital, and competitive advantages in the
production of highly valued goods“ [56]; a “single powerful
state that controls or dominates lesser states in the system”
[57]. Thus, a hegemon is omnipotent: it enjoys an absolute
ability to conduct policies it prefers and globally dictates
them.
The game assumes that the unipole is the state possessing
more resources than the sum of resources all other states
control in the system. However, the unipole’s extreme
resource superiority does not automatically translate into
absolute power, therefore, a hegemon. The control over
resources is distinct from the control over actors [58]. The
unipole is, in a sense, the hegemon in terms of resources but
not in terms of power.
The game implies a relationship between unipolarity and
hegemony: if bandwagoning is the ESS, so that all states
ultimately align with the unipole, the unipole faces no single
opposition and it transforms into a hegemon. Hence, if the
benefits are high and the cost of alignment with the unipole is
V a
v , then
sufficiently low, so that the environment is
2
the unipolar system transforms into a hegemonic one
whether states compare their resources randomly or
assortatively. A reason of bandwagoning is indeed argued to
be secondary state elites’ “subjective awareness” of benefits
the hegemonic order generates [59, 60]. If bandwagoners
32
Serdar Ş. Güner: An Evolutionary Game Analysis of Balancing and Bandwagoning in Unipolar Systems
obtain strictly higher resources among themselves than a
balancer among them, they become successful and balancing
is ultimately eradicated. States get fitter while the system
evolves into a hegemonic one. The bandwagoning of all with
the unipole becomes stable; if some states deviate to
balancing, evolutionary dynamics drives the population of
states back to the original equilibrium state.
Assortative matching refines the hegemony condition by
V a
v V a conducive
revealing the environment of
2
a V 2v
. A numerical example is
to hegemony if z
v
helpful here. Suppose that the maximum value of V is 1 and
the alignment cost a varies between 1 at maximum and 0 at
minimum. We have V > v > 0 by assumption. If V = 1, a =
V a
v V a
0.6, v = 0.3, so that the environment is
2
and z ≥ 0.66, so that at least two-third of states compare
resources only with similarly aligned states, the unipolar
system transforms into a hegemonic one. The result holds
even if there is a unique bandwagoner while the cost of
alignment with the unipole is twice as much the resource
available to balancers. However, if the cost increases and
attains, say, the value of 0.7, the unipole’s hegemony
becomes impossible. Thus, compared to random encounters,
assortative comparisons are more likely to lead to hegemony.
Benign Versus Malign Unipolarity
States’ alignment with and against a unipole can find
equal theoretical support assuming that are rational and react
against power or images of it [61]. The balance-of-threat
theory implies that states would oppose against a unipole
which constitutes a global threat; otherwise if the unipole is
benign, states would bandwagon with it to augment
resources and to improve security. Are there any chances that
a benign unipole becomes the hegemon?
The cost of bandwagoning with the benign unipole can be
zero or almost zero. Consequently, if the unipole is benign,
v
so that a 0 or a 0 , the environment V becomes
2
impossible by assumption: V v . The benign unipole never
faces global balancing efforts. Nevertheless, if states’
technology, development, and resource levels are relatively
high so that v satisfies the condition 2v V , it is possible
that balancers multiply if states randomly compare
v
V 2v and
their performances in the environments
2
v
V a 2v . Similarly, a benign unipole cannot become
2
a hegemon under assortative matching if the environment is
V
2v V
v V and z
as some states become
2
v
bandwagoners but some balancers. If, for example, V = 1, v
= 0.6, and a = 0, some states become bandwagoners and
some others balancers provided that less than one-third of the
states compares their resource levels with those that are
similarly aligned. The unipolar system evolves into a hybrid
one where it is better to bandwagon with the unipole when
balancers make up the majority and it is better to balance
against the unipole when bandwagoners are more numerous
than balancers in the non-homogeneous population under
random matching.12 The competition process does not single
out a unique successful strategy to increase fitness. States are
ultimately partitioned as balancers and bandwagoners.
Therefore, a zero or almost zero cost of bandwagoning is not
sufficient for a benign unipole to become a hegemon.
If the unipole is malign, that is, its interventions threaten
states, it takes advantage of its superior position by putting
no limits on own actions and does not necessarily promote
actions and institutions for joint gains. The resource of
bandwagoning would then be considerably cut down by a
high value of a. Will states balance against the malign
unipole? The answer is again negative. If the cost of
bandwagoning is so high that the unipolar system
v
V a , that is, half of balancers
environment is
2
resource becomes equal to the resource a bandwagoner
obtains matched with a balancer, a single bandwagoner can
disturb the stability of global balance against the malign
v
unipole. And if V a 2v under random matching, or
2
V a
a V 2v
v V a
z
if
and
under
2
v
assortative matching, some states would learn to align with a
malign unipole. These cases of polymorphic equilibria
confirm that opportunities of gain, even small, explain
alignments with threats [61]. Thus, states as adaptive
learners can become bandwagoners similar to those states
that decide to align with a threatening unipole to maximize
expected payoffs.
Socialization and Competition
Structural Realism
Structural realism contains strong evolutionary arguments
and is closely linked with social Darwinism [62]. 13 The
theory outlines an evolutionary framework without
specifying the population of states subject to evolution, the
environment, and the selection and success criteria in
international systems [63]. States, not assumed to be
necessarily rational, emulate successful practices through
socialization and competition processes which depend more
upon actions than preferences.14 Socialization is defined as
12
Deudney and Ikenberry (1999) provide an empirical approximation to this
case by indicating that even if states get institutionalized U.S. assurances, they
could defect frıom the United States. They qualify such systems as “punctuated
hegemony:” Deudney, Daniel and John Ikenberry (1999) “Realism, Structural
Liberalism and the Western Order,” in Ethan A. Kapstein and Michael
Mastanduno (eds), Unipolar Politics: Realism and State Strategies After the
Cold War New York: Columbia University Press, 103-137.
13
Elman (2006) in turn argues that structural realism is not an evolutionary
approach to international politics but can be used to analyze foreign policy of
individual states.
14
For an opposite view, see Morrow (1988: 89). Levy (1994: 298) argues that
socialization and selection require rational learning.
Journal of Game Theory 2017, 6(2): 21-37
the process through which states are affected back by their
own interactions. Competition is the process through which
states opt for the most successful practice [64]. We adopt the
same definition of competition. We define socialization as
the process of change in states’ fitness. 15 Both processes
indicate how structures of international systems shape and
shove state behavior and imply progressive similarity in
evolving policies and resources.
We first observe that each trajectory leading to
evolutionarily stable strategies in the figures describe
socialization and competition in unipolar systems. States’
adoption of specific actions affects their fitness. States get
progressively resourceful as the actions evolve in the
direction of the ESSs. If, for example, bandwagoning is the
V a
v , socialization
ESS as the unipolar environment is
2
constitutes a dynamic process of states getting more
resourceful. The fitness of bandwagoners improves as
balancers always obtain lower resources among them.
Structural realism indicates that states are expected to
balance against the unipole: balancing is the most successful
practice as states are watchful about relative gains. In
contrast, the ESS imply that the processes are not unique, as
there are infinitely many trajectories leading to overall
balancing or bandwagoning depending upon initial
conditions. The polymorphic equilibria imply that
competition among states for more resources can establish
two opposite but successful behaviors. Hence, the most
successful alignment policy can be bandwagoning, balancing,
or a mixture of them. Competition is not a one-way street.
Liberalism
Preferences constitute the central liberal variable and stem
from various sources ranging from societal ideas and
interests to institutions [65]. Liberalism does not either
necessitate that states are unitary or rational similar to the
foreign policy model assumptions. The game demonstrates
that not elites’ preferences but the dispersal of alignment
actions in the system drives evolutionary dynamics. States’
adjust their alignment policies toward the unipole through
elites’ learning processes under errors and mistakes which
can be related to domestic constraints such as the trade-off
between domestic and international needs, that is, the “guns
or butter” problem.
A connection between elites, socialization, and
competition processes for hegemonic systems is indeed
proposed [66]. States become socialized as their elites accept
values and norms the hegemon advances. National leaders
progressively accept the hegemon’s norms and values, and,
as a consequence, variations in policies towards the hegemon
are reduced. It is argued that secondary states’ elites’
internalization of hegemon’s value orientations have nothing
to do with forces at system-level ruling structure’s effect
15
Resende-Santos (1995) notes that Waltz’s definition of socialization is more
convenient for social systems than for a theory of international politics
considered as asocial: Resende-Santos, J. (1995). “Anarchy and the Emulation
of Military Systems,” Security Studies, 5 (3): 190-245.
33
upon socialization out and that there are “meager analytic
tools to understand the mechanisms and conditions” of
socialization [67]. The game proposes mechanisms and
conditions for socialization and demonstrates that elites’
learning processes make up forces at system level given the
distribution of alignment actions. Any ESS of the game
derives from elites’ learning processes and describes policy
dispersal across states. An alignment policy becomes
evolutionarily stable as system environments set up unipolar
constraints affecting elites’ learning (including organizations’
operations). Thus, structure’s effect upon socialization can
matter.
It is possible that elites’ learning and organizations’
operations imply mistakes, for example, a state may adopt
balancing while bandwagoning produce higher resource
amounts, or, making no mistakes and achieving a boost in
their state’s domestic resources. In this sense, socialization
and competition do not indicate the imitation of the unipole’s
values and orientations but the elites’ emulation of fellow
states’ elites who become successful by enhancing their
countries’ resource levels. The emulation of elites by others
can lead to a hegemonic system even when there is initially a
unique state that adopts for bandwagoning. Thus, an
evolutionary game produces results at system-level by
assuming states as boundedly-rational non-unitary agents
[68].
Constructivism
Constructivism provides assumptions for a dynamic
framework of repeated interactions and concentrates on
continuing processes and practices such as socialization.
Socialization is constructivism’s “home turf” and “the
process of inducting actors into the norms and rules of a
given community” [69, 70]. The theory implies that
interactions among lesser states generate endogenous
changes in states’ identities, norms, and foreign policies in a
unipolar system. States’ interactions can generate different
types of anarchies [71]. States can either bandwagon with or
balance against the unipole, as it is “appropriate” to do so due
to norms and rules on-going interactions generate.
Consequently, the central constructivist problem is to find
those rules and norms that prevail in unipolar systems
through time. Constructivism does not lend a methodological
tool to study complex dynamic systems, however.
The evolutionary game theory is such a method. The
theory implies the ESS as rules and conventions but not as
conscious human designs established over time [72]. If
bandwagoning is the ESS, it becomes strongly established
among states as a self-enforcing rule. The deviant balancers,
or, equivalently, mutants, revert to bandwagoning under
evolutionary pressures of environment and fitness. All states
bandwagoning with the unipole create a new type or “culture”
of anarchy, that is, anarchy among states that finally accept
the unipole’s superiority [73].
Each point on any trajectory leading to an ESS can be
interpreted as displaying the dispersal of states’ actions
toward the unipole at some period of time. Moreover, any
34
Serdar Ş. Güner: An Evolutionary Game Analysis of Balancing and Bandwagoning in Unipolar Systems
dispersal depends upon the previous one; none can be
separated from earlier distribution of actions. Thus, early
state practices constitute new ones: it is impossible to treat
dispersals independently on an individual basis. In fact, the
interpretation parallels structuration theory, as paths toward
an ESS demonstrate mutual constitution of evolutionary
phases [74]. Each resource comparison is based upon shared
knowledge about whether bandwagoners or balancers get
fitter in the system. Any ESS emerges as a consequence of
on-going comparisons and self-enforcing social conventions.
Identities also change during practices. If bandwagoners
get fitter, balancers become bandwagoners in the next round.
For example, the variants 3 and 4 demonstrate how unipole’s
friends transform into balancers and foes into
bandwagoners.16 Suppose that both of states initially are all
v
balancers, the environment is V a 2v , and that the
2
type-1 states are the traditional friends of the unipole and the
type-2 states are the unipole’s traditional foes. Once states
recognize that some of them are the friends and some others
are the foes of the unipole, their actions become dependent
on these traits playing different roles, for example: “if we are
the friends of the unipole, then we adopt bandwagoning, if
we are the foes of the unipole, we adopt balancing, or vice
versa.” The role is the expected behavior of states possessing
a given identity. Thus, identities are positions and roles refer
to behavior [75]. As a result, socialization and competition
become role specific and consequences of enactment of roles,
as states differentiate themselves and assign each among
them to a different role. States are distinguished as
balancer-friends and bandwagoner-foes or vice versa while
their actions towards the unipole evolve.17
Furthermore, the knowledge about the asymmetry helps to
explain the selection of the ESS depending upon the initial
dispersal of states as friends and foes. If “friends” are
expected to align with the unipole, the evolutionary
dynamics can demonstrate that “foes” rather than friends
align with the unipole to such an extent that all friends turn
into balancers if foe-bandwagoners are more numerous than
friend-bandwagoners. The foreign policy model implies that
individual states’ decision makers’ role-conceptions and
unipolar environments determine the ultimate dispersal of
actions toward the unipole.
Assortative matching goes even one step further by
assuming that the friends of the unipole do not compare their
resource performances with those of the enemies of the
unipole at the outset, therefore conditional actions and roles
do not matter. And even if some differentiate each other as
unipole’s allies or enemies but some do not, the resulting
16
Pape (2005, 2006) argues that France, Germany, and even Japan are
nowadays involved in soft-balancing measures targeting the United States. Soft
balancing covers the use of international institutions, economic measures, and
diplomacy to delay, frustrate, and undermine unilateral U.S. policies without
directly challenging the United States. These moves can be interpreted as
balancing acts.
17
Thies (2003: 546) notes that role playing constitutes a bridge between
constructivism and structural realism proving an ideational meaning for
socialization processes.
ESS would not differ from those implied by random
matching in non-homogeneous populations. Hence,
socialization processes can lead to hegemony or global
balance when states’ actions are all pre-determined so that
their identities are all the same or role-playing applies to
some group within the population.
6. Conclusions
We aimed rather to generate useful insights than to
conduct a rigorous test. Using a simple evolutionary
framework, we investigated the evolutionary directions that
states’ actions toward a unipole can take. Few parameters
used produced different paths of competition and
socialization in unipolar systems. It is found that the
direction of these evolutionary processes depends upon the
environment of unipolar systems, resources available to
bandwagoners and balancers, cost of complying with the
unipole, and, in some cases, initial composition of the
population in terms of bandwagoners and balancers.
There are possible extensions of the model. One is the
inclusion of taking no action as a third strategy. Other
plausible extensions include the use of different dynamics
and game parameters. It is possible to assume that resources
shared among balancers and among bandwagoners sum up to
a constant value such as V + v = l with V, v > 0 amounting to
a relationship between V and v. If l = 1, for example, one unit
increment in one implies one unit reduction in the other.
The discussion is limited to general security terms. The
implications help to organize insights and simple ideas about
how states react to the unipole. The values of model’s three
parameters would differ across different global issues. To
bandwagon with the unipole in protection of international
environment, finance, proliferation of nuclear weapons
technology, or counterterrorism would not produce the same
amount of benefit and loss of autonomy cost. Structural
constraints would vary across the multiplicity of issues the
global agenda contains; therefore evolutionary paths would
vary accordingly. The superposition of these processes hints
at complex evolutions of strategies states may adopt towards
the unipole.
Appendix
Variant 1
The evolution of bandwagoning and balancing is given by:
dx
v
a V v
f ( BAN ) f ( BAL)
x V a (9)
2
2
dt
Letting m
write:
a V v
2
and n V a
dx
mx n
dt
v
, we can
2
(10)
Journal of Game Theory 2017, 6(2): 21-37
The general solution of the differential equation (10) is:
n
n
x(t ) emt
m
m
(11)
n 2(a V ) v
, and α
where m 0 by assumption,
m
a V v
denotes any initial condition.
n 2(a V ) v
Let k
. There are three stable
m
a V v
equilibria:
Case 1 If k ≥ 1, that is, V a 2v , x(t) converges to 1.
v
Case 2 If k ≤ 0, that is, V a , x(t) converges to 0.
2
v
Case 3 If V a 2v , x(t) converges to k.
2
Variant 2
The following system of differential equations describe
linear dynamics
assumption:
dx
v
a V v
f 1 ( BAN ) f 1 ( BAL)
y V a (12)
2
2
dt
dy
v
a V v
f 2 ( BAN ) f 2 ( BAL)
x V a (13)
2
2
dt
Hence,
dx/dt = my + n
(14)
dy/dt = mx + n
(15)
The system in matrix form is:
dx
dt
0 m x n
m 0 y n
dy
dt
Case 2 (0, 1) is stable if
V − a > 2v, or V − a < 2v.
Case 3 (1, 0) is stable if
v
V a 2v , it is unstable if
2
v
V a 2v , it is unstable if
2
V − a > 2v, or V − a < 2v.
Case 4 (1, 1) is stable if V − a > 2v, it is unstable if V − a <
2v.
Variant 3
The evolution of bandwagoning and balancing is given by:
dx
f ( BAN ) f ( BAL)
dt
(V a)(2 z ) v
( z 1)(V a v)
x
2
2
Letting m
( z 1)(V a v )
2
we can write:
and n
dx
mx n
dt
(18)
(V a )(2 z ) v
2
,
(19)
The solution of (19) is similar to the one of (10) with
n (V a)(2 z ) v
implying additional conditions
k
m (V a v)(1 z )
on the probability z:
Case 1 If k ≥ 1, that is, V a v v(1 z ) , x(t)
converges to 1.
Case 2 If k ≤ 0, that is, V a v (1 z )(V a) , x(t)
converges to 0.
Case 3 If 0 k 1 , that is, (1 z)(V a) V a v
v(1 z ) , x(t) converges to k.
(16)
0 m
The Coefficient matrix is therefore C (x, y) =
.
m 0
Setting | C − λI | = 0, we obtain the characteristic roots λ1 = m,
λ2 = − m. The general solution is therefore:
x k 1 ( )m 2n mt 1 mt
y k ( 2 )m ( )m 2n e 2 e
x(0)
(17)
where
y (0)
The solution of the system implies that:
v
Case 1 (0, 0) is stable if V − a <
, it is unstable if
2
v
.
V−a>
2
35
REFERENCES
[1]
Kahler, Miles (1999). “Evolution, Choice, and International
Change,” in David A. Lake and Robert Powell (eds),
Strategic Choice and International Relations. Princeton, New
Jersey: Princeton University Press, 165-196.
[2]
Friedman, Daniel (1991). “Evolutionary Games in
Economics,” Econometrica 59 (3): 637- 666; Friedman,
Daniel (1998). “On Economic Applications of Evolutionary
Game Theory,” Journal of Evolutionary Economics 8 (1):
15-43.
[3]
Maynard-Smith, John (1982). Evolution and the Theory of
Games. Cambridge: Cambridge University Press.
[4]
Osborne, Martin J. (2004). An Introduction to Game Theory.
Oxford: Oxford University Press.
[5]
Samuelson, Larry (1997). Evolutionary Games and
Equilibrium Selection. Cambridge, Massachusetts: MIT
Press.
36
Serdar Ş. Güner: An Evolutionary Game Analysis of Balancing and Bandwagoning in Unipolar Systems
[6]
Waltz, Kenneth N. (1979). Theory of International Politics.
Reading, Massachusetts: Addison-Wesley, p. 118.
[7]
Waltz, Kenneth N. (1993). “The Emerging Structure of
International Politics,” International Security 18 (2): 44-79;
Waltz, Kenneth N. (2000). “Structural Realism After the Cold
War,” International Security 25 (1): 5-41.
[8]
[9]
Layne, Christopher (1993). “The Unipolar Illusion: Why New
Great Powers Will Rise,” International Security 17 (4): 5-51;
Layne, Christopher (2006). “The Unipolar Illusion Revisited:
The Coming End of the United States’ Unipolar Moment,”
International Security 31 (2): 7-41.
Waltz, Kenneth N. (1979), p. 126.
72-108, p.76.
[27] Powell, Robert (1999) In the Shadow of Power: States and
Strategies in International Politics. Princeton: Princeton
University Press, pp.23-39.
[28] Aumann, Robert J. (1997). “Rationality and Bounded
Rationality,” Games and Economic Behavior 21 (1): 2-14, 3.
[29] Niebuhr, Reinhold (1941). The Nature and the Destiny of
Man: A Christian Interpretation. New York: Charles
Scribner’s Sons, p.189.
[30] Morgenthau, Hans J. (1960). Politics among Nations, 3rd
edition. New York: Knopf, pp.10-12.
[10] Walt, Stephen (1988). “Testing Theories of Alliance
Formation,” International Organization 43 (2): 275-316;
Walt, Stephen (1987). The Origins of Alliances (Ithaca: New
York, Cornell University Press.
[31] Waltz, Kenneth N. (1979), pp. 91-92.
[11] Boulding, Kenneth E. (1961). Conflict and Defense. Boston:
Harper and Row.
[33] Krauthammer, Charles (2002). “The Unipolar Moment is
Revisited,” The National Interest 70 (1): 5-17, p.8.
[12] Quester, George H. (1977). Offense and Defense in the
International System. New York: Wiley.
[34] Pape, Robert A. (2006) “Soft-Balancing and the
Consequences of Serial Preventive War,” unpublished
manuscript, p.14.
[13] Walt, Stephen (2009). “Alliances in a Unipolar World,”
World Politics 61 (1): 86-120.
[32] Wohlforth William C. (2009). “Unipolarity, Status
Competition, and Great Power War,” World Politics 61 (1):
28-57.
[35] Waltz, Kenneth N. (1979), p.105.
[14] Jervis, Robert (2009). “Unipolarity: A Structural Perspective,”
[36] Powell, Robert (1991). “Absolute and Relative Gains in
World Politics 61 (1): 188-213.
International Relations Theory,” American Political Science
Review 85 (4): 1303-1320.
[15] Wohlforth, William C. (1999). “The Stability of a Unipolar
World,” International Security 24 (1): 5-41, p.25.
[37] Powell, Robert (1994). “Anarchy in International Relations
Theory: the Neorealist-Neoliberal Debate,” International
[16] Nye, Joseph S. (2002). The Paradox of American Power: Why
Organization 48 (2): 335-338.
the World’s Only Superpower Can’t Go It Alone, New York:
Oxford University Press.
[38] Elman, Colin (2003). “Appraising Balance of Power Theory,”
in John A. Vasquez and Colin Elman (eds), Realism and
[17] kenberry, John G. (1998/99). “Institutions, Strategic Restraint,
Balancing of Power. New Jersey: Prentice Hall: 1-22, p.8
and the Persistence of Postwar Order,” International Security
23 (3): 43-78.
[39] Morrow, James D. (1991). “Alliances and Asymmetry: An
Alternative to the Capability Aggregation Model of
[18] Ikenberry, John G. (2003) “Strategic Reactions to American
Alliances,” American Journal of Political Science 35 (4):
Preeminence: Great Power Politics in the Age of Unipolarity,”
904-933, p.909, 913.
report to the National Intelligence Council.
[19] Posen, Barry R. and Andrew L. Ross (1996/97). “Competing
Visions for U.S. Grand Strategy,” International Security 21
(3): 5-53, p.6.
[40] Altfeld, Michael F. (1984). “The Decision to Ally: A Theory
and Test,” The Western Political Quarterly 37 (4): 523-544,
p.526.
[20] Posen, Barry R. and Andrew L. Ross (1996/97), p.33.
[41] Reus-Smit, Christian (2006). “Unipolarity and Legitimacy,”
unpublished manuscript, 2006, pp.2-6.
[21] Lieber, Keir A. and Gerard Alexander (2005). “Waiting for
Balancing: Why the World is Not Pushing Back,”
International Security 30 (1): 109-139, p.133.
[22] Simes, Dimitri (2003). “America’s Imperial Dilemma,”
Foreign Affairs 82 (6): 91-102, p.95.
[23] Pape, Robert A. (2005) “Soft-Balancing against the United
States,” International Security 30 (1): 7-45, p.10.
[42] Kupchan, Charles A. (1998). “After Pax Americana: Benign
Power, Regional Integration, and the Sources of a Stable
Multipolarity,” International Security 23 (2): 40-79,
pp.42-47.
[43] Layne, Christopher (1993). “The Unipolar Illusion: Why New
Great Powers Will Rise,” International Security 17 (4): 5-51,
p.7
[25] Lieber, Keir A. and Gerard Alexander (2005). “Waiting for
Balancing: Why the World is Not Pushing Back,”
International Security 30 (1): 109-139, p.125.
[44] Mastanduno, Michael and Ethan B. Kapstein (1999).
“Realism and State Strategies after the Cold War,” in Ethan A.
Kapstein and Michael Mastanduno (eds), Unipolar Politics:
Realism and State Strategies After the Cold War New York:
Columbia University Press, 1-27.
[26] Brooks, Stephen G. and William C. Wohlforth (2005). “Hard
Times for Soft Balancing,” International Security 30 (1):
[45] Fearon, James D. (1998) “Domestic Politics, Foreign Policy,
and Theories of International Relations,” Annual Review of
[24] Pape, Robert A. (2005), p.35.
Journal of Game Theory 2017, 6(2): 21-37
Political Science 1 (1): 289-313, p.296.
[46] Allison, Graham T. (1971). Essence of Decision: Explaining
the Cuban Missile Crisis. Boston: Little, Brown.
[47] Bendor Jonathan and Thomas H. Hammond (1992).
“Rethinking Allison’s Models,” The American Political
Science Review 86 (2): 301-322.
[48] Jervis, Robert (1976). Perception and Misperception in
International Politics. Princeton: Princeton University Press.
[49] Bueno de Mesquita, Bruce and Lalman, David (1992). War
and Reason: Domestic and International Imperatives. New
Haven: Yale University Press.
[50] Levy, Jack S. (1994). “Learning and Foreign Policy:
Sweeping a Conceptual Minefield,” International
Organization 48 (2): 279-312.
[51] Hamilton, William D. (1964) “The Genetical Evolution of
Social Behavior,” Journal of Theoretical Biology 7 (1): 1-16.
[52] Osborne, Martin J. (2004). An Introduction to Game Theory.
Oxford: Oxford University Press, p.406.
[53] Samuelson, Larry (1997). Evolutionary Games and
Equilibrium Selection. Cambridge, Massachusetts: MIT Press,
p.41.
[54] Hargreaves Heap, Shaun P. and Yanis Varoufakis (1995).
Game Theory: A Critical Introduction. New York: Routledge,
p.200.
37
What Security Dilemma?” in Benjamin Frankel (ed), Realism:
Restatements and Renewal. London: Frank Cass, 90-121;
Elman, Colin and Miriam F. Elman (1995). “History vs.
Neorealism: A Second Look,” International Security 20 (1):
182-193, p.185.
[62] Modelski, George and Kazimierz Poznanski (1996).
“Evolutionary Paradigms in the Social Sciences,”
International Studies Quarterly 40 (3): 315-319, p.319.
[63] Kahler, Miles (1999). “Evolution, Choice, and International
Change,” in David A. Lake and Robert Powell (eds),
Strategic Choice and International Relations. Princeton, New
Jersey: Princeton University Press, 165-196, p.181.
[64] Waltz, Kenneth N. (1979), pp.74-77.
[65] Moravcsik, Andrew (1997). “Taking Preferences Seriously:
A Liberal Theory of International Politics,” International
Organization 51 (4): 513-53.
[66] Ikenberry John G. and Charles A. Kupchan (1990).
“Socialization and Hegemonic Power,” International
Organization 44 (3): 283-315, p.289.
[67] Ikenberry John G. and Charles A. Kupchan (1990), p.284,
290.
[68] Fearon, James D. (1998) “Domestic Politics, Foreign Policy,
and Theories of International Relations,” Annual Review of
Political Science 1 (1): 289-313, p.298-301.
[55] Levy, Jack S. (1985). “Theories of General War,” World
Politics 37 (3): 344-374, p.348.
[69] Zürn, Michael and Jeffrey T. Checkel (2005). “Getting
Socialized to Build Bridges: Constructivism and Rationalism,
Europe and the Nation-State,” International Organization 59
(4): 1045-1079.
[56] Keohane, Robert O. (1984). After Hegemony: Compliance
and Discord in the World Political Economy. Princeton:
Princeton University Press, p.32.
[70] Finnemore, Martha and Kathryn Sikkink (1998).
“International Norm Dynamics and Political Change,”
International Organization 52 (4): 887-917, p.914.
[57] Gilpin, Robert (1981) War and Change in World Politics.
Cambridge: Cambridge University Press, p.29.
[71] Wendt, Alexander (1992). “Anarchy is What States Make of
It: the Social Construction of Power Politics,” International
Organization 46 (2): 391-425.
[58] Hart, Jeffrey (1976). “Three Approaches to the Measurement
of Power in International Relations,” International
Organization 30 (2): 289-305.
[59] Keohane, Robert O. (1984). After Hegemony: Compliance
and Discord in the World Political Economy. Princeton:
Princeton University Press, p.45.
[60] Ikenberry John G. and Charles A. Kupchan (1990).
“Socialization and Hegemonic Power,” International
Organization 44 (3): 283-315, p.57.
[61] Schweller, Randall L. (1996). “Neorealism’s Status Quo Bias:
[72] Sugden, Robert (1989). “Spontaneous Order,” Journal of
Economic Perspectives 3 (4): 85-97.
[73] Wendt, Alexander (1992).
[74] Wendt, Alexander (1987). “The Agent-Structure Problem in
International relations Theory,” International Organization
41 (3): 335-370.
[75] Holsti, Kalevi J. (1970). “National Role Conceptions in the
Study of Foreign Policy,” International Studies Quarterly 14
(3): 233-309.