1. A Marketing Game:
Consumer Choice and the Ising Model
Matthew G. Reyes
mgreyes@umich.edu
Abstract—In this paper we present an outline for a market-
ing/consumer choice problem using graphical games and Markov
random fields. We consider a simple marketing scenario and cast
it as a coordination game on a network. We consider Glauber
dynamics for the strategy update dynamics and then consider
the corresponding equilibrium Ising distribution. The parameters
of the network are the interaction and marketing strengths, the
former being the exponential parameters on statistics describing
the relative strategies of the players on the network, the latter
being the exponential parameters applied to the strategies of the
players, representing an external field. We discuss how marketers
for the two brands can estimate the network parameter from data
and thereby estimate not only the interaction strengths, but also
his opponent’s marketing strength. Connecting an equilibrium
Gibbs distribution with the set of typical configurations, we
discuss how marketers can then use understanding of the
interaction and marketing strengths to select a new marketing
strength to help drive the current configuration into the typical
set for a new equilibrium Gibbs distribution. We briefly discuss
the possible influence of phase transitions on this approach.
I. HIGH LEVEL PICTURE
There is intense and heightening interest in the mechanisms
by and rates at which networks of interconnected individuals
converge to a sort of collective understanding or pattern in
their stated and confirmed opinions and preference [39], [30],
[11], [35], [28]. Furthermore, social scientists would like to
know how to use such understanding to effect desired political
or marketing goals1
[37], [14], among other concerns. The
abundance of available data, in terms of network structure as
gleaned from Facebook, LinkedIn and Instagram, for example,
as well as information on consumer purchases and preferences,
as collected by outfits such as Target, Wal-Mart and Google,
underlies this push to fathom and influence network behavior.
The self-organization and adaptivity of such systems have been
addressed from a range of scientific [20], [24], mathematical
[10], [18], [28], and philosophical [2], [19] perspectives. As
such there is much to be gained by casting the complexities
of interesting social network problems onto the rigorous and
general foundation of such systems. Reference [40] provides
a good introduction to the different lines of research into both
models of network behavior and algorithms for optimizing
marketing resources.
In this paper we consider a network of consumers engaged
in coordination games with each other. We cast the network
1Formerly with The University of Michigan and MIT Lincoln Laboratory,
the author is currently laying the groundwork for a marketing consulting
business based upon, among other things, analysis of social network data.
Check the author’s LinkedIn page starting September 1, 2015 for more details.
dynamics in the framework of Gibbs (i.e., Markov) random
fields and, using this framework, outline a hypothetical game
that marketers can use to influence the selection of preferences
across the network. To be sure, Gibbs fields have been
considered for modeling interaction games by a number of
researchers [8], [41], [28] and [35] has considered the selection
of optimal subsets of players to target to achieve a profit goal.
What sets the focus of the present work apart from previous
efforts is the emphasis on the typical set for the equilibrium
Gibbs distribution and drawing connections between patterns
in typical sets for different equilibrium Gibbs distributions
on the network and desired patterns of brand expression
that indicate certain marketing or sales milestones. Then by
parameterizing the original network game into an equilibrium
Gibbs distribution we can incorporate the marketing efforts
for the different companies into the model in the form of
an external field. Combining the influences of the social
network and marketing efforts into a single model allows
us to better understand the combined effects of these forces
on the evolution of norms and provides a more systematic
approach for marketers to select the allocation of resources.
For example, while [35] introduced the marketer’s selection
of an optimal subset of players to whom to market, it does
not take into account the influence that the marketer has on
the evolution of the equilibrium set of configurations. While
the contribution of this paper is more prose than proof, we
aim to present an approach to combining both the modeling
of network interactions with marketing efforts into a single
model and develop a story that provides a template for doing
data analytics. This paper is a first step toward that.
A. Social Network Interactions and Equilibrium Behavior
Game-theory is the study of strategic decision-making,
where a strategy is an action, belief, or preference2
that an
individual makes or holds and in response to which receives
a measure of utility called a payoff. For instance, in the
two-person, two-choice coordination game considered in this
paper, participants receive a higher payoff if they choose the
same strategy. This type of game applies to many real-world
situations, for example, phone plans where callers receive
discounts when talking to other customers with the same
carrier. A pair of individuals is engaged in this game if they
talk on the phone with enough regularity to influence each
2For example, the purchase of a particular brand; the belief that a brand
possesses a certain property; or preference for one brand over others.

2. others’ decisions. Two people engaged in a game are said to
be neighbors in the network. Depending on the strengths of
strengths of interaction between neighbors certain patterns will
appear in the steady state steady-state configurations . This is
because the combined effect of pairwise incentives imparts
a pattern to the global choices made. For different marketing
aims, certain patterns in strategy choices on the network could
represent a tipping point of sorts and is therefore sought
after. As the preferences on a social network change with
time the pattern may change in ways that would be too
tedious to catalog and too many in number, and uncertain
in scope, to attempt any real understanding of or hope to
have any influence over. The notion of a physical system
whose macroscopic behavior can be characterized through a
statistical treatment of the complex microscopic interactions
therein is that of a Gibbs random field [13]. The idea is that
we describe the network dynamics not by the specific decisions
at any one time, but rather by patterns in the configurations of
strategies that tend to appear the most in the evolution of the
dynamics. The rigorous treatment of Gibbs distributions arose
from the introduction of the Ising model as an explanation for
the spontaneous magnetization of iron [20].
A key feature of a social network interaction that determines
the influence of pairwise interactions on the emergence of
a global pattern in network choices is the topology of the
network. In other words, who’s connected to whom, where
two individuals are said to be connected if they are neighbors.
Properties such as the number steps to go from one person to
the next, the presence of cycles and other substructures, and
so on, are topological properties. Research has established that
social networks possess certain topological properties, such
as a degree distribution that follows a power law, termed
a scale-free topology; as well as short average path length
and high clustering coefficient [30], termed the Small World
topology [30]. Furthermore, it has been shown that social
networks consist of many small communities within which the
ties or connections are strong, with these small communities
connected through weaker ties [30], [11]. It has been argued
that these topological properties are key in the propagation of
norms through a social network [30], [11]. There are ways
to generate both a scale-free topology, a small-world starting
from a regular graph [30]. An example of both a 5 × 5 grid
graph and a small-world graph generated from it are shown
in Figure 1 (a) and (b), respectively. For convenience, this
examples shown in the paper will be on grid graphs.
In addition to the network topology, what determines the
spread of a particular idea or preference for a particular brand
are the dynamics on the network. The dynamics refer to
the influence of players’ strategies on those of other players
in the network. Two prominent types of dynamics models
considered in the literature are cascade models which describe
viral behavior where preferences percolate through a network
by probabilistically propagating from neighbor to neighbor
[30], [26]; and local interaction game models, in which
individuals update their preferences based on the preferences
of their neighbors and the respective games played between the
(a) (b) (c)
Fig. 1. (a) Original 5 × 5 grid graph; (b) New graph with edges re-wired
with probability .5; and (c) A 5 × 5 grid subgraph of an infinite graph.
neighbors [12], [41], [22]. Both cascade and local interaction
game models consider the time evolution of the configuration
of strategies on the network. In [8], the connection is made
between the temporal dynamics of local interaction games and
Gibbs distributions describing the equilibrium configurations
of strategies on the network.
There are two interesting phases to the dynamics on a
network. The transient phase and the equilibrium phase. With
regard to the spread of a preference throughout the network,
it is the transient phase that is the most interesting, in that
gives us an idea of how fast a preference will spread through
the network before an equilibrium is reached. Considerable
work has looked at the speed at which cascade models spread
through networks of different topologies [26], and the results
have cast doubt [39] on the supposition that the adoption
of a brand by certain key influentials triggers a cascade
of preference for that brand. Furthermore, recent research
shows that the rates of convergence for local interaction game
models on various topologies are diametric to those under
cascade models, suggesting that local interaction games may
be better suited to explain the spread of preference through a
network [28]. In this paper we consider local interaction games
with stochastic-response Glauber dynamics in which players
update their decisions randomly according to a probability
distributions that is log-linear in the payoff margins.
In this paper we are looking at the equilibrium Gibbs
distribution to which the Glauber dynamics converge. By
expressing the payoff margin as the product of a statistic
that measures whether the strategies of two players agree
or disagree, and an exponential parameter which quantifies
the strength of interaction between the two players, we can
cast the network coordination game in the Gibbs/Glauber
framework, which we will use in setting up the game that
marketers can play, discussed in Section IV. Part of this game
involves estimating the interaction strengths on the network. It
is well-known that Gibbs distributions are maximum-entropy
distributions over the space of strategy configurations, subject
to constraints on the expected values of the statistics. Thus
one can collect data and derive a Gibbs distribution which is
consistent with the data but assumes no more about the random
evolution of the strategy configuration on the network.
An equilibrium Gibbs distribution is stationary in the sense
that, once equilibrium is reached, the distribution on the set

3. (a) (b) (c)
Fig. 2. Typical images from an Ising distribution on a 200 × 200 4-pt. grid
with uniform interaction strength θ = (a) .4; (b) .44; and (c) .5.
of configurations is the same at each time. In general network
parameters will change with time [1]. Thus the idea presented
here is a first step in a more complicated model where transient
behavior is accounted for. An equilibrium Gibbs distribution
will put nearly all of its mass roughly uniformly on a relatively
small subset of configurations called the typical set [9]. In
Figure 2 we can see typical images from the Ising distributions.
Therefore we can associate to each set of interaction strengths,
which are exponential parameters in the Gibbs distribution, a
set of strategy configurations that ‘look like’ the distribution
in terms of the particular statistic that represents the game. For
example, Best-response dynamics is a limiting case of Glauber
dynamics with infinite interaction strength. The typical set for
this extreme Gibbs distribution consists of the configuration
where everyone chooses brand A and the configuration where
everyone chooses brand B. However, Glauber dynamics ap-
plied to one of these typical configurations will never reach the
other typical configuration which is related to reasons given
in [28] for why Nash equilibria may not be a good way to
understand network dynamics. This concept is related to phase
transitions in families of Gibbs distributions, will be discussed
briefly in Section V.
B. Gibbs Fields for Marketing Analytics
Beyond understanding the dynamics that contribute to the
spread of preferences on a network, there is a desire to use
this understanding to promote the widespread preference for a
particular brand or idea [40], [23], [35]. This generally takes
the form of selecting which players in the network to target
with marketing efforts. In [23], for example, cascade dynamics
are considered, while in [35] equilibrium Gibbs distributions
are considered. Thus our approach is aligned with [35] in
that we are casting it in the steady-state Gibbs frame work
yet still includes the temporal dynamics through the Glauber
specification, and as such is similar to [23].
At a given time we want to select a correct group of
players to target with marketing efforts. In Section IV we
propose a game that marketers from competing companies
can play against each other to influence the expression of
brand preferences on the network. To begin, add an external
field to the Gibbs distribution to represent marketing strength
applied to the certain players. An external biases the strategies
of the players [25]. Each of the two marketers will choose
a subset of sites to apply their marketing strength upon.
The combined marketing strength at a site will be the sum
of the marketing strengths for the two marketers. Where to
apply marketing strength and to what extent depends on a
number of factors. For certain interaction strengths, more or
less marketing strength may need to be applied to achieve a
desired effect on the network. Moreover, if we could learn or
estimate the marketing strength that our opponent is applying,
this would also allow us to allocate marketing resources
(near) optimally. Gibbs distributions are parameterized by dual
coordinate systems [3]. The first set of coordinate are the
interaction and marketing strengths. The second coordinate
system is the set of expected statistics and expected strategies
on the network. By collecting data in the form of sales infor-
mation we can determine averages for the relevant statistics
and estimate the network parameters. This estimate will allow
him to make a (near) optimal choice in marketing strength the
next time.
A marketer may express his goals in terms sets of config-
urations of strategies on the network that possess a desired
property. For instance, a uniform Ising model with a high
enough interaction strength will produce typical images that
have giant monotone clusters of sites. Such clustering of
sites, interpreted in the current marketing scenario, would
correspond to positive brand expression and is an example
of a property that a marketer may seek. Each marketer could
then define a subset of configurations that possess a favorable
property with respect to his brand. This subset of configura-
tions would then serve as a target set of configurations towards
which they would like to drive the evolution of the typical
set for the random strategy process. If a marketer associates
patterns within typical sets of different Gibbs distributions
with such a desired set of configurations, by modifying the
his marketing strength he may be able to tip the market in his
favor.
C. Infinite Networks and Multiple Equilibria
Though all networks are finite, it can helpful to illuminate
the behavior of a large network by considering the properties
of an infinite network. In the case of Gibbs measures on
infinite networks, they are specified by (conditional) Gibbs
distributions on all finite subsets given all possible boundary
configurations. Any probability measure whose conditional
probabilities agree with these conditional (Gibbs) distributions
will be called a Gibbs measure relative to the given specifi-
cation [13]. A Gibbs measure is a macroscopic equilibrium
distribution over the set of strategies. It is sometimes the
case that there are multiple equilibrium Gibbs measures for
a given specification. Each such equilibrium Gibbs measure
is considered to be a distinct ‘mode’ of the random field. A
convenient way of thinking about this is that nature chooses
one of the modes at a time minus infinity, and subsequently
each configuration on the network is drawn from that chosen
Gibbs measure. We just do not know which mode was chosen.
If there is only one mode for an equilibrium Gibbs distribution,
then the random process that converged to that distribution
is ergodic, which means empirical averages equal statistical

4. averages. If there are multiple modes, the dynamimcs are non-
ergodic and we cannot estimate statistical probabilities from
data [16].
For example the phenomenon of spontaneous magnetization
is an example of a phase transition. Even when there is no
external magnetic field applied, a bar of iron will sponta-
neously magnetize at a given temperature. When this happens
the distribution of atomic spins moves away from the fifty-fifty
split and a large clustering of sites will have the same spin.
We can think of this as the typical set splitting into subsets
such that once the process is in one of the subsets of these
configurations, the process will not reach any of the other
subsets. In terms of the random process of evolving strategies,
we say that two such typical sets do not communicate. We can
see typical images for a different values of the exponential
parameter for uniform Ising models in Figure 2. We can see
that as the exponential parameter is increased the sites become
grouped together in larger and larger clusters, though these
clusters remain sufficiently far apart from one another. Once
the exponential parameter reaches the critical value, solved
by Onsager in 1944, these smaller clusters merge with one
another an form a giant cluster that extends throughout the
network. Essentially, the influence of the initial pattern of
strategies in the network does not fade away and the particular
equilibrium Gibbs measure observe is determined by both the
initial configuration and the Glauber dynamics that specify the
Gibbs measure.
The problem of multiple equilibria in networks of local
interaction games has been considered in the literature [8],
[41], [12], [28]. Much analysis has centered on best-response
dynamics and associated Nash equilibria. A complaint of Nash
equilibria is given in [28] because of the multiple equilibria
of the best-response dynamics. Best-response dynamics is a
limiting example of Glauber dynamics in which all inter-
action strengths on the network are infinite. In the absence
of an external field, best-response dynamics is non-ergodic
and converging to one of multiple non-communicating and
degenerate typical sets. Our model includes marketing in the
Gibbs distribution so it will be interesting in the future to see if
applying an external field influences the observed degeneracy
of best-response dynamics. In either case, with regards to our
approach, care will need to be taken in understanding the
existence of multiple ‘disjoint’ typical sets associated with a
given interaction and marketing strength, and in how to take
advantage or or guard against any interesting consequences of
the phase transition on the proposed game. We do not take up
this matter in the present paper, though we do introduce the
notion of a phase transition in Section V.
D. Outline of Paper
In Section II we define graphs and a network coordination
game with Glauber dynamics. In Section III we discuss Gibbs
distributions and typical sets. In Section IV we introduce a
game using an external field to represent marketing strength
and drive the typical set of an equilibrium Gibbs distribution
to a new typical set possessing desired properties. In Section
V we discuss the presence of phase transitions and what this
may mean for the marketing game we have in mind, and in
Section VI we discuss some experiments.
II. NETWORK COORDINATION GAME
Consider a set of sites V where a site corresponds to a
person in a social network, a network consisting of peo-
ple whose neighbor relations are determined through direct
contact through the telephone. We will first consider the
case where V is finite, the case where V is infinite will
be considered in Section V. A set of edges E consists of
pairs of adjacent players i and j, regarded as neighbors. The
object G = (V, E) is a graph with sites or players V and
undirected edges E. For each site i ∈ V , Ωi is the finite
alphabet or strategy space for player i. Here, the strategy
space is the same Ωi = Ω = {−1, 1} for all players i ∈ V .
For concreteness, think of two phone carriers, the two values
in Ω corresponding to the two carriers. Let xi = −1 indicate
that player i prefers or is a customer of the brand assigned −1,
and likewise if xi = 1. Time will be taken to be discrete and
will be denoted by n, taking values in the sequence of non-
negative integers {0, 1 . . . , n, . . .}. For i ∈ V , X
(n)
i denotes
the random strategy of player i, where x
(n)
i is the actual
value assumed at time n. At time n, the collection of random
variables X(n)
= {X
(n)
i } is a random field and x(n)
= {x
(n)
i }
is the image or configuration on the network at time n.
For {i, j} ∈ E, players i and j are engaged in a two-choice
coordination game. The payoff matrix for this game is Pij and
is denoted by
Pij =
aij bij
bij aij
, (1)
where aij > bij indicates a higher payoff when the players
agree versus disagree. In this case Pij(xi, xj) indicates the
payoff when player i plays strategy xi and player j plays
strategy xj.
Consider the symmetric coordination game being played
between all pairs of nodes i and j, with {i, j} ∈ E. For
each edge {i, j} ∈ E, the payoff matrix Pij from (1) can
be rewritten as
Pij =
aij +bij
2 +
aij −bij
2
aij +bij
2 −
aij −bij
2
aij +bij
2 −
aij −bij
2
aij +bij
2 +
aij −bij
2
= pij +
aij −bij
2 −
aij −bij
2
−
aij −bij
2
aij −bij
2
= pij +
θij −θij
−θij θij
, (2)
Since a term of the payoff, pij, is constant over all images
on V , we can ignore it, since the interaction between players
i and j is in the margin between a guaranteed payoff and the
maximum payoff. This is easy to see by considering the case
that a = b. Regardless of how large the value is, if the payoff is
the same no matter what strategies are played by i and j, then

5. players i and j are not really playing a game as their combined
decision do not affect their payoffs. Therefore, we can now say
that when player i picks choice xi and player j picks choice
xj the payoff is Pij(xi, xj) = θijtij(xi, xj). A more concise
representation is given by Pij(xi, xj) = θijtij(xi, xj), letting
tij(Xi, Xj) denote a random statistic (measurement) given by
tij(xi, xj) =
1 xi = xj
−1 xi = xj
We assume that at each time instant n a single player
updates his strategy. This assumption is more or less universal
and is based on the idea that each player updates his strategy
according to a Poisson clock and the well-known fact that
the probability of two arrivals at the same time instant is
essentially zero. Therefore, we can view our time variable
n as incrementing when a player updates his strategy. In
terms of the original and real continuous clock, the strategies
of the players remains constant between times i and i + 1.
When player i updates or revises his strategy based on the
strategies played by his neighbors, he does so according to the
stochastic-response Glauber dynamics, where the probability
of selecting strategy xi conditioned on the current strategies
of his neighbors, after cancelation of common terms, is given
by
p(xi|x∂i; θ) =
exp{
j∈∂i
θijtij(xi, xj)}
xi
exp{
j∈∂i
θijtij(xi, xj)}
. (3)
The point is not that player i observes the choices of his
neighbors, computes this probability distribution, then through
a sequence of coin flips chooses his next strategy. Rather, the
use of random choice is to account for the fact that any one
particular decision depends on too many things to enumerate,
and trying to determine the appropriate mapping between all
of these factors and a decision would be too tedious. The
use of a log-linear distribution owes to its maximum entropy
properties. In the case of the two-choice coordination game
statistic, (3) simplifies even further to
p(−xi|x∂i; θ) =
1
1 + exp{−
j∈∂i
θijtij(xi, xj)}
. (4)
For further simplification we can let the each interaction
strength θij be the same value β. Under this Ising model with
uniform interaction strength, the strategy update for player i
is given by the uniform Ising model
p(xi|x∂i; β) =
1
1 + exp{β
j∈∂i
xixj}
. (5)
We refer to the temporal sequence X(0)
, X(1)
, . . . , X(n)
, . . .
as a random process, and x(0)
, x(1)
, . . . , x(n)
, . . . as the se-
quence of strategy configurations on the network. Plainly
stated, game-theory considers the temporal updates from the
image x(n)
at the current time to the next image x(n+1)
at
the next time based on decisions by the players, where the
decision by player i is a (random) rule for updating his choice
x
(n)
i at the current time to his choice x
(n)
i+1 at the next time.
III. TYPICAL CONFIGURATIONS
The Glauber dynamics converge to an equilibrium Gibbs
distribution. The typical set for this equilibrium Gibbs distri-
bution will consist of configurations which statistically ‘look
like’ the distribution in some sense. A consequence of the
Asymptotic Equipartition Property [9] is that the equilibrium
Gibbs distribution places all of its mass uniformly on the
typical set. In this way we can associate a particular Gibbs
distribution with a set of configurations of strategies that
presumably shared certain patterns or properties.
If the players update their respective strategies according the
dynamics of (3) for a sufficiently long time3
, then the random
process {X(n)
} will converge to the following equilibrium
Gibbs distribution:
p(x; θ) = exp{
{i,j}∈E
θijtij(xi, xj) − Φ(θ)}
= exp{ θ, t(x) − Φ(θ)},
where
Φ(θ) = log
x∈X
exp{ θ, t(x) }
is the log-partition function.
The vector statistic t = (tij) defines a family of exponential
(Gibbs) distributions F = {p(·; θ) | θ ∈ Θ} based on t.
The parameter θ representing the interaction strengths of the
network indexes a particular Gibbs distribution in F. the set
Θ = {θ ∈ R
|E|
+ | Φ(θ) < ∞}
is the set of admissable exponential parameters and is a
coordinate system for the family of Gibbs distributions on G
based on t. The statistic for the equilibrium Gibbs distribution
towards which the network coordination game of the previous
section converges is that of the Ising model and can be
expressed as
tij(xi, xj) = xixj.
in which case the equilibrium distribution is
p(x; β) = exp{β
{i,j}∈E
xixj − Φ(β)},
This distribution can be reexpressed in terms of the number
of odd bonds, or edges where the corresponding players
3Convergence times for Glauber and other metropolis type algorithms is an
area of active research.

6. disagree in their strategy. When the interaction strengths are
not uniform, odd and even bonds are weighted according to
the respective interaction strengths. For a stationary (equi-
librium) distribution over the set of possible configurations
of strategies on a finite network, the equilibrium distribution
actually places its mass roughly uniformly on a relatively
small subset of configurations called the typical set. There are
multiple definitions of typicality. For example, weak or entropy
typicality of a configuration is satisfied if the probability of
the configuration under the stationary distribution is roughly
2 to the negative logarithm of the entropy of the process, i.e..
We denote the typical set as ΩT
(θ). Typical configurations
for the Ising model, and thus the network coordination game
that it models can then be classified according to the numbers
of odd bonds they possess. In this way it makes sense to
think of configurations ‘looking like’ a particular distribution
or interaction strength. In Figure 2 we see typical image from
different uniform interaction strength Ising models.
Moreover, the statistic for the Ising model has the property
that for positive exponential parameters, any two components
of the statistic tij and tkl have positive covariance. This was
shown by Griffiths [17] in the case of the Ising model. In [31],
we termed this property positive correlation and have showed
[31], [32], [33] that families of Gibbs distributions defined by
a statistic having this property have certain monotonicity prop-
erties. Looking at coordination game dynamics as a stochastic-
response with interaction strength θ, it is clear that best-
response dynamics that tend towards Nash equilibria is the
limiting case where θ = ∞. The limiting assumption of best-
response dynamics, then, may be appropriate for the dynamics
within smaller communities, though when considering ties
between these small communities, it is likely too extreme.
Conclusions from [28] make a compelling argument against
Nash equilibria because the different equilibria are rather
diverse. We believe that looking at a local interaction game
on a network in the Gibbs/Glauber framework permits focus
not on the extreme. This makes sense, because as θ → ∞,
the mass of the equilibrium Gibbs distribution is on the all
‘-1’s configuration and the all ‘+1’s configuration. Thus the
configurations one would expect to be ‘drawn from’ this
distribution will be close to a uniform configuration. That such
configurations are so diverse is explored later in Section V.
As actual social networks consist of many small communities
connected by ‘strong ties’, with these small communities
connected to each other through ‘weak ties’, best-response
dynamics may be appropriate on small subsets of nodes but on
the larger scale, general Glauber dynamics is more compelling.
IV. MARKETING GAME:
DRIVING THE EVOLUTION OF THE TYPICAL SET
In this section we introduce a game that two opposing
marketers, A and B, can engage in to influence brand ex-
pression on the network. The version of the game that we
present here is a simplified version of a more complicated
game that can be played. Marketers A and B will apply
external fields to select players in the network. The external
field represents marketing strength that influences the strategy
update dynamics in addition to the interaction strengths that
represent the influence due to social affiliations. To do this we
introduce an identity statistic ti(·) for each player. Moreover,
we look at the exponential parameter θ as now consisting
of a portion θE corresponding to interaction strengths, and
a portion θV corresponding to marketing strengths. The in-
teraction strengths θE are assumed to be constant, but the
marketing strengths θ
(n)
V will be updated by marketers A and
B and will thus be superscripted as such. The marketing
strength from player A at time n is denoted ψ
(n)
V |A, and
likewise for player B and ψ
(n)
V |B. The total marketing strength
is θ
(n)
V = ψ
(n)
V |A + ψ
(n)
V |B. Without loss of generality we let
θA 0 and θB 0. Thus the marketing strength ψ
(n)
V |A
biases the strategies of the players to -1, whereas ψ
(n)
V |B biases
the strategies of the players to 1. When displaying update
equations for the players we will omit the time index for
convenience.
The Glauber dynamics for this situation are now
p(−xi|x∂i; θ) =
1
1 + exp{θixi) +
j∈∂i
θijxixj}
,
where we can think of θi has the payoff (cost) for agreeing
(disagreeing) with the external field. These dynamics will
converge to the equilibrium distribution given by
p(x; θ) = exp{
i∈V
θixi +
{i,j}∈E
θij ixj − Φ(θ)},
and configurations will be typical for these interaction and
marketing strengths.
We assume that at time 0, the random process given
by the Glauber dynamics has converged to its equilib-
rium distribution. As the sequence of network configurations
x(0)
, x(1)
, . . . , x(n−1)
evolves marketers A and B collect a
corresponding sequence of data ¯x(0)
, ¯x(1)
, . . . , ¯x(n−1)
. The ¯x
are in general noisy versions of the true x. Noise could repre-
sent missed collects due to limited resources, or corruption of
signal due to channel error. Here we will consider the noiseless
case. While in general we would want to distinguish between
¯x collected by marketer A versus ¯x collected by marketer B,
we will not do so here for simplicity. The data of interest is the
sequence of statistics t(0)
= t(¯x(0)
), t(1)
= t(¯x(1)
), . . . , t(1)
=
t(¯x(n−1)
). From this we compute the statistic
¯t(n)
=
1
n
m−1
i=0
t(x(n−i)
).
From this information the marketers would like to estimate
θ(n)
, from which they have an estimate of the interaction
strengths θ
(n)
E . This would also give them an estimate of the
marketing strengths θ
(n)
V . And since they know the value of
ψV |A (ψV |B), they could estimate ψV |B (ψV |A).

7. (a) (b) (c)
(d) (e) (f)
Fig. 3. Typical images from an Ising distribution on a 200 × 200 4-pt.
grid with uniform interaction strength β = (a) .2; (b) .3; and (c) .4, and zero
external field; and with uniform interaction strength and marketing strength
α = .05 in (d), (e), and (f).
A. Parameter Estimation
For each θ ∈ Θ, the expected value of the statistic t is the
vector Eθ [t] = µ, which is referred to as the moments of the
MRF under θ. The set of all moments corresponding to MRFs
based on t is
M = {µ ∈ R|E|
| ∃θ ∈ Θ, Eθ[t] = µ}.
If the components of t are affinely independent, the statistic
is said to provide a minimal representation of F. In this case,
it is well-known that the mapping p : Θ −→ F is one-
to-one, in which case F is a statistical manifold with dual
coordinate systems Θ and M [3], [4]. This dual coordinate
system provides, among other things, a method for estimating
the exponential parameters from measured averages collected
from typical configurations. There are well-known algorithms
for performing (approximating) these algorithms, depending
as it were on the topology of the network [3], [38].
We let ˆθ|A(¯t(n)
) be the parameter estimator for marketer
A and likewise for ˆθ|B(¯t(n)
) and marketer B. Marketer A
forms the estimate ˆθ
(n)
|A = ˆθ|A(¯t(n)
) from the measurements
¯t(n)
of the configurations x(n−m+1)
, x(n−m+2)
, . . . , x(n)
. He
then forms the estimate of θ(n)
given the data and estimation
algorithm of marketer A and from this estimate of the total
interaction and marketing strength, form an estimate
ˆψ
(n)
V |B = ˆθ
(n)
|A − ψ
(n)
V |A
of his opponent’s marketing strength. Likewise, marketer B
estimates θ
(n)
|B with his estimation algorithm, and from this
derives an estimate ˆψ
(n)
V |A. Since he knows ψ
(n)
V |B, he under-
stands not only the interaction strengths of the network, but
where he and his opponent are investing their resources.
B. Desired Configurations
The idea is to connect this desired set of images with typical
sets for different values of the of the exponential parameters
or interaction strengths. Then, we can look at subspaces of the
parameter set Θ as indexing sets of configurations with certain
statistical properties and patterns. By finding a correspondence
between the patterns of images in different typical sets and
the desired patterns for the expression of a particular brand,
one can use knowledge of the interaction strengths to decide
how much marketing strength to apply in order to drive the
current typical set to a desired typical set. For example, in
Ising models with no external field, there is a certain value,
called the critical value, such that typical images contain
a ‘giant cluster’ of sites having the same strategy. Thus a
desired set of configurations could defined in terms of a large
monotone cluster having some geometric properties of the
image representing significant representation on the network,
for example. Then by drawing a connection between this
desired set of patterns and the patterns depicted in typical
sets for different Gibbs distributions, we can identify possible
places where the addition of marketing strength could drive
the Glauber dynamics into a new typical expressing desired
properties.
For instance, marketer A can define his set Ω∗
A of desired
configurations as
Ω∗
A =
θ ∈ΘA
ΩT
(θ )
where
ΘA = {θ ∈ Θ : ΩT
(θ) ⊂ Ω∗
A}
is the set of interaction and marketing strengths such that the
set of configurations with desired property for A contains the
typical set for these interaction and marketing strengths. This is
just an example of how one might relate elements to {ΩT
(θ) :
θ ∈ Θ}. An alternative definition of ΘA could be
ΘA = {θ ∈ Θ : ΩT
(θ) ⊃ Ω∗
A}
.
In addition to considering containment based relationships,
we could define relationships based on proximity in some
sense. For some distance function d : Ω × Ω −→ R+
on
the product space of configurations on the network, we could
use
d ΩT
(θ), Ω∗
A < ρ
as the membership criterion. The point is by connecting
exponential parameters to typical sets of configurations that
are likely to possess desired properties, we can adjust the
marketing strength applied in different regions.
C. Playing a Marketing Strength
With estimates ˆθ
(n)
|A and ˆθ
(n)
|B and with recent data collects,
marketers A and B now update their own strategies, which

8. are their respective marketing strengths. We assume that the
desired sets of configurations Ω∗
A and Ω∗
B are constant for
the entire game. Marketers A and B updating their marketing
strengths to ψ
(n+1)
V |A and ψ
(n+1)
V |B , respectively. At time n,
marketer A will chooses to ‘play’ marketing strength ψ
(n)
V |A,
and likewise, marketer B plays ψ
(n)
V |B. At the next time instant
n+1 players A and B choose new marketing strengths ψ
(n+1)
V |A
and ψ
(n+1)
V |B according to some functions
ψ
(n+1)
V |A = F x(n(m))
, ˆθ
(n)
E|A, ˆψ
(n)
V |B, ψ
(n)
V |A, Ω∗
A
and
ψ
(n+1)
V |A = F x(n(m))
, ˆθ
(n)
E|B, ˆψ
(n)
V |A, ψ
(n)
V |B, Ω∗
B
that takes into account the most recent block of configurations,
the estimated interaction strengths, the estimated marketing
strength for his opponent, and his known marketing strength.
These functions may also depend on the respective desired
sets of configurations Ω∗
A and Ω∗
B.
If x(n)
∈ Ω∗
A, then marketer can seek to buttress his
advantage through appropriately updating (or maintaining) his
marketing strength. This may be accomplished, for instance,
by moving the typical set ‘deeper into‘ the desired set Ω∗
A. If
x(n)
∈ Ω∗
A marketer A wants to choose a marketing strength
ψ
(n+1)
V |A that drives the typical set toward Ω∗
A. For example,
marketer A may choose the target Gibbs distribution, i.e. the
next typical set, with a criterion such as
min
θ :(θ )E =θE
max
x ∈ΩT (θ)
d(x(n)
, x )
for some appropriately defined distance measure, for example
the Earth Mover’s Distance [36]. In setting, though we do not
have in mind a database of images with which to compare the
current configuration x(n)
, but rather a way to determine from
x(n)
how far it is from possessing the respective patterns of
different typical sets.
V. PHASE TRANSITIONS AND ERGODIC COMPONENTS
Let V be infinite, for instance the sites being the set of
ordered pairs of natural numbers, and again let E consist of all
pairs of horizontally and vertically adjacent sites. Furthermore,
let the players on the infinite network G = (V, E) update
their strategies according to the Glauber dynamics of (3) and
(5). Let L be the set of all finite subsets of V . For Λ ∈ L
and boundary configuration x∂Λ, the random process (X
(n)
Λ )
converges to the (conditional) Gibbs distribution
p(xΛ|x∂Λ; θ) =
exp{ΨΛ(xΛ) + ΨΛ|∂Λ(xΛ, x∂Λ)}
ZΛ|∂Λ(θ)
(6)
where ΨΛ(x) = sΛ, tΛ(xΛ) and ΨΛ|∂Λ(x) =
sΛ,∂Λ, tΛ,∂Λ(xΛ, x∂Λ) .
Moreover,
(a) (b) (c)
(d) (e) (f)
Fig. 4. In (a), (b), and (c) typical configurations from Ising models with
uniform interaction strength β = .3, .4, and .5, respectively; In (d), (e), and
(f), typical configurations from Ising models with the same uniform interaction
strengths as in (a), (b), and (c), but with a uniform external field of α = .05
applied.
ZΛ|∂Λ(θ) =
xΛ∈ΩΛ
exp{ΨΛ(x) + ΨΛ|∂Λ(x)}
is the conditional partition function on Λ given the boundary
configuration xΛ. Likewise, the set
ΘΛ|∂Λ = {s ∈ R
|E|
+ | ΦΛ|∂Λ(θ) < ∞}
is the set of admissable interaction strengths in Λ with
boundary configuration x∂Λ, while FΛ|∂Λ = {p(·; θ) | θ ∈
ΘΛ|∂Λ} is the family of conditional Gibbs distributions on Λ
with boundary configuration x∂Λ, based on t. The collection
{pΛ|∂Λ : Λ ∈ L} of these conditional Gibbs distributions is
called a specification, specifically a Gibbsian specification, in
that they specify the dependence type for a probability measure
on the infinite random field {Xi : i ∈ V }. A probability
measure on {Xi : i ∈ V } that agrees with the specification
{pΛ|∂Λ : Λ ∈ L} is called a Gibbs measure relative to the
specification [13]. Let G(θ, t) be the set of Gibbs measures
relative to the specification given by the payoffs θ and the
statistic t. If |G(θ, t)| = 1 the Glauber dynamics are ergodic,
while if |G(θ, t)| > 1, the dynamics are non-ergodic and
a phase transition is said to occur. The equilibrium Gibbs
measure is a discrete stationary random process and as such
is a mixture of stationary ergodic random processes [16], with
each mode of the process being a stationary ergodic process.
This is referred to as the Ergodic Decomposition of discrete
stationary random processes [15], and we can think of this
decomposition as being a decomposition of the space Ω of all
strategy configurations into subsets of configurations which are
typical sets for the different modes of the Glauber dynamics.
The presence of a phase transition will affect a marketer’s
ability to estimate θ(n)
from data. If If |G(θ, t)| = 1 dynamics
are ergodic and sample averages ¯t(n)
equal statistical averages.

9. In fact, in the parameter estimation portion of the marketing
game above, our estimates of θ(n)
were using ¯t(n)
as a
proxy for µ(n)
, the true statistical average. All conditional
distributions given by the equilibrium Gibbs distribution match
the specification, but because we do not know the initial
configuration, then we do can use ¯t(n)
as a proxy for µ(n)
.
If we think of the Glauber dynamics as the evolution of the
typical set associated with the equilibrium measure, then we
can think of a phase transition as the splitting of the typical
set ΩT
(θ) into subsets of configurations that share certain
patterns and are all typical in the sense of having similar
and typical likelihoods in the overall distribution on Ω; but
these subsets, which are in fact typical sets for the different
modes of the Glauber dynamics, do not communicate in the
sense that if Glauber dynamics. Because multiple equilibrium
typical sets are consistent with a single type of dynamics
care will have to be taken in pursuing this idea further, to
know how we may influence which typical set for interaction
and marketing strengths at a phase transition. Moreover, the
issues with the Nash equilibria concept mentioned in [28]
seem are likely related to the fact that best-response dynamics
is Glauber dynamics with infinite interaction strength. And
with no marketing strength incorporated into the model, the
dynamics are non-ergodic and the specification corresponds to
a phase transition.
We now briefly discuss what is about certain combinations
of interaction strength θE and marketing strength θV in terms
of the existence of phase transitions. Most research into phase
transitions for the two-dimensional Ising model has focused on
the uniform external field case [5]. Recently, there has been
attention paid to the case of non-uniform external fields [6],
[7], and [29]. As in the external field case, phase transitions
correspond to a degeneracy of the ground state, or in other
words, a splitting of the evolving typical set of the random
process. While coordination games with uniform non-zero
external fields do not exhibit phase transitions, non-uniform
external fields in which the self-statistics all have the same
polarity can show a phase transition [6]. Likewise, it has
also been shown that non-uniform external fields (of a fixed
polarity) in which the magnitude decays with distance from a
given site may also possess a phase transition [7]. If we are
interested in marketing applications in which the polarity of
the external field indicates bias towards one brand/idea over
others, then we need to consider external fields of non-uniform
polarity. In [29] they consider a chessboard pattern pattern of
alternating polarities where the magnitude of the external field
is constant. They show that in this case a phase transition can
occur. In each of these scenarios a common assumption is
that the magnitude of the external field is small relative to the
interaction strength.
VI. EXPERIMENTS
In Figure 4 (a), (b) and (c) we see typical configurations
for equilibrium Ising distributions specified by three different
uniform interaction strengths. In (d), (e), and (f), we see a
typical configuration after applying a uniform external field
(a) (b) (c)
Fig. 5. In (b) a typical configuration from an Ising model with uniform
interaction strength β = .42. In (a), the same uniform interaction strength
but with an external field of α = .1 applied in the indicated region, and in
(c), an external field of α = −.1 applied.
inside the indicated region of interest. What we see is that
when the interaction strength is greater, a given amount of
marketing strength has a greater impact on the adoption of
the brand favored by the external field.
In Figure 5 (b) we have a typical image from an Ising
model and in (a) and (c), we see typical images from Ising
models with the same interaction strength as in (b) but with,
respectively, positive and negative uniform external fields
applied within specified regions of interest. We can see from
(b) that the regions of interest were chosen to be areas of the
‘current’ typical configuration where neither white nor black
had a clear advantage. In both (a) and (c) there appears to be
a ‘fan out’ region where the influence from the external field
applied within the ROI impacts the brand bias immediately
outside the region of interest.
VII. DISCUSSION AND FUTURE CONSIDERATIONS
Previous [31], [32], [33] and ongoing [34] work of ours
has examined the relationship between the statistic defining a
family of Markov random fields (MRFs) and the behavior of
information-theoretic quantities within that family of MRFs.
For example, entropy is monotone increasing in positive in-
teraction strengths for the class of Gibbs distributions defined
by positively correlated statistics, which includes the family of
Ising models with positive statistics [17], [31]. In pursuing the
ideas of this paper, it will be interesting to see if information
geometry, for example, can provide interesting insights into a
marketing scenario. Moreover, implicit in said previous work
is the delineation of information-theoretic behavior in terms of
the type of statistic defining the family of Gibbs distributions.
Heeding the conclusions of [8], we should understanding what
properties of a statistic associated with a game admits a Gibbs
distribution, or for that matter, whether the non existence
of a Gibbs state for a game imply that no such meaningful
statistic can be define? While the scenario presented here was
certainly simple, our hope is that connecting the influence of
the social network with the influence of marketing into a single
framework will provide a solid template from which to pursue
meaningful data analytics in the real world.
ACKNOWLEDGMENT
The author would like to thank Bob Gray for helpful
discussions regarding ergodicity of random processes.

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