Conditional Game Theory as a Model for Coordinated Decision Making
Wynn C. Stirling
1
and Luca Tummolini
2
1
Department of Electrical and Computing Engineering, Brigham Young University, Provo, Utah, U.S.A.
2
Institute of Cognitive Sciences and Technologies, Italian National Research Council, Rome, Italy
Keywords:
Game Theory, Coordination, Social Influence, Social Utility, Network Theory.
Abstract:
Standard game theory is founded on the premise that choices in interactive decision situations are strategically
rational—best reactions to the expected actions of others. However, when studying groups whose members are
responsive to one another’s interests, a relevant notion of behavior is for them to coordinate in the pursuit of
coherent group behavior. Conditional game theory provides a framework that facilitates the study of coordina-
ted rational behavior of human social networks and the synthesis of artificial social influence networks. This
framework comprises three elements: a socialization model to characterize the way individual preferences
are defined in a social context; a diffusion model to define the way individual preferences propagate through
the network to create an emergent social structure; and a deduction model that establishes the structure of
coordinated individual choices.
1 INTRODUCTION
Coordinated decision making is one of the fundamen-
tal attributes of intelligent behavior. Indeed, the word
intelligent comes from the Latin roots inter (between)
+ leg
˘
ere (to choose). Accordingly, much effort has
been devoted to defining what it means for a choice to
be “rational.” And appending the modifier “coordina-
ted” adds a level of complexity that moves beyond the
hypothesis that each of the individual decision makers
should behave as if it were solving a constrained max-
imization problem without overt regard for the wel-
fare of others.
Coordination, as used in this paper, has a precise
meaning, as expressed by the Oxford English Dictio-
nary:
[To coordinate is] to place or arrange (things)
in proper position relative to each other and to
the system of which they form parts; to bring
into proper combined order as parts of a whole
(Murray et al., 1991).
Coordination is a principle of behavior on a paral-
lel with, but different from, performance. Individuals
perform; the group coordinates. Performance deals
with operational measures of efficiency and effecti-
veness of individual behavior in terms of individual
payoffs. Coordination, however, is an attribute of or-
ganizational structure regarding how the members of
a group function together.
An important class of collectives comprises enti-
ties that possess the ability to respond to the social
influence that they exert on one another. Examples in-
clude cooperative groups, such as teams and business
entities, mixed organizations such as families, which
can encompass both cooperative and conflictive influ-
ence, and adversarial groups such as tennis players
who exert conflictive influence on each other. Team
members coordinate by cooperating in the pursuit of
a common goal, business partners coordinate by di-
viding the labor, family members coordinate by re-
specting (or not) each other’s opinions and priorities,
and tennis players (an anti-team?) coordinate by op-
posing each other in some systematic way.
In terms of overall functionality, it is often the case
that the propensity of a group to coordinate is more
relevant than the propensity of the individuals to op-
timize. It is more relevant for a team to win the game
than for each player to maximize the number of points
he or she scores. It is more relevant for a business
entity to settle on a productive division of labor than
for each partner to maximize individual control. It is
more relevant for a family to function in a civil and
equitable way than for the members to focus exclu-
sively on what is individually best for themselves. It
is more relevant to the conducting of a war for each
opponent to seek victory rather than simply to destroy
as many enemy resources as possible.
Focusing on performance without considering
Stirling, W. and Tummolini, L.
Conditional Game Theory as a Model for Coordinated Decision Making.
DOI: 10.5220/0006956702950302
In Proceedings of the 10th International Joint Conference on Knowledge Discovery, Knowledge Engineering and Knowledge Management (IC3K 2018) - Volume 2: KEOD, pages 295-302
ISBN: 978-989-758-330-8
Copyright © 2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
295

coordination is an incomplete characterization of
group behavior. Similarly, focusing on coordination
without considering performance is an incomplete
characterization of individual behavior. A football
team may possess the organizational structure requi-
red to win the game, but that structure is useless if
the players do not attempt to maximize the number of
goals scored. A business firm may be well organized
in terms of individual responsibilities, but unless the
partners exert control, the entity will not prosper. A
family may possess fair and equitable rules of con-
duct but will still be dysfunctional if the members do
not pursue their individual goals within that context.
Tennis players may collectively understand the rules
and best practices of the game, but unless each is able
to execute those practices, playing the game will be
unrewarding. Coordination without performance is
unproductive, and performance without coordination
is equivocal. A full understanding of the functiona-
lity of a group requires the assessment of both attri-
butes. Coordination occurs when individual contri-
butions appropriately fit together to form a coherent
organizational structure.
Coordination requires individuals to possess some
notion of social connectivity in addition to concerns
for their own material welfare. There are two extreme
methodologies for incorporating coordination into a
multilateral decision scenario. One way is for the par-
ticipants to come to a social engagement with a global
view of the way the group is intended to behave. Un-
der this view, coordination is built-in: Each partici-
pant performs its ex ante assigned part. Another way
is for participants to come to the engagement with lo-
cal views of how they will behave as they interact with
others. Under this view, coordination is emergent: It
occurs (or not) as each participant responds to the so-
cial influence exerted by others. We argue that the
latter approach is the appropriate way to design a col-
lective of autonomous decision makers (agents), and
present a general framework for the analysis of human
social networks and the design and synthesis of artifi-
cially intelligent networks. For coordination to be de-
signed into such a network, however, the social relati-
onships must be defined operationally—they must be
characterized via mathematical expressions that ex-
plicitly model social influence.
A social influence network comprises a group of
agents whose choices can depend on the attitudes and
opinions of others as well as their own welfare. More
precisely, it is a collective of agents who are empo-
wered to make individual choices under the following
conditions: a) the combination of the choices of all
generates an outcome that affects the welfare of each,
and b) the preferences over outcomes for each can be
influenced by the preferences of others. The first con-
dition is the usual scenario for standard game theory,
but the second condition introduces a social compo-
nent that is not explicitly modeled by the standard the-
ory. With a social influence network, there can be a
difference between what constitutes rational behavior
when viewing the anticipated behavior of others as a
constraint on the pursuit of narrow self-interest (e.g.,
material benefit) and what constitutes rational beha-
vior from a socially oriented perspective of viewing
oneself as a part of a coordinated whole—a society.
Thus, the ability of the individuals to make their choi-
ces in a way that responds to social influence, while
at the same time retaining their individuality and con-
cern for their own welfare, is of prime importance.
This position paper argues that conditional game
theory, introduced by (Stirling, 2012) and (Stirling
and Felin, 2013), provides a framework within which
to model social influence networks. Conditional game
theory comprises three components: a socialization
mechanism by which indviduals may incorporate the
interests of others into their own self-interest without
compromising their individuality; a diffusion mecha-
nism by which the preferences resulting from an ex-
panded view of self-interest can be conglomerated to
produce a comprehensive social model that accounts
for all social interrelationships; and a deduction me-
chanism by which coordinated individual decisions
may be deduced from the social model. A criti-
cal feature of this theory is that it is consistent with
the fundamental assumptions of game theory; in fact,
conventional noncooperative game theory is a special
case of this extended theory.
2 SOCIALIZATION
With conventional game theory, preferences are
categorical—fixed, immutable, and unconditional.
The mathematical mechanism used to express cate-
gorical preferences is a payoff function. Given a col-
lective of agents {X
1
,...,X
n
} for n ≥ 2, let A
i
denote
a finte set of actions for X
i
, and let the Cartesian pro-
duct set A = A
1
× · ·· × A
n
denote the outcome set.
The function u
i
: A → R quantifies the payoff to X
i
as a
function of the combined actions of the collective. Of
course, X
i
is free to define its preferences in whatever
way it chooses, be it egocentric, altruistic, or other.
Once defined, u
i
is the formal expression of X
i
’s no-
tion of self-interest. The payoff function establishes
a global ordering of the outcome set. The most well-
known solution concept associated with such payoffs
is to juxtapose them into a payoff array and identify
Nash equilibria—the set of outcomes such that, if any
KEOD 2018 - 10th International Conference on Knowledge Engineering and Ontology Development
296

agent were to make a unilateral change, it’s payoff
would either decrease or remain unchanged.
The innovation provided by conditional game the-
ory is to allow agents to possess conditional payoffs.
Establishing this concept requires the application of
graph theory. A network graph G(X,E) comprises a
set of vertices X = {X
1
,. ..,X
n
} (the set of agents) and
a set E ⊂ X × X of pairs of vertices such that there is
an explicit connection between them that serves as the
medium by which influence is propagated between X
i
and X
j
. Specifically, the expression X
i
−→ X
j
means
that the influence propagates in only one direction—a
directed edge from X
i
to X
j
. A path from X
j
to X
i
is
a sequence of directed edges from X
j
to X
i
, denoted
X
j
7→ X
i
. A path is a cycle if X
j
7→ X
j
. A graph is said
to be a directed acyclic graph, or DAG, if all edges are
directed and there are no cycles. For each X
i
, its pa-
rent set is pa (X
i
) = {X
i
1
,. ..,X
i
q
i
}, where X
i
k
−→ X
i
,
k = 1,.. ., q
i
. If pa(X
i
) = ∅ then X
i
is a root vertex.
A conjecture profile a
i
= (a
i1
,. ..,a
in
) ∈ A is a
profile hypothesized by X
i
as the outcome to be actua-
lized. The expression X
i
|=
a
i
means that X
i
conjectu-
res a
i
. The element a
ii
is X
i
’s self-conjecture, denoted
X
i
|=
a
ii
, and a
i j
, j 6= i, is an other-conjecture by X
i
for
X
j
, denoted X
i
|=
a
i j
. The array (a
1
,. ..,a
n
) is termed
a joint conjecture set.
A conditioning conjecture profile by X
i
for X
i
k
, de-
noted a
i
k
= (a
i
k
1
,. ..,a
i
k
n
), is a profile that X
i
hypot-
hesizes that X
i
k
|=
a
i
k
, k = 1, .. ., q
k
. A conditioning
conjecture set α
α
α
pa(i)
= (a
i
1
,. ..,a
i
q
i
) ∈ A
q
i
by X
i
for
pa(X
i
) is the set of conditioning conjecture profiles
by X
i
for its parents, denoted pa(X
i
)
|=
α
α
α
pa(i)
.
A conditional payoff given α
α
α
pa(i)
, denoted
u
i|pa(i)
(·|α
α
α
pa(i)
): A → R, is an ordering function such
that, given the antecedent pa(X
i
)
|=
α
α
α
pa(i)
, then
u
i|pa(i)
(a
i
|α
α
α
pa(i)
) ≥ u
i|pa(i)
(a
0
i
|α
α
α
pa(i)
) (1)
if X
i
prefers the conjecture profile a
i
to a
0
i
or is indif-
ferent, given that pa(X
i
)
|=
α
α
α
pa(i)
. If pa(X
i
) = ∅, then
u
i|pa(i)
(a
i
|α
α
α
pa(i)
) = u
i
(a
i
), a categorical payoff.
A conditional network game is a triple {X,A ,U},
where X = {X
1
,. ..,X
n
}, A = A
1
× ··· × A
n
, and
U = {u
i|pa(i)
(·|α
α
α
pa(i)
) ∀α
α
α
pa(i)
∈ A
q
i
, i = 1,...,n}.
(2)
A conditional network game degenerates to a stan-
dard noncooperative normal-form game when U =
{u
i
, i = 1,. .. ,n}. Thus, conditional game theory is
an extension of standard noncooperative game theory.
The conditional structure of the preferences ena-
bles agents to extend their spheres of interest beyond
strategic self-interest without surrendering individua-
lity, and therefore differs fundamentally from the ca-
tegorical preference structure of a standard noncoope-
rative game. The players of a standard game react to
the fixed categorical preferences; there is no opportu-
nity for adaptation as the game is played—the prefe-
rences are static. A conditional game enables play-
ers to adapt to the social environment, since they are
able to respond to the preferences of others as they
interact—the preferences are dynamic. The represen-
tation of a conditional network game as a graph with
agents as vertices and linkages as conditional payoffs
fully integrates the individual preferences into an or-
ganizational structure that enables the synthesis of a
comprehensive model of the way individual preferen-
ces interact. An example of a three-agent social influ-
ence network is
X
1
u
2|1
~~
X
2
u
3|12
//
X
3
(3)
where X
1
is a root vertex and thus must have a catego-
rical payoff u
1
(a
1
), pa (X
2
) = {X
1
} with conditional
payoff u
2|1
(a
2
|a
1
), and pa(X
3
) = {X
1
,X
2
} with condi-
tional payoff u
3|12
(a
3
|a
1
,a
2
).
3 DIFFUSION
There is a distinct operational difference between ca-
tegorial and conditional preferences. Given that X
i
categorically prefers a
i
to a
0
i
, X
i
has sufficient infor-
mation to choose between the two conjectures. But
if X
i
only conditionally prefers a
i
to a
0
i
, X
i
does not
have sufficient information to choose between them
without entering into the conditioning social relati-
onships. As the conditional preferences propagate
through the group, emergent social interrelationships
are established between its members. This process,
termed diffusion, involves conglomerating the indivi-
dual conditional payoffs to form a social model that
provides a comprehensive expression of the emergent
social structure. Conglomeration is superficially re-
lated to the concept of aggregation as employed by
social choice theory, but serves a different purpose.
With social choice theory, the votes of the individuals
are aggregated to form a group-level decision. Con-
glomeration, by contrast, is a process of combining a
collective of parts to form a whole while remaining
distinct entities.
Given a conditional game {X,A ,U}, a coordina-
tion functional is a mapping F: U → [0,1] that gene-
rates a social model u
1:n
: A
n
→ [0,1] of the form
u
1:n
(a
1
,. ..,a
n
) = F[u
i|pa(i)
(a
i
|a
i
1
,. ..,a
i
q
i
), i = 1,...,n].
(4)
The intended role of the coordination functional is to
provide a measure of the degree of compatibility of
Conditional Game Theory as a Model for Coordinated Decision Making
297

?
6
X
1
X
2
`
r
r
`
Figure 1: The doorway game.
the agents as they conjecture their various outcomes.
To illustrate the manifestation of coordination, con-
sider a scenario involving {X
1
,X
2
}, who approach a
doorway from opposite directions, as illustrated in Fi-
gure 1. Suppose the doorway is just wide enough for
two agents to pass simultaneously if they both move
either to their respective right (r) or left (`) sides of
the doorway. Let A
1
= A
2
= {r,`}, yielding the four-
element outcome set
A = A
i
× A
j
= {(`,`),(`, r),(r.`),(r,r)}, (5)
with influence relations
X
1
u
2|1
//
X
2
X
1
X
2
u
1|2
oo
(6)
for i, j,∈ {1, 2}, i 6= j. Suppose X
i
possesses a condi-
tional payoff u
i| j
(a
i1
,a
i2
|a
j1
,a
j2
) defined over A × A ,
and views X
j
as a root vertex possessing a categorical
payoff u
j
(a
j1
,a
j2
) defined over A.
Let u
i j
(a
i
,a
j
) be a social model as defined by
(4) and consider the joint conjecture sets [a
i
,a
j
] =
[(`,`),(r,r)] and [a
0
i
,a
0
j
] = [(`,`), (`,r)]. The former
joint conjecture set corresponds to a scenario where,
although the players do not agree regarding which
way they should turn, they do agree that they should
cooperate, whereas the latter joint conjecture set cor-
responds to a scenario where X
i
conjectures coope-
ration and X
j
conjectures conflict. Assuming that
u
i| j
(`,`|r,r) > u
i| j
(`,`|`,r), it would be reasonable
that
u
i j
[(`,`),(r,r)] > u
i j
[(`,`),(`,r)], (7)
meaning that, the joint conjecture set [(`,`),(r,r)] is
more coordinated than [(`,`), (`,r)]]. However, both
of these joint conjecture sets are less coordinated than
[(`,`),(`,`)].
The choice of a suitable coordination functional
is a critical component of conditional game theory.
To motivate such a choice, it is instructive to recog-
nize the analogical relationship between conditional
payoffs and conditional probabilities. Indeed, syn-
tax of conditional payoffs and conditional probabili-
ties are in the form of hypothetical propositions of the
form “If p then q”, where p is the antecedent and q is
the consequent. Furthermore, the topology of the net-
work illustrated by (3) is similar to that of a Bayesian
network. Thus, the structure and syntax of a social
influence network will be identical to that of a Baye-
sian network if the conditional payoffs are expressed
using the mathematical structure of probability the-
ory, namely, that the conditional payoffs are conditio-
nal mass functions, that is,
u
i|pa(i)
(a
i
|α
α
α
pa(i)
) ≥ 0 for all a
i
∈ A
∑
a
i
u
i|pa(i)
(a
i
|α
α
α
pa(i)
) = 1 for all α
α
α
pa(i)
∈ A
q
i
(8)
Furthermore, the analogy with a Bayesian network
can be made exact by defining the coordination functi-
onal according to the fundamental theorem of Baye-
sian networks, namely,
u
1:n
(a
1
,. ..,a
n
) = F[u
i|pa(i)
(a
i
|α
α
α
pa(i)
), i = 1,..., n]
=
n
∏
i=1
u
i|pa(i)
(a
i
|α
α
α
pa(i)
).
(9)
This structure is attractive for three key reasons.
First, it takes advantage of one of the great strengths
of probabilistic reasoning, which has long been re-
cognized as an important model of human reasoning.
Indeed, as Glenn Shafer has noted, “Probability is not
really about numbers; it is about the structure of rea-
soning” (Pearl, 1988, quoted by). Second, the social
model is analogous to the joint distribution of a set
of random variables. Analogous to the way a joint
probability mass function that captures all of the sta-
tistical relationships that exist among a collective of
random variables, the social model captures all of the
social influence relationships that exist among a col-
lective of agents. Third, adopting (9) as the diffusion
functional ensures that no agent can be categorically
subjugated by the group in that whatever it chooses
as its most preferred outcome is socially unaccepta-
ble to the collective. To explain this concept, suppose
X
i
possesses a categorical payoff u
i
, and let u
1:n
be a
social model defined by (4). X
i
is subjugated if, for
every fixed a
i
∈ A,
u
i
(a
i
) > u
i
(a
0
i
) for all a
0
i
6= a
i
(10)
holds, then
u
1:n
(a
1
,. ..,a
i−1
,a
i
,a
i+1
,. ..,a
n
)
< u
1:n
(a
1
,. ..,a
i−1
,a
0
i
,a
i+1
,. ..,a
n
) (11)
for all joint conjecture sets
(a
1
,. ..,a
i−1
,a
0
i
,a
i+1
,. ..,a
n
) ∈ A
n
with a
0
i
6= a
i
.
If X
i
is subjugated, then, no matter which outcome
KEOD 2018 - 10th International Conference on Knowledge Engineering and Ontology Development
298

it most prefers, all joint conjecture sets with X
i
conjecturing its most preferred outcome have lower
coordination than all joint conjecture sets with X
i
not
conjecturing its most preferred outcome. In other
words, X
i
’s participation in the group is so toxic that
the very fact that it even has a preference destroys
the functionality of the group. Thus, avoiding even
the potential for any agent to be subjugated is an im-
portant consideration for the design of a coordination
functional.
The notion of subjugation is mathematically equi-
valent to the notion of a sure loss gambling scena-
rio; that is, a Dutch book, where the gambler loses
more than the entry fee regardless of the outcome.
1
The Dutch book theorem establishes that a sure loss
is impossible if, and only if, the gambler’s beliefs and
actions conform to the axioms of probability theory.
Thus, subjugation is impossible if, and only if, the
preferences and actions of the agents also conform to
the probability axioms.
4 DEDUCTION
The ordering provided by the social model is with re-
spect to joint conjecture sets α
α
α
1:n
= (a
1
,. ..,a
n
), with
each conjecture profile of the form a
i
= (a
i1
,. ..,a
in
),
where a
ii
∈ A
i
is a self-conjecture by X
i
and a
i j
∈ A
j
is
an other-conjecture for X
j
by X
i
. This model is com-
prehensive in the sense that it captures all of the social
relationships that exist among the individuals. It con-
tains all of the information necessary for each agent
to deduce the actions that are consistent with its need
for individual performance as well as the social influ-
ence exerted by others. The deduction process com-
prises two phases: First, the extraction of an explicit
measure of the degree of coordination associated with
each conjecture outcome a ∈ A,and, second, an orde-
ring over its own self-conjecture a
ii
∈ A
i
.
Vilfredo Pareto understood the distinction bet-
ween individual preference and group sociality. Indi-
vidual behavior is expressed in terms of the way one
makes choices according to one’s preferences over al-
ternatives, and group behavior is expressed in terms of
the way its members interact as a consequence of their
preferences. He employs the notion of “social utility”
as a characterization of the degree of satisfaction asso-
ciated with an alternative. For individuals, the utility
of an alternative can be expressed economically with
1
To establish this equivalence, suppose one were to
place a $1 bet on the event (10), with an fair entry fee of
p > 1/2 and, simultaneously, to place a bet on the event (11)
with a fair entry fee of q > 1/2. Regardless of the outcome,
the gambler wins $1 but pays p + q > 1—a sure loss.
operational measures such as payoffs or other mani-
festations of individual benefit. According to Pareto,
however, the utility of a group should be analyzed so-
ciologically, and may not coincide with the economic
payoffs of its individual members.
In pure economics a community cannot be re-
garded as a person. In sociology it can be
considered, if not as a person, at least as a
unit. There is no such thing as the opheli-
mity of a community; but a community uti-
lity can roughly be assumed. So in pure eco-
nomics there is no danger of mistaking the
maximum of ophelimity for a community for
a non-existent maximum of ophelimity of a
community. In sociology, instead, we must
stand watchfully on guard against confusing
the maximum of utility for a community with
the maximum utility of a community, since
they both are there [emphasis in original] (Pa-
reto, 1935, pp. 1471, par. 2133).
Coser elaborates on Pareto’s distinction between
economic utility and social utility.
By making his distinction between the utility
for and the utility of a community, Pareto mo-
ved from classical liberal economics, where it
was assumed that total benefits for a commu-
nity simply involved a sum total of the bene-
fits derived by each individual (“the greatest
happiness of the greatest number”), to a soci-
ological point of view in which society is trea-
ted as a total unit and sub-groups or individu-
als are considered from the viewpoint of their
contribution to the overall system as well as in
terms of their peculiar wants and desires. Sy-
stem needs and individual or sub-group needs
are distinguished [emphasis in original]. (Co-
ser, 1971, p. 401)
Although the social model provides a ranking of
the sociality of the network with respect to the joint
conjecture sets of the network, each X
i
has direct
control over only a
ii
, its own self-conjecture. Thus,
what is most relevant with respect to coordination
is a ranking of how individual self-conjectures a
ii
,
i = 1,. .. ,n combine to form a notion of coordination.
Given a joint conjecture set α
α
α
1:n
= (a
1
,. ..,a
n
),
form the coordination profile a := (a
11
,. ..,a
nn
) com-
prising the set of self-conjectures, and compute the
marginal of the social model u
1:n
[(a
11
,. ..,a
1n
),...,
(a
n1
,. ..,a
nn
)] with respect to the coordination pro-
file by summing the social model over all elements of
each a
i
except the self-conjectures to form the social
Conditional Game Theory as a Model for Coordinated Decision Making
299

utility w
1:n
for {X
1
,. ..,X
n
}, yielding
w
1:n
(a
11
,. ..,a
nn
)
=
∑
¬a
11
·· ·
∑
¬a
nn
u
1:n
[(a
11
,. ..,a
1n
),
.. ., (a
n1
,. ..,a
nn
)], (12)
where the not-sum notation
∑
¬a
11
means that the sum
is taken over all elements in the argument list except
a
ii
.
Social utility as a measure of coordination serves
as an operational manifestation of the sociologic no-
tion of utility introduced by Pareto, and is distinct
from the economic concept of utility expressed via in-
dividual payoffs. The relation
w
1:n
(a) > w
1:n
(a
0
) (13)
means that the degree to which the set of self-
conjectures {a
ii
,i = 1,. .. ,n} (the parts) fit together
to form systematic group-level behavior (a whole) is
greater than the degree to which {a
0
ii
,i = 1,...,n} ge-
nerates a whole.
Once the coordination function has been defined,
the final deduction step is to identify the payoffs for
each member of the collective. The coordinated pa-
yoff for X
i
is the i-th marginal of w
1:n
, that is,
w
i
(a
ii
) :=
∑
¬a
ii
w
1:n
(a
11
,. ..,a
1n
). (14)
The relationship between social utility w
1:n
and
coordinated payoffs w
i
for a collective of agents
{X
1
,. ..,X
n
} is analogous to the relationship between
a joint probability mass function p
1:n
and marginal
mass functions p
i
for a collective of random varia-
bles {Y
1
,. ..,Y
n
}. p
1:n
(y
1
,. ..,y
n
) is the degree of pro-
bability of the simultaneous realization of the joint
event {Y
i
= y
1
,. ..,Y
n
= y
n
}, and p
i
(y
i
) is the proba-
bility of the single event {Y
i
= y
i
} for each Y
i
. If the
Y
i
’s are mutually independent, then p
1:n
(y
1
,. ..,y
n
) =
∏
n
i=1
p
i
(y
i
). The “difference” between p
1:n
(y
1
,. ..,y
n
)
and
∏
n
i=1
p
i
(y
i
) is a measure of the degree of statisti-
cal dependence that exists among the random varia-
bles.
Similarly, for (a
11
,. ..,a
nn
) ∈ A
1
,× ·· · × A
n
, the
social utility w
1:n
(a
11
,. ..,a
nn
) is the degree of coordi-
nation of the simultaneous actualization of the joint
event {X
i
|=
a
11
,. ..,X
i
|=
a
nn
}, and w
i
(a
ii
) is the coor-
dinated payoff of the single event {X
i
|=
a
ii
} for each
X
i
. If the X
i
’s are mutually socially independent, then
w
1:n
(a
11
,. ..,a
nn
) =
∏
n
i=1
w
i
(a
ii
). The “difference” be-
tween w
1:n
(a
11
,. ..,a
nn
) and
∏
n
i=1
w
i
(a
ii
) is a measure
of the degree of social dependence that exists among
the agents. Intuitively, the greater the social depen-
dence, the more the group is able to coordinate.
5 RELATION TO PREVIOUS
RESEARCH
Social psychologists and mathematicians have stu-
died social influence network theory since the 1950s,
with much of the research focusing on the organiza-
tional structure of so-called small groups, defined as
loosely coupled collectives of mutually interacting in-
dividuals (Weick, 1995). Specifically, much of the
emphasis has been placed on the structure of such or-
ganizations (cf. (French, 1956; DeGroot, 1974; Fried-
kin, 1986; Arrow et al., 2000; Friedkin and Johnson,
2011)). A basic model is that an individual’s soci-
ally adjusted payoff is a convex combination of its
own categorical payoff and a weighted sum of the
categorical payoffs of those agents who influence it.
(Hu and Shapley, 2003a; Hu and Shapley, 2003b) ap-
ply a command structure to model player interactions
by simple games. The subject of influence has also
been extensively studied in the context of voting ga-
mes where the individuals must vote yes or no on a
given proposition. (Hoede and Bakker, 1982) intro-
duce the concept of decisional power as a measure of
the degree of influence of an individual or coalition of
other voters to alter their vote from their original incli-
nation (cf. (Grabisch and Rusinowska, 2010)). (Gale-
otti et al., 2013) establish conditions for reaching an
equilibrium for social influence networks.
Other approaches to the issue of coordination fo-
cus on models drawn from biological and social evo-
lutionary processes ((Axelrod, 1984; Bicchieri, 2003;
Fefferman and Ng, 2007; Goyal, 2007; Gintis, 2009;
Bossert et al., 2012)). Coordination is addressed by
studying repeated games, where players replay the
same game multiple times. The argument supporting
these approaches is that players gain insight regarding
the social dispositions of the other players through re-
peated interaction. They may learn to recognize be-
havioral patterns and predict the behavior of others.
Through this process, they can establish their own re-
putations and gain the trust of others. Coordination,
therefore, is viewed as the end result of social evo-
lution. Such approaches provide important models
of the emergence of social relationships in repeated-
play environments where individual fitness for long-
term survival is taken into consideration in addition to
short-term material payoffs. Coordination issues are
also central to the study of multiagent systems and
general network theory (Jackson, 2008; Shoham and
Leyton-Brown, 2009; Easley and Kleinberg, 2010).
Social scientists have long recognized the need to
expand notions of preference beyond egocentric inte-
rest. Behavioral game theory (cf. (Bolton and Ocken-
fels, 2005; Fehr and Schmidt, 1999; Henrich et al.,
KEOD 2018 - 10th International Conference on Knowledge Engineering and Ontology Development
300

2004; Camerer et al., 2004; Henrich et al., 2005))
is a response to the desire to introduce psychologi-
cal realism and social influence into game theory by
incorporating notions such as fairness and reciprocity
into preferences in addition to considerations of ma-
terial benefit. The closely related field of psychologi-
cal game theory (cf. (Geanakoplos et al., 1989; Duf-
wenberg and Kirchsteiger, 2004; Colman, 2003; Bat-
tigalli and Dufwenberg, 2009; Gilboa and Schmeid-
ler, 1988)) also employs preferences that account for
beliefs as well as actions and takes into consideration
belief-dependent motivations such as guilt aversion,
reciprocity, regret, and shame. The concept of “team-
reasoning” has been promoted by (Sugden, 2015) and
(Bacharach, 2006), where individuals view themsel-
ves as members of a team, and therefore are moti-
vated to modify their behavior to conform with team
aspirations. (Hedahl and Huebner, 2018) focus on va-
lue sharing and discuss processes for providing nor-
mative grounding for pursuing shared ends. (Reisch-
mann and Oechssler, 2018) introduce a mechanism
for public good provision using conditional offers ba-
sed on the willingness of others to contribute.
A thread common to these approaches is that they
rely on ex ante linear preference orderings that are sta-
tic, immutable, global, and unconditional—they are
categorical. We argue that this single thread must be
replaced by a richer interweave of preference relati-
onships that involve explicit social influence.
The perspectives that comport most closely with
this paper are the views held by (Ross, 2014) and
(Bratman, 2014). Ross asserts that individual prefe-
rences are not formed in a social vacuum; rather, they
are the consequence of social processes, and must the-
refore be dependent on the social environment. Brat-
man argues similarly, and introduces a notion of aug-
mented individualism, where the intentions of an indi-
vidual are composed of relevant interrelated attitudes,
leading to a notion of shared agency. Essentially, con-
ditional game theory is the operationalization of these
two perspectives.
6 CONCLUSIONS
Conditional game theory offers a significant extension
of standard game theory as a framework for both the
analysis of human networks and the design and synt-
hesis of artificial social influence networks.
• Social influence is ex ante incorporated endoge-
nously into the payoffs rather than exogenously
imposed via an ex post solution concept.
• An operational definition of coordination is gene-
rated as a group-level attribute that is considered
parallel to the individual-level attribute of prefe-
rence.
• Individual coordinated decisions are deduced as a
consequence of the diffusion of social influence
throughout the network.
REFERENCES
Arrow, H., McGrath, J. E., and Berdahl, J. L. (2000). Small
Groups as Complex Systems. Sage Publications, Inc.,
Thousand Oaks, CA.
Axelrod, R. (1984). The Evolution of Cooperation. Basic
Books, New York.
Bacharach, M. (2006). Beyond Individual Choice: Teams
and Frames in Game Theory. Princeton University
Press, Princeton, NJ.
Battigalli, P. and Dufwenberg, M. (2009). Dynamic psycho-
logical games. Journal of Economic Theory, 144:1–
35.
Bicchieri, C. (2003). Rationality and Coordination. Cam-
bridge University Press, Cambridge, UK.
Bolton, G. E. and Ockenfels, A. (2005). A stress test of
fairness measures in models of social utility. Econo-
mic Theory, 24(4).
Bossert, W., Qi, C. X., and Weymark, J. A. (2012). Mea-
suring group fitness in a biological hierarchy: an ax-
iomatic social choice approach. Economics and Phi-
losophy, 29:301–323.
Bratman, M. (2014). Shared Agency. Oxford University
Press, Oxford, UK.
Camerer, C., Lowenstein, G., and Rabin, M., editors (2004).
Advances in Behavorial Economics. Princeton Univ.
Press, Princeton, NJ.
Colman, A. M. (2003). Cooperation, psychological game
theory, and limitations of rationality in social inte-
raction. Behavioral and Brain Sciences, 26:139–198.
Coser, L. A. (1971). Masters of Sociological Thought. Har-
court Brace Jovanovich, Inc., New York, NY.
DeGroot, M. H. (1974). Reaching a consensus. Journal of
the American Statistical Association, 69:118–121.
Dufwenberg, M. and Kirchsteiger, G. (2004). A theory of
sequential reciprocity. Games and Economic Beha-
vior, 47:268–298.
Easley, D. and Kleinberg, J. (2010). Networks, Crowds, and
Markets: Reasoning about a Highly Connected World.
Cambridge University Press, Cambridge.
Fefferman, N. H. and Ng, K. L. (2007). The role of in-
dividual choice in the evolution of social complexity.
Annales Zoologici Fennici, 44:58–69.
Fehr, E. and Schmidt, K. (1999). A theory of fairness, com-
petition, and cooperation. Quarterly Journal of Eco-
nomics, 114:817–868.
French, J. R. P. (1956). A formal theory of social power.
The Psychology Review, 63:181–194.
Friedkin, N. E. (1986). A formal theory of social power.
Journal of Mathematical Sociology, 12:103–136.
Conditional Game Theory as a Model for Coordinated Decision Making
301

Friedkin, N. E. and Johnson, E. C. (2011). Social Influence
Network Theory. Cambridge University Press, Cam-
bridge, UK.
Galeotti, A., Ghiglino, C., and Squintani, F. (2013). Stra-
tegic information transmission networks. Journal of
Economic Theory, 148(5):1751–1769.
Geanakoplos, J., Pearce, D., and Stacchetti, E. (1989). Psy-
chological games and sequential rationality. Games
and Economic Behavior, 1:60–79.
Gilboa, I. and Schmeidler, D. (1988). Information depen-
dent games. Economics Letters, 27:215–221.
Gintis, H. (2009). The Bounds of Reason: Game Theory and
the Unification of the Behavioral Sciences. Princeton
University Press, Princeton, NJ.
Goyal, S. (2007). Connections. Princeton University Press,
Princeton, NJ.
Grabisch, M. and Rusinowska, A. (2010). Different approa-
ches to influence based on social networks and simple
games. In Deeman, A. V. and Rusinowska, A., edi-
tors, Collective Decision Making: Views from Social
Choice and Game Theory. Springer-Verlag, Berlin.
Hedahl, M. and Huebner, B. (2018). Sharing values. The
Southern Journal of Philosophy, 56:240–272.
Henrich, J., Boyd, R., Bowles, S., Camerer, C., Fehr, E.,
and Gintis, H., editors (2004). Foundations of Human
Sociality: Economic Experiments and Ethnographic
Evidence from Fifteen Small-scale Societies. Oxford
University Press, Oxford, UK.
Henrich, J. et al. (2005). “Economic Man” in Cross-
culturative Perspective: Behavorial Experiments in 15
Small-scale Societies. Behavioral and Brain Sciences,
28(6):795–855.
Hoede, C. and Bakker, R. (1982). A theory of decisional
power. Journal of Mathematical Sociology, 8:309–
322.
Hu, X. and Shapley, L. S. (2003a). On authority distributi-
ons in organizations: Controls. Games and Economic
Behavior, 45:153–170.
Hu, X. and Shapley, L. S. (2003b). On authority distribu-
tions in organizations: Equilibrium. Games and Eco-
nomic Behavior, 45:132–152.
Jackson, M. (2008). Social and Economic Networks. Prin-
ceton University Press, Princeton, NJ.
Murray, J. A. H., Bradley, H., Craigie, W. A., and Onions,
C. T., editors (1991). The Compact Oxford English
Dictionary, Oxford, UK. The Oxford Univ. Press.
Pareto, V. (1935). A Treatise on General Sociology, Volumes
Three and Four. Dover, New York, NY. edited by A.
Livingston.
Pearl, J. (1988). Probabilistic Reasoning in Intelligent Sys-
tems. Morgan Kaufmann, San Mateo, CA.
Reischmann, A. and Oechssler, J. (2018). The binary con-
ditional contribution mechanism for public good pro-
vision in dynamic settings—theory and experimental
evidence. Journal of Public Economics, 159:104–115.
Ross, D. (2014). Philosophy of Economics. Palgrave Ma-
cmillan, Houndmills, Basingstoke, UK.
Shoham, Y. and Leyton-Brown, K. (2009). Multiagent Sys-
tems. Cambridge University Press, Cambridge, UK.
Stirling, W. C. (2012). Theory of Conditional Games. Cam-
bridge University Press, Cambridge, UK.
Stirling, W. C. and Felin, T. (2013). Game theory, conditio-
nal preference, and social influence. PLos ONE 8(2):
e56751. doi:10.1371/journal.pone.0056751.
Sugden, R. (2015). Team reasoning and intentional coope-
ration for mutual benefit. Journal of Social Onto-
logy, 1(1):143–166. ISSN (Online) 2196-9963, ISSN
(Print) 2196-9655, DOI: 10.1515/jso-2014-0006, No-
vember 2014.
Weick, K. E. (1995). Sensemaking in organizations. Sage
Publications, Thousand Oaks, CA.
KEOD 2018 - 10th International Conference on Knowledge Engineering and Ontology Development
302