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.

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