Power Indices in the Context of Social Learning Behaviour ...downloads.hindawi.com/journals/ddns/2019/4532042.pdf · ResearchArticle Power Indices in the Context of Social Learning

Research ArticlePower Indices in the Context of Social LearningBehaviour in Social Networks

Ruili Shi 12 Chunxiang Guo 1 and Xin Gu 1

1Business School Sichuan University Chengdu 610065 China2Chongqing Business Vocational College Chongqing 401331 China

Correspondence should be addressed to Chunxiang Guo guochunxiangscueducn

Received 30 March 2019 Revised 30 May 2019 Accepted 2 June 2019 Published 25 June 2019

Academic Editor Marko Robnik

Copyright copy 2019 Ruili Shi et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper puts forward the concept of integrated power synthetically measures the votersrsquo ability to influence the results ofdecision-making by influencing others through social learning considering the interactions between decision-makers in socialnetworks and offers a method for measuring integrated power Based on the theory and model of social learning we analyze theinfluence of social learning on the voting process and power indices from the perspective of individualsrsquo professional level positionwithin the social network structure relationship closeness and learning efficiency A measurement model of integrated power isconstructed and the variation in integrated power compared with that of the Banzhaf index is analyzed by numerical simulationThe results show that when the individualrsquos professional level is higher and closeness with neighboring decision-makers is greaterthen the integrated power index is higher An individualrsquos integrated power index may decrease when heshe changes from anisolated node to a nonisolated node and then hisher integrated power will increase with the increases of neighbor nodes Sociallearning efficiency can promote the integrated power of individuals with lower social impact and relationship closeness but it isnot beneficial for the core and influential members of the social network

1 Introduction

A core issue in group decision-making problems such asvoting games is how to construct a preference sequencebased on individualsrsquo preferences The power distribution ofvoters is the basis for the prioritization and selection of thebestmdashor at least a satisfactorymdashscheme which is related tothe credibility of the voting results Therefore power analysisbecame one of the main issues in the analysis of a votingsituation

Voting power that is a voterrsquos capacity to influence thevoting procedure can reflect the fairness and rationality ofthe voting rules and is always the focus of scholarly study Sev-eral remarkable power indices such as the ShapleyndashShubikindex [1] and the Banzhaf index [2] have been introducedin the literature on simple games and have been applied tomany practical problems such as decision conflict resolutionresponsibility sharing benefit distribution equity allocationand so on [3 4] The classical power indices measure voting

power under the assumption that voters cast their votes inde-pendently and that each voter is equally likely to vote ldquoyesrdquoor ldquonordquo These measures are not supported empirically andfail to account for the preferences of and interactions betweenvoters A standard power index is therefore considered as apriori power related only to the voting rules [5 6] and canbest be described as a constitutional voting power that isthe power that is inherent in the formal rules of the votinggame only and that does not consider the behaviour in anactual voting system [7] These models of voting power donot seem to capture actual a posteriori voting power wherepreferences and interdependence may exist Therefore manylater literatures give up the assumptions of independenceand random voting and propose corresponding voting powerindices such as spatial power indices [8 9] empirical powerindices [5] behavioural power indices [7] causal dependencepower indices [10 11] probabilistic power indices for votingwith abstention [12] and other modified indices [13 14]These mentioned power indices consider votersrsquo preferences

HindawiDiscrete Dynamics in Nature and SocietyVolume 2019 Article ID 4532042 13 pageshttpsdoiorg10115520194532042

2 Discrete Dynamics in Nature and Society

and behaviour in actual voting systems however they arepowerless to reflect managerial phenomena caused by com-plex social interactions within organizations

Group interactions are indeed inseparably linked withour increasingly interconnected world and interactionsamong human societies are more complex [15ndash17] As com-plex network structures are more representative for humansocieties group interactions on complex network had beenwidely studied and present how social diversity affects humanbehaviour (for a review see [15]) Human experiments ofcooperation have also suggested that social relationship net-work structure is an important factor of human behaviours[16] In group decision-making process the structured rela-tionships between voters may also influence their behavioursand judgments [18 19] Thus later literatures have partiallyconsidered heterogeneous voters and their social structurewhen measuring power indices The most representativeof these models are opinion leadership models which arebased on the theory of the two-step flow of communication[20]The properties and axiomatization of opinion formationunder a leader-follower power dynamic have been provided[21ndash23] but voter heterogeneity in such models takes asimple form (leader and follower) Such relationships are notgeneral as relations among members are usually equal suchas in citizensrsquo elections and do not exist an ldquoinstitutionalpowerrdquo In addition the voting process is usually a processof information exchange and learning among members thatis based on social relationships [24]

Relationships among members can be described bysocial networks which provide a new way to study thebehaviour and influence of relationships among individualsand organizations in society Thus the theory and methodsof social networks have been introduced into problems ofgroup decision-making [25ndash28] Network analysts implicitlyassume that power is related to network centrality [29ndash33]but they do not consider the voting rule that is in effect whenanalyzing group decisions with social networks

This paper focuses on a power analysis based on socialnetworks The problem is complex on the one hand a socialnetwork enhances the interaction between decision-makersand the network effect may enlarge or reduce the influenceof decision-makers on the results on the other hand inthe social networks people can observe and obtain othersrsquobehavioural information more conveniently and update orstrengthen their own judgments which increase the variabil-ity in individual preferenceThese characteristics require thatpower indices not only reflect the direct influence that votershave on results but also consider the indirect influence onvoting result through social interactions

In this paper we analyze the process of group interactionsinformed by social learning theory [34] and construct apower measurement model based on social networks Sociallearning theory has been widely used in the fields of soci-ology psychology pedagogy and management science [2435 36] and holds that individualrsquos behaviours are deeplyinfluenced by the social environment and that individualswilldetermine or change their views and behaviours accordingto the observed behaviours [34ndash36] The group interactionsduring voting process can be regarded as a process of

behavioural observation and social learning Therefore westudy the influence group interactions have on voting basedon social learning theory and analyze the influence of theseinteractions on power indices

Thepaper is organized as follows Section 2 introduces theconcepts of power indices social network and social learningtheory Section 3 describes the integrated power measuremodel informed by social learning theory based on socialnetworks next in Section 4 we present a numerical exampleto analyze the influence of social learning on power indicesand finally Section 5 provides a brief summary

2 Preliminaries

This section presents a brief review of the main conceptsused in this research We introduce classic power indicesbriefly and explain the indexmeasure method that ourmodelextends Since our model is based on social networks thistheory is briefly reviewed Social learning theory is anotherkey concept used in this research therefore the concepts ofsocial learning and social impact are reviewed briefly and thephases in social learning modeling are introduced A briefdescription of a power indicesmodel based on social learningis also presented at the end of this section

21 Power Indices Power is a voterrsquos capacity to influence thevoting procedure Several important and now standard powerindices such as the ShapleyndashShubik index [1] and the Banzhafpower index [2] have been introduced in the literature forsimple games Their axiomatizations have been provided byamong others Shapley [8] andOwen [9]These power indiceshave been modified and extended in various ways [10ndash14 37]

A group decision-making problem can be described as anordered binary (119883 V) = [119902119908] in which 119883 = 1 2 3 sdot sdot sdot 119899is the set of voting members with weights of 119882 = 1199081198941times119899and 119902 is the winning threshold The two most well-knownpower indices for binary voting games (voting betweenldquoYesrdquo and ldquoNordquo) are the ShapleyndashShubik index [1] and theBanzhaf index [2] The difference between these two indicesis that the Shapley index calculates the winning probabilityof an individual based on sorting while the Banzhaf indexcalculates the same probability based on combinations ofindividuals [38] This paper analyzes the influence of sociallearning on a power distribution based on Banzhaf index

Let V be a voting game that consists of119883 = 1 2 3 sdot sdot sdot 119899voters with two alternatives (ie vote ldquoYesrdquo or ldquoNordquo) andsatisfies V(119862119884) = 1 if 119908(119862119884) ge 119902 V(119862119873) = 0 if 119908(119862119884) lt 119902where 119862119897 is the set of voters who choose alternative 119897 (119897 =119884119873) and 119908(119862119884) is the voting weight of voters in 119862119897 In thiscase 119862 = (119862119884 119862119873) is called an arrangement of the playersamong the two alternatives The function V 2119883 997888rarr 0 1is monotonic (119862119884 sube 119878119884 997904rArr V(119862119884) le V(119878119884)) and satisfiesV(120601) = 0 V(119883) = 1 Then the Banzhaf index of voter 119894 is

119861119894 (V) = 12119899minus1 times sum119862119894isin119862119884

(V (119862119884) minus V (119862119884 119894)) (1)

Discrete Dynamics in Nature and Society 3

and the relative power index is

1198611015840119894 (V) = 119861119894 (V)sum119894 119861119894 (V) (2)

22 Social Network Analysis Due to the overwhelmingevidence indicating that complex network promotes thedescription of the complex relationship in human society[15] social network analysis (SNA)mdashan important branchof complex network sciencemdashhas become one of the mostpopular methodologies for investigating social relationshipsamong group members It provides an effective way toanalyze interconnection structures that more representativefor human societies [15ndash17] and has been introduced intoanalysis of group decision-making [26ndash28]

Social networks have been modeled as a graph of rela-tionships among individuals In a voting group the socialrelations between members can be described as a socialnetwork 119866 = (119883 120581 119864119872) (shown in Figure 1) with votersas nodes where 119883 = 1 2 3 sdot sdot sdot 119899 is the set of voters120581 = 119896119894 where 119896119894 is the weight of node 119894 representingits attributes (eg level of expertise level of professionalknowledge etc) 119864 = 119890119894119895 is the set of edges representingthe social relationship of members and 119872 = 119898119894119895 is the setof 119890119894119895rsquos weights representing the closeness of 119894 119895

Previous research shows that the structure and charac-teristics of a social network determine the behaviours andopinions of humans [15 16] To measure the importanceor influence of decision-makers in a social network theconcept of power was introduced into network analysis [2728] Network analysis assume that power is related to nodecriticality such as network centrality [30 31] or connectivity[39 40] More recently new ideas have been proposedMorone and Makse [32] mapped the problem onto optimalpercolation in random networks to identify the minimalset of influencers and introduced the collective-influencealgorithm To adapt the Banzhaf index and discover thecritical node in a network Asif et al [41] proposed thecombined Banzhaf amp diversity index Such network centralitymeasures incorporate network structures but do not considervoting rules

23 Social Learning eory Power indices in the context ofsocial network should include not only the direct impact ofthe individual on the outcome of the decision (direct power)but also the influence gained through influence on otherindividuals (indirect power) In a group decision-makingproblem the group interaction process is essentially theprocess of obtaining information through empirical searchor behavioural observation and learning [34] People oftenbelieve that they can obtain useful information from othersrsquoactions and use that information to improve the correctnessof their own decisions ie they can experience social learn-ing

Social learning theory has been widely used in sociologypsychology and behavioural economics [35 36] The theoryof social learning shows that social learning efficiency is

m46

m16

m15

m26 m36

m34

k1

k2 k3

k6

k4k5

Figure 1 Social network of voting groups

related to the attributes of the individual that is their socialinfluence

119901 (119874 (119894) | (119904 (119894) 119878119894)) =lt 119904 (119894) 119878119894 119868119894 gt (3)

where 119874(119894) is the postbehaviour of individual 119894 119904(119894) is thepersonal information (behaviour or opinion) of 119894 119878119894 is thebehaviour of other individuals observed by individual 119894 and119868119894 is the social impact on individual 119894

Social impact is a common social phenomenon that refersto the effect on peoplersquos behaviours of the surrounding socialenvironment According to the social impact theory whichwas formulated by Latane (Latane 1981) such impact dependson three factors [42] (1) the social distance of sources fromthe subject which could be spatial proximity or closeness inspace time or an abstraction of the personal relationship(2) the sourcesrsquo strength or power of persuasionsupportand (3) the number of sources Thus the total influence thatindividual 119894 experiences from the society is

119868119894 = [[119899sum119895=1

120585119895 ∙ 120574119894119895 (1 minus 120590 (119894) 120590 (119895))]]minus [[119899sum119895=1

120577119895 ∙ 120574119894119895 (1 + 120590 (119894) 120590 (119895))]] (4)

where 120585119894 is 119894rsquos persuasiveness and 120577119894 is 119894rsquos supportivenessdescribing the agentrsquos capability to convince another agentto change or persist in its opinion respectively 120574119894119895 is thecloseness between 119894 119895 and 120590(119894) = plusmn1 respects the binaryopinions of individuals When the pressure to change onersquosopinion is greater than the pressure to maintain onersquos point ofview that is when 119868119894 gt 0 then the opinion changes

This paper aims to analyze the influence of individualinteractions on voting processes and voting power It isassumed that the interaction between individuals in groupdecision-making is essentially a process for individuals torenew their opinions through social learning The interactionprocess in group decision-making can be described as inFigure 2


Decision group

Individual attribute

Decision-making Behavior

Social viewpoint

Social factorsSocial network

Social relations

Learning

Assimilate

Judgement

Individual

Initial Status Behavior Observation Social Learning Update decision

Figure 2 The social learning process of voting groups based on social networks

Individual viewpoint Combination

social learning

Decision resultUpdate Decision

affect

Voting power

YesNo

Figure 3 The influence of social learning on the voting process

Based on social learning theory the process of decision-makers influencing voting results can be described as inFigure 3

3 Proposed Model

Our proposed model of power indices is based on sociallearning theory The interactive voting game is essentially aprocess of behavioural observation learning based on a socialnetwork

31 Social Learning Model The model consists of 119883 =1 2 3 sdot sdot sdot 119899 voters The voters share their opinions 119904(119894) =plusmn1 (119894 = 1 2 sdot sdot sdot 119899) based on social network119866 = (119883 120581 119864119872)Let 1198620 = (1198621198840 1198621198730) be the initial distribution of grouppreference and let 119878119894 = 119904(119895) | 119895 isin Γ119894 be the preferencecombination observed by individual 119894 119878119894 = 119904(119895) | 119895 isin Γ119894where Γ119894 is the collection of 119894rsquos neighbor nodes

119904 (119894) = 1 119894 isin 1198620119884minus1 119894 isin 1198620119873 (5)

The total influence that individual 119894 experiences from thesociety is

119868119894 = 120572minussum119895

120582119895 times 120574119894119895 times (1 minus 119904 (119894) 119904 (119895)) minus 120572+sum119895

120582119895 times 120574119894119895times (1 + 119904 (119894) 119904 (119895))

(6)

where 120572+ 120572minus are the influence parameters of the same or thedifferent views respectively 120572+ 120572minus ge 0 120572+ gt 120572minus means thatthe individual is more receptive to the same point of view 120582119894

is the social influence (persuasiveness or supportiveness) and120574119894119895 is the closeness between 119894 119895Social influence can be defined as ldquochange in the belief

attitude or behaviour of a person that results from theaction or presence of another personrdquo [43] In the originalsocial impact model the strengths of persuasiveness andsupportiveness are assigned to the agents according to auniform random distribution However in the real worldsocial influence that is social power depends not onlyon the agentrsquos own professionalism which is the basis ofsocial power but also on hisher structural position in thesocial network Researches on network science have shownthat social power depends on the individualrsquos connectionstructure in the network [44] and suggested that the well-connected nodes should be considered influential people[30 39 40]Therefore social power could be given as follows

Definition 1 (social power) If in a social network 119866 =(119883 120581 119864119872) 119889119894 is the node degree of 119894 119896119894 is 119894rsquos knowledge leveland the network centrality of 119894 is

119862119863 (119894) = 119889119894119899 minus 1 (7)

then the social power of 119894 is120582119894 = 1198961015840119894 times 119862119863 (119894) (8)

where 1198961015840119894 is the normalized value of 119896119894 1198961015840119894 = 119896119894max119899119894=1(119896119894)The relationship closeness is another factor in social

behaviour learning In traditional social impact modelsthe closeness of any pair of agents is usually measured bythe distance function In fact the closeness degree can bedetermined by the frequency of communication


Definition 2 (relationship closeness) In a social network 119866 =(119883 120581 119864119872) the closeness of any pair of nodes 119894 119895 can beassigned as

120574119894119895 = 119898119894119895max (119872) (9)

According to (6) when the pressure of persuasiveness is lessthan that of supportiveness that is 119868119894 lt 0 the individualtends to keep his or her original view if the opposite istrue and the pressure of supportiveness is less than that ofpersuasiveness the agent tends to change his or her ownopinion Especially when an individual has nonneighbornode then its social impact 119868119894 = minusinfin

Based on the analysis above (3) could be rewritten for themodel as

119901 (119874 (119894) = minus119904 (119894) | (119904 (119894) 119878119894)) = 11 + 119890minus2(119868119894119887)119901 (119874 (119894) = 119904 (119894) | (119904 (119894) 119878119894)) = 11 + 1198902(119868119894119887)

(10)

where 119887 is a coefficient of social impact expressing the sociallearning efficiency of the group 119887 gt 0

For individual 119894 119901(119874(119894) = 119900 | (119904(119894) 119878119894)) isin [0 1] andsum119900 119901(119874(119894) = 119900 | (119904119894 119878119894)) = 1 (119900 = minus1 1)32 Power Indices Model The ability of an individual toinfluence the outcome changes when social learning takesplace Let 119862119884 | 1198620 be a set of decision-makers who vote ldquoYesrdquoafter an opinion interaction when the initial arrangement ofthe voters is 1198620 = (1198621198840 1198621198730) If voter 119894 isin 1198621198840 and after theopinion interaction the ldquoYesrdquo alternative is chosen and whenthe initial inclination of voter 119894 is changed from ldquoYesrdquo to ldquoNordquoand then after the opinion interaction the outcome of group isldquoNordquo that is V(119862119884 | 1198620) = 1 and V(119862119884 | (1198621198840119894 119862cup119894)) = 0if 119894 isin 1198620119884 we say that the individual 119894 influences the outcomeof voting through both direct voting and influence on othersrsquovoters which is called the integrated power of voters

Definition 3 (integrated power index) In a group decision-making problem (119883 V) = [119902119908] with the social network119866 = (119883 120581 119864119872) let 120583119894(1198620) = 119901(119874(119894) = 1 | (119904119894 119878119894)) bethe probability of voting ldquoYesrdquo after social learning then theprobability of a combination after interaction is

120587 (119862119884 | 1198620) = prod119894isin119862119884|119862

0

120583119894 (1198620) times prod119895notin119862119884|119862

0

(1 minus 120583119895 (1198620)) ) (11)

Let the set of winning combinations be 119881 then 119894rsquos integratedpower index is

119877119894 (V) = 12119899minus1 sum1198620 119894isin1198620

119884

( sum119862119884|1198620isin119881

120587 (119862119884 | 1198620)

minus sum119862119884|(119862119884

0119894119862cup119894)isin119881

120587 (119862119884 | (1198621198840 119894 119862 cup 119894))) (12)

and its relative integrated power index is

1198771015840119894 (V) = 119877119894 (V)sum119894isin119873 119877119894 (V) (13)

Compared to traditional Banzhaf indices (119861119894) the proposedintegrated power index of voters in a voting gamewith a socialnetwork has the following properties

Property 4 For any 119894 isin 119883 119877119894 = 0 997904rArr 119861119894 = 0 and 119861119894 = 0 997904rArr119877119894 = 0 if and only if it is an isolated node in the social networkWhen 119877119894 = 0 it shows that the direct power and indirect

power of the voter are both zero which is easy to represent as119861119894 = 0 Conversely a voter who has no ability to influencethe voting result is called a ldquodummyrdquo in the traditionalpower measure and hisher power index is 0 ie 119861119894 = 0Although a ldquodummyrdquo cannot affect the outcome directlyheshe may influence the behaviour of other voters throughsocial interactionThus hisher integrated power index is notnecessarily 0 In addition when the ldquodummyrdquo is an isolatednode in the social network heshe does not interact withother nodes and cannot generate power by influencing othernodes Therefore hisher recessive power index must be 0

Property 5 For any 119894 isin 119883 119861119894 = 1 997904rArr 119877119894 = 1 if and only if itis an isolated node in the social network

Similar to Property 4 when 119861119894 = 1 it is a dictator in thetraditional Banzhaf index A dictatorrsquos power index weakensbecause of the influence of other individuals and hisherrecessive power index is not necessarily 1 In contrast whenthe dictator is an isolated node the recessive power indexmust be 1 because heshe is not affected by other individuals

4 Simulation Analysis and Results

Now we can analyze the influence of social learning onpower indices The most important dependent variables ofinterest are the individual attribute (professional level) socialnetwork structure relationship closeness and social learningefficiency Thus an application scenario and simulationresults are described in this section

41 Setup A voting group with 9 members votes for anenvironmental protection project following a simple majorityvoting rule Suppose the social relationships among the groupcan be described as a social network shown in Figure 4 Thevoting weight and social information of the voters are shownin Table 1

42 Results and Analysis We analyze the power indices usingthe proposed model introduced in Section 3 Assuming thatthe social impact coefficient (social learning efficiency) 119887 = 1and coefficient 120572+ = 120572minus = 1 then votersrsquo Banzhaf indicesand integrated power indices can be calculated The resultsare shown in Table 2 and the comparison of the integratedpower indices and Banzhaf indices is shown in Figure 5

As seen from Table 2 and Figure 5 due to the influenceof social learning behaviour the integrated power indices of


Table 1 Voting weight and votersrsquo social information

Voter Voting weight Node weight Neighbor nodes (edge weight)V1 5 9 V3(1) V9(5)V2 2 8 V3(2) V5(7) V9(6)V3 1 5 V1(1) V2(2) V8(3)V4 4 3V5 3 2 V2(7) V7(5) V8(9)V6 5 8 V9(4)V7 2 9 V5(5)V8 3 6 V3(3) V5(9)V9 3 2 V1(5) V2(6) V6(4)

Table 2 Banzhaf indices and integrated power indices for voters

Voters V1 V2 V3 V4 V5 V6 V7 V8 V9119861119894 04102 01523 00742 03242 02305 04102 01523 02305 02305119877119894 02796 04569 02272 04499 04547 02719 02394 02819 045121198611015840119894 01852 00688 00335 01464 01041 01852 00688 01041 010411198771015840119894 00898 01468 00730 01445 01461 00874 00769 00906 01449

9

6

9

3

85

2

8

2

51

9

4

7

6

v5

v1

v3

v2v4

v8

v6

v7

2

v9

5

3

Figure 4 The social network in a voting group The node V119894 repre-sents voter 119894 the nodersquo weight represents the knowledge professionallevel 119896119894 and the weighted edge represents the relationship closeness(119898119894119895) among voters

voters have varied changes compared to the Banzhaf indicesVoter 2 has the highest integrated power in the voting processeven though his or her Banzhaf index is only slightly higherthan that of the least powerful (Banzhaf index) voter 3whereas the integrated power indices of voter 1 and voter 6who originally had the highest Banzhaf index ratings becomerelatively small As shown in Figure 5 the power indices ofvoter 1 voter 2 and voter 6 changed considerably Specificallyvoter 4 is the ldquoisolatedrdquo node in the social network andits integrated power index varies slightly compared to theBanzhaf index

421 e Influence of Individual Professionalism on thePower Index Professional level is an important factor affect-ing a voterrsquos social power which definitely influence the

02

01801601401201

008006004002

0

1 2 3 4 5 6 7 8 9

voters

pow

er

Integrated powerBanzhaf index

Figure 5 Comparison of the integrated power and Banzhaf indices

social learning behaviour and voting power We discusshow a voterrsquos professional level affects power indices byanalyzing the relationship between professional level 119896119894 and119863 -119901119900119908119890119903(119894) (119863-119901119900119908119890119903(119894) = 1198771015840(119894) minus 1198611015840(119894)) For any voter119894 isin 119883 let voter 119894 have a professional level 119896119894 = 1 2 3 sdot sdot sdot 9in turn while keeping the other parameters the changes inpower indices are shown in Figure 6

Figure 6 shows that for any voter with the exception ofan isolated node the greater the value of 119896119894 is the higher thevalue of 119863 -119901119900119908119890119903(119894) is This result means that in the samevoting environment and conditions the higher the voterrsquosprofessional level is the higher the integrated power index is

Overwhelming evidences suggest that the structure ofsocial relationship network affects the group interactions andbehaviours of individuals [15 16] It is necessary to test therobustness when a varied number of players are involved

The robustness is discussed when the number of playerschanges based on random network structure Figure 7 shows


voter1voter3voter5

voter7voter9

minus01

minus005

0

005

01

015

02

D-p

ower

83 6 92 51 4 7Professional level (k)

Figure 6 Influence of professionalism on power indices Voters in the social network shown in Figure 4 play a repeated voting gamemdashinwhich they have social learning behaviours through observing their neighborsmdashwith voter 119894rsquos professional level 119896119894 rising from 119896119894 = 1 to 119896119894 = 9To make it clearer only changes in power indices (119863-119901119900119908119890119903(119894) = 1198771015840(119894) minus 1198611015840(119894)) of voter 1 voter 3 voter 5 voter 7 and voter 9 are shown Thechanges in power indices improve with the professional level increasing except for the isolated nodes

voter1voter2voter3

voter4voter5

minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

025

D-p

ower

3 951 7Professional level (k)

(a) The results when N=5

voter1voter3voter5voter7

voter9voter11voter13

minus006

minus004

minus002

0

002

004

006

008

01

012

014

D-p

ower


(b) The results when N=13

Figure 7 Robustness of professionalism influence Subgraph (a) shows the simulation results of 5 players (ie119873 = 5) who play a repeatedvoting game in a social network with a 30 random connection probability accompanied by voter 119894rsquos professional level 119896119894 rising from 119896119894 = 1to 119896119894 = 9 while leaving other parameters unchanged Subgraph (b) shows the results of 13 players (ie 119873 = 13) with the same simulationconditions The changes in power indices (119863-119901119900119908119890119903(119894) = 1198771015840(119894) minus 1198611015840(119894)) improve with the professional level increasing in both cases exceptfor the isolated nodes

the simulation results when the number of players is 5 and13 Voters play a repeated voting game in a social networkwith a 30 connection probability accompanied by voter119894rsquos professional level 119896119894 rising from 119896119894 = 1 to 119896119894 = 9 Asshown in Figure 7 regardless of the number of players the

integrated power index increases with the professional levelrising except for the isolated nodes

The finding that the integrated power index increaseswith the professional level rising is in line with peoplersquosinstincts the higher the individualrsquos professional knowledge


voter1voter3voter5

voter7voter9

minus01

minus005

0

005

01

015

02

025

D-p

ower

63 80 521 4 7Number of neighbours (d)

Figure 8 Impact of social network structure on the power indexVoters in the social network shown in Figure 4play a repeated votinggame with the number of voter 119894rsquos neighbors 119889119894 increasing from 119889119894 =0 to 119889119894 = 8while keeping the other parametersWhile the integratedpower index may decrease when the number of neighbors changesfrom0 to 1 the changes in power indices (119863-119901119900119908119890119903(119894) = 1198771015840(119894)minus1198611015840(119894))increase with the number of neighbors increasing from 1 to 8

or technical level is the higher the authority or discoursepower in the group is and the greater the influence overothers in the interaction is in this case the individual willhave higher integration power Therefore in an interactivevoting game problem if the decision-makers want to obtainmore power especially those at the low professional level oneof the important ways to improve authority is through theprofessional field

422 e Influence of Social Network Structure on the PowerIndex Another important parameter in which we are inter-ested is the structure of the social network which can bedescribed using the network centrality of voters To avoidinterference with relationship closeness we assume that forany pair of voters 119894 119895 the relationship closeness 119898119894119895 = 1For any voter 119894 isin 119883 let the number of voter 119894rsquos neighborschanges as 119889119894 = 1 2 3 sdot sdot sdot 9 while keeping other parametersor conditions unchanged The changes in power indices areshown in Figure 8

As we can see from Figure 8 the value of 119863 -119901119900119908119890119903(119894)increases with the number of neighbors increasing exceptwhen the voterrsquos number of neighbors is 0 This means thatthe voter may have a higher integrated power index whenheshe is an isolated node in the social network comparedwith having relatively fewer neighbors

To analyze the robustness of the above finding we discussthe influence of voter 119894rsquos social structure based on the randomnetwork with a 30 connection probability Figure 9 showsthe simulation results when the number of players is either119873 = 5 or 119873 = 13 with voter 119894rsquos neighbors 119889119894 increasing from119889119894 = 0 to119889119894 = 8while keeping the other parameters As shownin Figure 9 the simulation results are the same with the abovefindings

The analysis results show that the integrated power indexof decision-makers may decrease when heshe changes froman isolated node to a nonisolated node and then hisherintegrated power will increase with the number of neighbornodes increasing This finding is inconsistent with peoplersquosintuitions and the main reasons for such inconsistency arethat when an individual is an isolated node in a socialnetwork he or she is not disturbed by any other viewpointand the influence on the result depends entirely on hisor her own judgment ie direct power When there is asocial learning process the direct power is reduced by theinfluence of other individuals However as the extensivenessof individual social relations increases the influence on otherindividualswill also gradually increase and his or her indirectpower index will gradually increase When indirect powerincreases to a certain extent his or her integrated powerwill gradually increase Therefore if the extensiveness ofan individualrsquos social relations is at a relatively low levelsocial learning will reduce his or her power (compared withnoninteractive nodes) only when the extensiveness of socialrelations reaches a certain high level does social learningimprove his or her integrated power

423 e Influence of Relationship Closeness on the PowerIndex Now we examine the relationship between the close-ness 120574119894119895 and power index For any voter 119894 isin 119883 let therelationship closeness among voter 119894 and all of his neighborvoters be equal and 119898119894119895 = 119898119894119896 = 1 2 3 sdot sdot sdot 9 (forall119895 119896 isin Γ119894)while keeping other parameters or conditions unchanged thechanges in the power index are shown in Figure 10

Figure 11 shows the simulation results when the numberof players is either 119873 = 5 or 119873 = 13 within a social networkconstructed by a 30 connection probability and the resultsshows the same that the integrated power improves with thevoterrsquos closeness accompanied by neighbors increasing

The analysis results show that the greater the closeness isin the relationship between individual and neighbor nodesthe higher the integrated power index is The relationshipcloseness is related to the social influence on the individualthe individual is more susceptible to the behaviour of theperson who is closer to her or him Therefore in votinggames individuals with high closeness to other individualsare more likely to gain higher integration power

424e Influence of Social Learning Efficiency Social learn-ing efficiency is an important parameter that reflects thedegree of interactionWe explore the impact of social learningefficiency on voting power by assuming social learningefficiency 119887 = 1 2 3 sdot sdot sdot 9 (119895 isin Γ119894) while keeping the otherparameters or conditions and the result is shown in Figure 12

As we can see from Figure 12 with the improvement insocial learning efficiency 119887 the integrated power index ofvoter 5 and voter 9 decreased gradually while the integratedpower index of the other individuals increased gradually Themain reason is the difference in the social network structureFigure 4 shows that voters 5 and 9 are people with extensiverelationships and high relationship closeness in the social net-work and the social influence they experience is at a higherlevel Thus the improvement in social learning efficiency



voter1voter2voter3

voter4voter5

minus04

minus02

0

02

04

06

08

1

D-p

ower


D-p

ower

0 2 4 6 8 10 12Number of neighbours (d)

minus02

minus01

0

01

02

03

04

05

06




Figure 9 Robustness of voterrsquos neighbors influence Subgraphs (a) and (b) show the simulation results of five players (ie 119873 = 5) and13 players (ie 119873 = 13) respectively in which voters play a repeated voting game in a social network with a 30 connection probabilityaccompanied by voter 119894rsquos neighbors 119889119894 increasing from 119889119894 = 0 to 119889119894 = 8 while keeping the other parameters

D-p

ower

minus01minus008minus006minus004minus002

0002004006008

01

voter1voter3voter5

voter7voter9

83 6 92 51 4 7Closeness (m)

Figure 10 The influence of relationship closeness on the powerindex Votersmdashwithin the social network shown in Figure 4mdashplaya repeated voting game with the closeness among voter 119894 and hisherneighbor voters 119898119894119895 = 119898119894119896 (forall119895 119896 isin Γ119894) increasing from 119898119894119895 = 1 to119898119894119895 = 9 while leaving other parameters or conditions unchangedThe changes in power indices increase (119863-119901119900119908119890119903(119894) = 1198771015840(119894) minus 1198611015840(119894))with the voterrsquos closeness with neighbors increasing

affects themmore than it does with other individuals and theintegrated power index will be relatively lower In contrastother individuals gained relatively high integrated powerThesimulation experiment results shown in Figure 13 also presentthe phenomenon that a higher social learning efficiency may

lead to a decrease in the integrated power index of voters withhigher social impact or closeness and the voters with lowersocial impact or closeness are the opposite while both casesare possible for voters with middle social impact or closeness

This finding shows that the effect of social learningefficiency on the integrated power index is closely relatedto the social network structure of an individual for peoplewith strong communicative competence a higher sociallearning efficiency is not conducive to the promotion of theirintegrated power

In brief integrated power is directly or indirectly affectedby multiple factors such as professional knowledge levelsocial network structure relationship closeness and sociallearning efficiency Only when all these factors are at higherlevels can they have higher integrated power than otherdecision-makers Limited social relations in a group mayrestrict the integrated power of an authority figure with ahigh level of professional knowledge However individualswith normal professional knowledge levels have higher powerif they engage in extensive communication and garner highinfluence The measurement of the integrated power indexcan help us find the recessive power in management and thekey factors in problems and it is beneficial for improvementsin management efficiency

5 Conclusions

Based on the interaction among views in a voting group theconcept of integrated power is put forward to simultaneously


voter1voter2voter3

voter4voter5

minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

D-p

ower

3 951 7Closeness (m)





minus008

minus006

minus004

minus002

0

002

004

006

008

D-p

ower


Figure 11 Robustness of voterrsquos closeness influence Subgraphs (a) and (b) show the simulation results of five players (ie 119873 = 5) and13 players (ie 119873 = 13) respectively Voters play a repeated voting game in a social network with a 30 random connection probabilityaccompanied by voter 119894rsquos closeness with his neighbors 119898119894119895 = 119898119894119896 (forall119895 119896 isin Γ119894) increasing from 119898119894119895 = 1 to 119898119894119895 = 9 while keeping the otherparameters

D-p

ower

voter1voter3voter5

voter7voter9

minus01

minus005

0

005

63 8 942 51 7coefficient (b)

Figure 12 Impact of social learning efficiency on the power indexVotersmdashwithin the social network shown in Figure 4mdashplay arepeated voting game with the social learning efficiency parameters119887 rising from 119887 = 1 to 119887 = 9 while leaving other parameters orconditions unchanged

measure the direct and indirect power of voters and amethodfor measuring integrated power is given We explored theinfluence of interaction on power indices using the theoryand model of social learning The influence of interactionon voting processes and voting results is analyzed from the

perspective of votersrsquo professional level social network struc-ture relationship closeness and social learning efficiencyWe construct a measurement model of integrated powerand analyze the change in integrated power relative to theBanzhaf index by numerical simulation The results show thefollowing

(a) When the individualrsquos professional level is higherand the relationship closeness with the associateddecision-maker is stronger the integrated powerindex is higher and the value added is higher relativeto the Banzhaf index

(b) The integrated power index of decision-makers maydecrease when heshe changes from an isolated nodeto a nonisolated node and then hisher integratedpower will increase with the number of neighbornodes increasing

(c) The influence of social learning efficiency on theintegrated power index is closely related to theindividualrsquos social network structure and for thosewith a strong communicative competence within thesocial network higher social learning efficiency is notconducive to the promotion of integrated power andthe voters with lower social impact or closeness arethe opposite while both cases are possible for thevoters with middle communicative competence


social powercloseness

93 86 10 135 122 111 4 7Voter

0

05

1

15

2

25

Valu

e

(a) Votersrsquo social relationship structure

D-p

ower

minus004

minus002

0

002

004

006



3 951 7coefficient (b)

(b) Influence of learning efficiency

Figure 13 Impact of social learning efficiency and social relationship structure Voters play a voting game in a social network with 13 playersand a 30 random connection probability Subgraph (a) shows the simulation results of votersrsquo social power and relationship closeness withneighbors Subgraph (b) shows the simulation results of changes in power indices (119863-119901119900119908119890119903(119894) = 1198771015840(119894)minus1198611015840(119894)) in which voters play a repeatedvoting game with the social learning efficiency (119887) increasing from 119887 = 1 to 119887 = 9 while leaving other parameters unchangedThe integratedpower of voters with higher social impact or closeness decreases accompanied by the increases of social learning efficiency (119887) and the voterswith lower social impact or closeness are the opposite while both cases are possible for the voters with middle social impact or closeness

Compared to the previous methods of power measurementthe proposed method is no longer confined to the deter-mination of power based on voting results but takes intoaccount the social relationships of decision-makers and theeffect that interaction has on their views The proposedpower measure includes both the direct power and theindirect power gained through a social network and canbetter reflect the distribution of power in the voting problemthe obligations undertaken and the degree of contributionto the results This measure is valuable for the identificationof key decision-makers prediction and guidance regardingdeveloping trends in public opinion in online networks aswell as for network recommendations etc

The ideas explored in this paper however are quitepreliminary There are many important problems in need offurther research such as the heterogeneity of the learningefficiency of each voting individual in the analysis of thesocial learning The characteristics and properties of recessivepower are also a key area for further research

Data Availability

All the data used to support the findings of this study areincluded or described within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (Projects nos 71471123 7187115071571126) Sichuan University Innovation Spark Project(2018hhs-35) and the Fundamental Research Foundation forthe Central Universities (Project no skqy201621)

References

[1] L S Shapley and M Shubik ldquoA method for evaluation thedistribution of power in a committee systemrdquo e AmericanPolitical Science Review vol 48 no 3 pp 787ndash792 1988

[2] J F Banzhaf ldquoWeighted voting doesnrsquot work a mathematicalanalysisrdquo Rutgers Law Review vol 19 pp 317ndash343 1964

[3] J Gao X Yang and D Liu ldquoUncertain Shapley value ofcoalitional game with application to supply chain alliancerdquoApplied So13 Computing vol 56 pp 551ndash556 2017

[4] R R Yager ldquoOn using the shapley value to approximatethe choquet integral in cases of uncertain argumentsrdquo IEEE


Transactions on Fuzzy Systems vol 26 no 3 pp 1303ndash1310 2018[5] A Gelman J N Katz and J Bafumi ldquoStandard voting power

indexes do not work an empirical analysisrdquo British Journal ofPolitical Science vol 34 no 4 pp 657ndash674 2004

[6] D S Felsenthal andMMachover ldquoA priori voting power whatis it all aboutrdquo Political Studies Review vol 2 no 1 pp 1ndash232016

[7] S Kaniovski and D Leech ldquoA behavioral power indexrdquo PublicChoice vol 141 no 1-2 pp 17ndash29 2009

[8] L S Shapley ldquoA comparison of power indices and a non-symmetric generalizationrdquoDevelopment vol 115 no 4 pp 957ndash971 1977

[9] G Owen and L S Shapley ldquoOptimal location of candidates inideological spacerdquo International Journal of Gameeory vol 18no 3 pp 339ndash356 1989

[10] C Beisbart and L Bovens ldquoA power measure analysis ofAmendment 36 in Coloradordquo Public Choice vol 134 no 3-4pp 231ndash246 2008

[11] L Bovens and C Beisbart ldquoMeasuring voting power fordependent voters through causal modelsrdquo Synthese vol 179 no1 pp 35ndash56 2011

[12] J Freixas ldquoProbabilistic power indices for voting rules withabstentionrdquoMathematical Social Sciences vol 64 no 1 pp 89ndash99 2012

[13] S Courtin and B Tchantcho ldquoA note on the ordinal equivalenceof power indices in games with coalition structurerdquo eoryand Decision An International Journal for MultidisciplinaryAdvances in Decision Science vol 78 no 4 pp 617ndash628 2015

[14] S Courtin Z Nganmeni and B Tchantcho ldquoThe shapley-shubik power index for dichotomousmulti-type gamesrdquoeoryand Decision vol 81 no 3 pp 1ndash14 2016

[15] M Perc J Gomez-Gardenes A Szolnoki L M Florıa andY Moreno ldquoEvolutionary dynamics of group interactions onstructured populations a reviewrdquo Journal of the Royal SocietyInterface vol 10 no 80 Article ID 20120997 2013

[16] M Perc J J Jordan D G Rand Z Wang S Boccaletti andA Szolnoki ldquoStatistical physics of human cooperationrdquo PhysicsReports vol 687 pp 1ndash51 2017

[17] G Cao Y Shi and Q Li ldquoStructure characteristics of theinternational stock market complex network in the perspectiveof whole and partrdquoDiscrete Dynamics in Nature and Society vol1 pp 1ndash11 2017

[18] A E Sokhey and S D McClurg ldquoSocial networks and correctvotingrdquo Journal of Politics vol 74 no 3 pp 751ndash764 2012

[19] O Lesser L Naamani-Dery M Kalech and Y Elovici ldquoGroupdecision support for leisure activities using voting and socialnetworksrdquo Group Decision and Negotiation vol 26 no 3 pp473ndash494 2017

[20] E Katz and P F Lazarsfeld Personal Influence e Part Playedby People in e Flow of Mass Communications TransactionPublishers Piscataway NJ USA 2006

[21] R V D Brink A Rusinowska and F Steffen ldquoMeasuring powerand satisfaction in societies with opinion leaders dictator andopinion leader propertiesrdquoHomo Oeconomicus vol 28 no 1-2pp 161ndash185 2011

[22] R van den Brink A Rusinowska and F Steffen ldquoMeasuringpower and satisfaction in societies with opinion leaders anaxiomatizationrdquo Social Choice and Welfare vol 41 no 3 pp671ndash683 2013

[23] R van den Brink A Rusinowska and F Steffen ldquoSatisfaction insocieties with opinion leaders andmediators properties and anaxiomatizationrdquo Computer Science and Gameeory 2014

[24] J Feng and Z Yao ldquoConsumer-generated reviews based onsocial learning theory implications for purchase decisionrdquoChinese Journal of Management Science vol 24 no 9 pp 106ndash114 2016

[25] B Kittel G Kanitsar and S Traub ldquoKnowledge power and self-interestrdquo Journal of Public Economics vol 150 pp 39ndash52 2017

[26] CGuoR-L Shi andM Jin ldquoGroupdecision opinion evolutionand simulation based on community and individual influencepowerrdquo Journal of Intelligent amp Fuzzy Systems vol 33 no 5 pp2667ndash2676 2017

[27] J Wu F Chiclana H Fujita and E Herrera-Viedma ldquoA visualinteraction consensus model for social network group decisionmaking with trust propagationrdquo Knowledge-Based Systems vol122 pp 39ndash50 2017

[28] J Wu L Dai F Chiclana H Fujita and E Herrera-Viedma ldquoAminimum adjustment cost feedback mechanism based consen-susmodel for group decisionmaking under social networkwithdistributed linguistic trustrdquo Information Fusion vol 41 pp 232ndash242 2018

[29] Y T Mohmand and A Wang ldquoWeighted complex networkanalysis of pakistan highwaysrdquo Discrete Dynamics in Nature ampSociety vol 4 pp 1ndash5 2013

[30] L C Freeman ldquoCentrality in social networks conceptual clari-ficationrdquo Social Networks vol 1 no 3 pp 215ndash239 1978

[31] S B Seidman ldquoNetwork structure andminimumdegreerdquo SocialNetworks vol 5 no 3 pp 269ndash287 1983

[32] F Morone and H A Makse ldquoInfluence maximization incomplex networks through optimal percolationrdquo Nature vol524 no 7563 pp 65ndash68 2015

[33] C Simon de Blas J Simon Martin and D Gomez GonzalezldquoCombined social networks and data envelopment analysis forrankingrdquoEuropean Journal ofOperational Research vol 266 no3 pp 990ndash999 2018

[34] A Bandura Social Learning eory vol 1 ReCAPP ScottsValley Calif USA 2000

[35] P Kristjanson B Harvey M Van Epp and P K ThorntonldquoSocial learning and sustainable developmentrdquo Nature ClimateChange vol 4 no 1 pp 5ndash7 2014

[36] L Qiu and A BWhinston ldquoPricing strategies under behavioralobservational learning in social networksrdquoProduction Engineer-ing Research and Development vol 26 no 7 pp 1249ndash1267 2017

[37] S Kurz ldquoMeasuring Voting Power in Convex Policy SpacesrdquoEconomies vol 2 no 1 pp 45ndash77 2014

[38] P Hu and S Z Huang ldquoAnalysis and measuring model ofvoting individuals power on weighted majority rulesrdquo Journalof management science in China vol 4 no 6 pp 66ndash70 2001

[39] TNDinh Y XuanM TThai PM Pardalos andT Znati ldquoOnnew approaches of assessing network vulnerability hardnessand approximationrdquo IEEEACM Transactions on Networkingvol 20 no 2 pp 609ndash619 2012

[40] Y Shen N P Nguyen Y Xuan and M T Thai ldquoOn thediscovery of critical links and nodes for assessing networkvulnerabilityrdquo IEEEACM Transactions on Networking vol 21no 3 pp 963ndash973 2013

[41] WAsif H K QureshiM Rajarajan andM Lestas ldquoCombinedbanzhaf amp diversity index (CBDI) for critical node detectionrdquoJournal of Network and Computer Applications vol 64 pp 76ndash88 2016

[42] H C Kelman ldquoInterests relationships identities Three centralissues for individuals and groups in negotiating their socialenvironmentrdquo Annual Review of Psychology vol 57 no 1 pp1ndash26 2006


[43] W P Erchul and B H Raven ldquoSocial power in school consulta-tion a contemporary view of Frenchand Ravenrsquos bases of powermodelrdquo Journal of School Psychology vol 35 no 2 pp 137ndash1711997

[44] M Jalili ldquoSocial power and opinion formation in complexnetworksrdquo Physica A Statistical Mechanics and Its Applicationsvol 392 no 4 pp 959ndash966 2013

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and behaviour in actual voting systems however they arepowerless to reflect managerial phenomena caused by com-plex social interactions within organizations

Group interactions are indeed inseparably linked withour increasingly interconnected world and interactionsamong human societies are more complex [15ndash17] As com-plex network structures are more representative for humansocieties group interactions on complex network had beenwidely studied and present how social diversity affects humanbehaviour (for a review see [15]) Human experiments ofcooperation have also suggested that social relationship net-work structure is an important factor of human behaviours[16] In group decision-making process the structured rela-tionships between voters may also influence their behavioursand judgments [18 19] Thus later literatures have partiallyconsidered heterogeneous voters and their social structurewhen measuring power indices The most representativeof these models are opinion leadership models which arebased on the theory of the two-step flow of communication[20]The properties and axiomatization of opinion formationunder a leader-follower power dynamic have been provided[21ndash23] but voter heterogeneity in such models takes asimple form (leader and follower) Such relationships are notgeneral as relations among members are usually equal suchas in citizensrsquo elections and do not exist an ldquoinstitutionalpowerrdquo In addition the voting process is usually a processof information exchange and learning among members thatis based on social relationships [24]

Relationships among members can be described bysocial networks which provide a new way to study thebehaviour and influence of relationships among individualsand organizations in society Thus the theory and methodsof social networks have been introduced into problems ofgroup decision-making [25ndash28] Network analysts implicitlyassume that power is related to network centrality [29ndash33]but they do not consider the voting rule that is in effect whenanalyzing group decisions with social networks

This paper focuses on a power analysis based on socialnetworks The problem is complex on the one hand a socialnetwork enhances the interaction between decision-makersand the network effect may enlarge or reduce the influenceof decision-makers on the results on the other hand inthe social networks people can observe and obtain othersrsquobehavioural information more conveniently and update orstrengthen their own judgments which increase the variabil-ity in individual preferenceThese characteristics require thatpower indices not only reflect the direct influence that votershave on results but also consider the indirect influence onvoting result through social interactions

In this paper we analyze the process of group interactionsinformed by social learning theory [34] and construct apower measurement model based on social networks Sociallearning theory has been widely used in the fields of soci-ology psychology pedagogy and management science [2435 36] and holds that individualrsquos behaviours are deeplyinfluenced by the social environment and that individualswilldetermine or change their views and behaviours accordingto the observed behaviours [34ndash36] The group interactionsduring voting process can be regarded as a process of

behavioural observation and social learning Therefore westudy the influence group interactions have on voting basedon social learning theory and analyze the influence of theseinteractions on power indices

Thepaper is organized as follows Section 2 introduces theconcepts of power indices social network and social learningtheory Section 3 describes the integrated power measuremodel informed by social learning theory based on socialnetworks next in Section 4 we present a numerical exampleto analyze the influence of social learning on power indicesand finally Section 5 provides a brief summary

2 Preliminaries

This section presents a brief review of the main conceptsused in this research We introduce classic power indicesbriefly and explain the indexmeasure method that ourmodelextends Since our model is based on social networks thistheory is briefly reviewed Social learning theory is anotherkey concept used in this research therefore the concepts ofsocial learning and social impact are reviewed briefly and thephases in social learning modeling are introduced A briefdescription of a power indicesmodel based on social learningis also presented at the end of this section

21 Power Indices Power is a voterrsquos capacity to influence thevoting procedure Several important and now standard powerindices such as the ShapleyndashShubik index [1] and the Banzhafpower index [2] have been introduced in the literature forsimple games Their axiomatizations have been provided byamong others Shapley [8] andOwen [9]These power indiceshave been modified and extended in various ways [10ndash14 37]

A group decision-making problem can be described as anordered binary (119883 V) = [119902119908] in which 119883 = 1 2 3 sdot sdot sdot 119899is the set of voting members with weights of 119882 = 1199081198941times119899and 119902 is the winning threshold The two most well-knownpower indices for binary voting games (voting betweenldquoYesrdquo and ldquoNordquo) are the ShapleyndashShubik index [1] and theBanzhaf index [2] The difference between these two indicesis that the Shapley index calculates the winning probabilityof an individual based on sorting while the Banzhaf indexcalculates the same probability based on combinations ofindividuals [38] This paper analyzes the influence of sociallearning on a power distribution based on Banzhaf index

Let V be a voting game that consists of119883 = 1 2 3 sdot sdot sdot 119899voters with two alternatives (ie vote ldquoYesrdquo or ldquoNordquo) andsatisfies V(119862119884) = 1 if 119908(119862119884) ge 119902 V(119862119873) = 0 if 119908(119862119884) lt 119902where 119862119897 is the set of voters who choose alternative 119897 (119897 =119884119873) and 119908(119862119884) is the voting weight of voters in 119862119897 In thiscase 119862 = (119862119884 119862119873) is called an arrangement of the playersamong the two alternatives The function V 2119883 997888rarr 0 1is monotonic (119862119884 sube 119878119884 997904rArr V(119862119884) le V(119878119884)) and satisfiesV(120601) = 0 V(119883) = 1 Then the Banzhaf index of voter 119894 is

119861119894 (V) = 12119899minus1 times sum119862119894isin119862119884

(V (119862119884) minus V (119862119884 119894)) (1)



1198611015840119894 (V) = 119861119894 (V)sum119894 119861119894 (V) (2)






m46

m16

m15

m26 m36

m34

k1

k2 k3

k6

k4k5



119901 (119874 (119894) | (119904 (119894) 119878119894)) =lt 119904 (119894) 119878119894 119868119894 gt (3)



119868119894 = [[119899sum119895=1

120585119895 ∙ 120574119894119895 (1 minus 120590 (119894) 120590 (119895))]]minus [[119899sum119895=1

120577119895 ∙ 120574119894119895 (1 + 120590 (119894) 120590 (119895))]] (4)




Decision group



Social viewpoint


Social relations

Learning

Assimilate

Judgement

Individual




social learning


affect

Voting power

YesNo



3 Proposed Model



119904 (119894) = 1 119894 isin 1198620119884minus1 119894 isin 1198620119873 (5)


119868119894 = 120572minussum119895


120582119895 times 120574119894119895times (1 + 119904 (119894) 119904 (119895))

(6)





119862119863 (119894) = 119889119894119899 minus 1 (7)






120574119894119895 = 119898119894119895max (119872) (9)



119901 (119874 (119894) = minus119904 (119894) | (119904 (119894) 119878119894)) = 11 + 119890minus2(119868119894119887)119901 (119874 (119894) = 119904 (119894) | (119904 (119894) 119878119894)) = 11 + 1198902(119868119894119887)

(10)




120587 (119862119884 | 1198620) = prod119894isin119862119884|119862

0

120583119894 (1198620) times prod119895notin119862119884|119862

0

(1 minus 120583119895 (1198620)) ) (11)


119877119894 (V) = 12119899minus1 sum1198620 119894isin1198620

119884

( sum119862119884|1198620isin119881

120587 (119862119884 | 1198620)

minus sum119862119884|(119862119884

0119894119862cup119894)isin119881

120587 (119862119884 | (1198621198840 119894 119862 cup 119894))) (12)


1198771015840119894 (V) = 119877119894 (V)sum119894isin119873 119877119894 (V) (13)
















9

6

9

3

85

2

8

2

51

9

4

7

6

v5

v1

v3

v2v4

v8

v6

v7

2

v9

5

3




02

01801601401201

008006004002

0

1 2 3 4 5 6 7 8 9

voters

pow

er








voter1voter3voter5

voter7voter9

minus01

minus005

0

005

01

015

02

D-p

ower



voter1voter2voter3

voter4voter5

minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

025

D-p

ower





minus006

minus004

minus002

0

002

004

006

008

01

012

014

D-p

ower








voter1voter3voter5

voter7voter9

minus01

minus005

0

005

01

015

02

025

D-p

ower















voter1voter2voter3

voter4voter5

minus04

minus02

0

02

04

06

08

1

D-p

ower


D-p

ower


minus02

minus01

0

01

02

03

04

05

06





D-p

ower


0002004006008

01

voter1voter3voter5

voter7voter9

83 6 92 51 4 7Closeness (m)






5 Conclusions



voter1voter2voter3

voter4voter5

minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

D-p

ower






minus008

minus006

minus004

minus002

0

002

004

006

008

D-p

ower



D-p

ower

voter1voter3voter5

voter7voter9

minus01

minus005

0

005










93 86 10 135 122 111 4 7Voter

0

05

1

15

2

25

Valu

e


D-p

ower

minus004

minus002

0

002

004

006








Data Availability




Acknowledgments


References























































Journal of












Journal of







Volume 2018






Volume 2018










1198611015840119894 (V) = 119861119894 (V)sum119894 119861119894 (V) (2)






m46

m16

m15

m26 m36

m34

k1

k2 k3

k6

k4k5



119901 (119874 (119894) | (119904 (119894) 119878119894)) =lt 119904 (119894) 119878119894 119868119894 gt (3)



119868119894 = [[119899sum119895=1

120585119895 ∙ 120574119894119895 (1 minus 120590 (119894) 120590 (119895))]]minus [[119899sum119895=1

120577119895 ∙ 120574119894119895 (1 + 120590 (119894) 120590 (119895))]] (4)




Decision group



Social viewpoint


Social relations

Learning

Assimilate

Judgement

Individual




social learning


affect

Voting power

YesNo



3 Proposed Model



119904 (119894) = 1 119894 isin 1198620119884minus1 119894 isin 1198620119873 (5)


119868119894 = 120572minussum119895


120582119895 times 120574119894119895times (1 + 119904 (119894) 119904 (119895))

(6)





119862119863 (119894) = 119889119894119899 minus 1 (7)






120574119894119895 = 119898119894119895max (119872) (9)



119901 (119874 (119894) = minus119904 (119894) | (119904 (119894) 119878119894)) = 11 + 119890minus2(119868119894119887)119901 (119874 (119894) = 119904 (119894) | (119904 (119894) 119878119894)) = 11 + 1198902(119868119894119887)

(10)




120587 (119862119884 | 1198620) = prod119894isin119862119884|119862

0

120583119894 (1198620) times prod119895notin119862119884|119862

0

(1 minus 120583119895 (1198620)) ) (11)


119877119894 (V) = 12119899minus1 sum1198620 119894isin1198620

119884

( sum119862119884|1198620isin119881

120587 (119862119884 | 1198620)

minus sum119862119884|(119862119884

0119894119862cup119894)isin119881

120587 (119862119884 | (1198621198840 119894 119862 cup 119894))) (12)


1198771015840119894 (V) = 119877119894 (V)sum119894isin119873 119877119894 (V) (13)
















9

6

9

3

85

2

8

2

51

9

4

7

6

v5

v1

v3

v2v4

v8

v6

v7

2

v9

5

3




02

01801601401201

008006004002

0

1 2 3 4 5 6 7 8 9

voters

pow

er








voter1voter3voter5

voter7voter9

minus01

minus005

0

005

01

015

02

D-p

ower



voter1voter2voter3

voter4voter5

minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

025

D-p

ower





minus006

minus004

minus002

0

002

004

006

008

01

012

014

D-p

ower








voter1voter3voter5

voter7voter9

minus01

minus005

0

005

01

015

02

025

D-p

ower















voter1voter2voter3

voter4voter5

minus04

minus02

0

02

04

06

08

1

D-p

ower


D-p

ower


minus02

minus01

0

01

02

03

04

05

06





D-p

ower


0002004006008

01

voter1voter3voter5

voter7voter9

83 6 92 51 4 7Closeness (m)






5 Conclusions



voter1voter2voter3

voter4voter5

minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

D-p

ower






minus008

minus006

minus004

minus002

0

002

004

006

008

D-p

ower



D-p

ower

voter1voter3voter5

voter7voter9

minus01

minus005

0

005










93 86 10 135 122 111 4 7Voter

0

05

1

15

2

25

Valu

e


D-p

ower

minus004

minus002

0

002

004

006








Data Availability




Acknowledgments


References























































Journal of












Journal of







Volume 2018






Volume 2018









Decision group



Social viewpoint


Social relations

Learning

Assimilate

Judgement

Individual




social learning


affect

Voting power

YesNo



3 Proposed Model



119904 (119894) = 1 119894 isin 1198620119884minus1 119894 isin 1198620119873 (5)


119868119894 = 120572minussum119895


120582119895 times 120574119894119895times (1 + 119904 (119894) 119904 (119895))

(6)





119862119863 (119894) = 119889119894119899 minus 1 (7)






120574119894119895 = 119898119894119895max (119872) (9)



119901 (119874 (119894) = minus119904 (119894) | (119904 (119894) 119878119894)) = 11 + 119890minus2(119868119894119887)119901 (119874 (119894) = 119904 (119894) | (119904 (119894) 119878119894)) = 11 + 1198902(119868119894119887)

(10)




120587 (119862119884 | 1198620) = prod119894isin119862119884|119862

0

120583119894 (1198620) times prod119895notin119862119884|119862

0

(1 minus 120583119895 (1198620)) ) (11)


119877119894 (V) = 12119899minus1 sum1198620 119894isin1198620

119884

( sum119862119884|1198620isin119881

120587 (119862119884 | 1198620)

minus sum119862119884|(119862119884

0119894119862cup119894)isin119881

120587 (119862119884 | (1198621198840 119894 119862 cup 119894))) (12)


1198771015840119894 (V) = 119877119894 (V)sum119894isin119873 119877119894 (V) (13)
















9

6

9

3

85

2

8

2

51

9

4

7

6

v5

v1

v3

v2v4

v8

v6

v7

2

v9

5

3




02

01801601401201

008006004002

0

1 2 3 4 5 6 7 8 9

voters

pow

er








voter1voter3voter5

voter7voter9

minus01

minus005

0

005

01

015

02

D-p

ower



voter1voter2voter3

voter4voter5

minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

025

D-p

ower





minus006

minus004

minus002

0

002

004

006

008

01

012

014

D-p

ower








voter1voter3voter5

voter7voter9

minus01

minus005

0

005

01

015

02

025

D-p

ower















voter1voter2voter3

voter4voter5

minus04

minus02

0

02

04

06

08

1

D-p

ower


D-p

ower


minus02

minus01

0

01

02

03

04

05

06





D-p

ower


0002004006008

01

voter1voter3voter5

voter7voter9

83 6 92 51 4 7Closeness (m)






5 Conclusions



voter1voter2voter3

voter4voter5

minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

D-p

ower






minus008

minus006

minus004

minus002

0

002

004

006

008

D-p

ower



D-p

ower

voter1voter3voter5

voter7voter9

minus01

minus005

0

005










93 86 10 135 122 111 4 7Voter

0

05

1

15

2

25

Valu

e


D-p

ower

minus004

minus002

0

002

004

006








Data Availability




Acknowledgments


References























































Journal of












Journal of







Volume 2018






Volume 2018










120574119894119895 = 119898119894119895max (119872) (9)



119901 (119874 (119894) = minus119904 (119894) | (119904 (119894) 119878119894)) = 11 + 119890minus2(119868119894119887)119901 (119874 (119894) = 119904 (119894) | (119904 (119894) 119878119894)) = 11 + 1198902(119868119894119887)

(10)




120587 (119862119884 | 1198620) = prod119894isin119862119884|119862

0

120583119894 (1198620) times prod119895notin119862119884|119862

0

(1 minus 120583119895 (1198620)) ) (11)


119877119894 (V) = 12119899minus1 sum1198620 119894isin1198620

119884

( sum119862119884|1198620isin119881

120587 (119862119884 | 1198620)

minus sum119862119884|(119862119884

0119894119862cup119894)isin119881

120587 (119862119884 | (1198621198840 119894 119862 cup 119894))) (12)


1198771015840119894 (V) = 119877119894 (V)sum119894isin119873 119877119894 (V) (13)
















9

6

9

3

85

2

8

2

51

9

4

7

6

v5

v1

v3

v2v4

v8

v6

v7

2

v9

5

3




02

01801601401201

008006004002

0

1 2 3 4 5 6 7 8 9

voters

pow

er








voter1voter3voter5

voter7voter9

minus01

minus005

0

005

01

015

02

D-p

ower



voter1voter2voter3

voter4voter5

minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

025

D-p

ower





minus006

minus004

minus002

0

002

004

006

008

01

012

014

D-p

ower








voter1voter3voter5

voter7voter9

minus01

minus005

0

005

01

015

02

025

D-p

ower















voter1voter2voter3

voter4voter5

minus04

minus02

0

02

04

06

08

1

D-p

ower


D-p

ower


minus02

minus01

0

01

02

03

04

05

06





D-p

ower


0002004006008

01

voter1voter3voter5

voter7voter9

83 6 92 51 4 7Closeness (m)






5 Conclusions



voter1voter2voter3

voter4voter5

minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

D-p

ower






minus008

minus006

minus004

minus002

0

002

004

006

008

D-p

ower



D-p

ower

voter1voter3voter5

voter7voter9

minus01

minus005

0

005










93 86 10 135 122 111 4 7Voter

0

05

1

15

2

25

Valu

e


D-p

ower

minus004

minus002

0

002

004

006








Data Availability




Acknowledgments


References























































Journal of












Journal of







Volume 2018






Volume 2018













9

6

9

3

85

2

8

2

51

9

4

7

6

v5

v1

v3

v2v4

v8

v6

v7

2

v9

5

3




02

01801601401201

008006004002

0

1 2 3 4 5 6 7 8 9

voters

pow

er








voter1voter3voter5

voter7voter9

minus01

minus005

0

005

01

015

02

D-p

ower



voter1voter2voter3

voter4voter5

minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

025

D-p

ower





minus006

minus004

minus002

0

002

004

006

008

01

012

014

D-p

ower








voter1voter3voter5

voter7voter9

minus01

minus005

0

005

01

015

02

025

D-p

ower















voter1voter2voter3

voter4voter5

minus04

minus02

0

02

04

06

08

1

D-p

ower


D-p

ower


minus02

minus01

0

01

02

03

04

05

06





D-p

ower


0002004006008

01

voter1voter3voter5

voter7voter9

83 6 92 51 4 7Closeness (m)






5 Conclusions



voter1voter2voter3

voter4voter5

minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

D-p

ower






minus008

minus006

minus004

minus002

0

002

004

006

008

D-p

ower



D-p

ower

voter1voter3voter5

voter7voter9

minus01

minus005

0

005










93 86 10 135 122 111 4 7Voter

0

05

1

15

2

25

Valu

e


D-p

ower

minus004

minus002

0

002

004

006








Data Availability




Acknowledgments


References























































Journal of












Journal of







Volume 2018






Volume 2018









voter1voter3voter5

voter7voter9

minus01

minus005

0

005

01

015

02

D-p

ower



voter1voter2voter3

voter4voter5

minus025

minus02

minus015

minus01

minus005

0

005

01

015

02

025

D-p

ower





minus006

minus004

minus002

0

002

004

006

008

01

012

014

D-p

ower








voter1voter3voter5

voter7voter9

minus01

minus005

0

005

01

015

02

025

D-p

ower















voter1voter2voter3

voter4voter5

minus04

minus02

0

02

04

06

08

1

D-p

ower


D-p

ower


minus02

minus01

0

01

02

03

04

05

06





D-p

ower


0002004006008

01

voter1voter3voter5

voter7voter9

83 6 92 51 4 7Closeness (m)






5 Conclusions



voter1voter2voter3

voter4voter5

minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

D-p

ower






minus008

minus006

minus004

minus002

0

002

004

006

008

D-p

ower



D-p

ower

voter1voter3voter5

voter7voter9

minus01

minus005

0

005










93 86 10 135 122 111 4 7Voter

0

05

1

15

2

25

Valu

e


D-p

ower

minus004

minus002

0

002

004

006








Data Availability




Acknowledgments


References























































Journal of












Journal of







Volume 2018






Volume 2018









voter1voter3voter5

voter7voter9

minus01

minus005

0

005

01

015

02

025

D-p

ower















voter1voter2voter3

voter4voter5

minus04

minus02

0

02

04

06

08

1

D-p

ower


D-p

ower


minus02

minus01

0

01

02

03

04

05

06





D-p

ower


0002004006008

01

voter1voter3voter5

voter7voter9

83 6 92 51 4 7Closeness (m)






5 Conclusions



voter1voter2voter3

voter4voter5

minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

D-p

ower






minus008

minus006

minus004

minus002

0

002

004

006

008

D-p

ower



D-p

ower

voter1voter3voter5

voter7voter9

minus01

minus005

0

005










93 86 10 135 122 111 4 7Voter

0

05

1

15

2

25

Valu

e


D-p

ower

minus004

minus002

0

002

004

006








Data Availability




Acknowledgments


References























































Journal of












Journal of







Volume 2018






Volume 2018










voter1voter2voter3

voter4voter5

minus04

minus02

0

02

04

06

08

1

D-p

ower


D-p

ower


minus02

minus01

0

01

02

03

04

05

06





D-p

ower


0002004006008

01

voter1voter3voter5

voter7voter9

83 6 92 51 4 7Closeness (m)






5 Conclusions



voter1voter2voter3

voter4voter5

minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

D-p

ower






minus008

minus006

minus004

minus002

0

002

004

006

008

D-p

ower



D-p

ower

voter1voter3voter5

voter7voter9

minus01

minus005

0

005










93 86 10 135 122 111 4 7Voter

0

05

1

15

2

25

Valu

e


D-p

ower

minus004

minus002

0

002

004

006








Data Availability




Acknowledgments


References























































Journal of












Journal of







Volume 2018






Volume 2018









voter1voter2voter3

voter4voter5

minus02

minus015

minus01

minus005

0

005

01

015

02

025

03

D-p

ower






minus008

minus006

minus004

minus002

0

002

004

006

008

D-p

ower



D-p

ower

voter1voter3voter5

voter7voter9

minus01

minus005

0

005










93 86 10 135 122 111 4 7Voter

0

05

1

15

2

25

Valu

e


D-p

ower

minus004

minus002

0

002

004

006








Data Availability




Acknowledgments


References























































Journal of












Journal of







Volume 2018






Volume 2018










93 86 10 135 122 111 4 7Voter

0

05

1

15

2

25

Valu

e


D-p

ower

minus004

minus002

0

002

004

006








Data Availability




Acknowledgments


References























































Journal of












Journal of







Volume 2018






Volume 2018


























































Journal of












Journal of







Volume 2018






Volume 2018


















Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

Power Indices in the Context of Social Learning Behaviour ...downloads.hindawi.com/journals/ddns/2019/4532042.pdf · ResearchArticle Power Indices in the Context of Social Learning