Decision Making using Game Theory: An introduction for managers

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159Indices of power: measuring influence∑ two Education and Library Board representatives (ELB)∑ two parents∑ one teacherModel C: Out-of-state schools in the Republic of Ireland also educate pupilsfrom Northern Ireland. Typically, they have boards of governors consisting of:∑ three Education and Library Board representatives (ELB)∑ three parents∑ two majority religious representatives∑ one minority religious representative∑ two teachersWhat real power does each faction have on each of the three types ofgoverning body?Irrespective of religious aYliation or constitution, schools everywherehave become more complex organisations as a result of the increasedparticipation of stakeholders. This has manifested itself in a proliferationof committee-like structures, such as boards of governors, whichmediate between society and the organisation. Representation on thesecommittees usually reXects an imputational entitlement to power andmany assumptions about the relative voting strengths of factions onsuch committees are frequently made. Take the case of the Model Cschool described above. At Wrst sight, it appears that the voting powerof the majority religious body is twice that of the minority religiousbody. Similarly, the ELB and parent factions are assumed to be 50%more ‘powerful’ than the teaching or majority religious bodies. Thesecasual assumptions are dangerous on two counts. Firstly, they aresimply wrong; and secondly, they support an illusion of participativedemocracy and empowerment that distracts from the need to bringabout more fundamental change. The advantage of a game theoreticapproach is that it can model majority voting situations, such as existon school governing boards, so that these inherent fallacies becomeapparent.An analysis of power on ‘voluntary maintained’ boards (Model A)Let the Wve factions on the board of governors be denoted as follows:
160Multi-person games, coalitions and powerChurch nominees (four votes): CEducation and Library Board (two votes): LParent body (one vote): PTeaching staV (one vote): TDepartment of Education (one vote): DFor any one particular order of C, L, P, T and D, there are:5 one-faction, single coalitions20 two-faction coalitions60 three-faction coalitions120 four-faction coalitions120 Wve-faction, grand coalitionsA little consideration reveals the following:∑ None of the single coalitions is a winning one.∑ Eight two-faction coalitions are winning ones and they all include C.∑ Some 36 three-faction coalitions are winning ones and they allinclude C.∑ All four-faction and grand coalitions are winning ones.In summary, there are 284 winning coalitions out of a possible 325,though the three-faction, four-faction and grand coalitions requirefurther investigation.Three-faction coalitionsThere are 36 winning three-faction coalitions. If C is Wrst, the secondvoter will be pivotal; if C is second or third (12 occasions each) then Citself will be pivotal. Therefore, C is pivotal 24 times and L, P, T and Dare pivotal on three occasions each.Four-faction coalitionsThere are 120 four-faction coalitions and they are all winning. If C isWrst in the voting, then the second faction to vote will be pivotal (sixtimes each for L, P, T and D). If C is second, third or fourth in thevoting, then C itself will be pivotal. In the case of the 24 coalitions thatdo not include C, the last faction will always be the pivotal one.Therefore, C will be pivotal 72 times and L, P, T and D will be pivotalon 12 occasions each.Grand coalitionsIn grand coalitions, the last faction voting will never be pivotal, even if
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Decision Making Using Game TheoryAn
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CAMBRIDGE UNIVERSITY PRESSCambridge
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viContents4 Sequential decision mak
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MMMM
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xPreface∑∑∑To Wnd new solutio
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2Introductionvying for business fro
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5TerminologyTable 1.1 The union’s
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7Classifying gamesGAME THEORYGames
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9A brief history of game theoryIn 1
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11A brief history of game theoryano
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13A brief history of game theoryIt
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15Layoutexplained. Games involving
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2 Games of skillIt is not from the
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19Linear programming, optimisation
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27The Lagrange method of partial de
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33An introduction to basic probabil
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35An introduction to basic probabil
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37Games of chance involving riskV(X
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39Games of chance involving riskthe
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41Games of chance involving riskyu(
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43Games of chance involving riskExa
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45Games of chance involving uncerta
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47Games of chance involving uncerta
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49Representing sequential decision
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51Representing sequential decision
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53Sequential decision making in sin
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67Sequential decision making in two
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73Cooperative two-person gamesapply
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75Cooperative two-person games∑it
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5 Two-person zero-sum games ofstrat
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79Representing zero-sum gamesPlayer
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81Games with saddle pointsFigure 5.
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83Games with saddle pointsSurgeonSt
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85Games with saddle pointsPlayer 1a
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87Games with no saddle pointsPlayer
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89Games with no saddle pointsStrate
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91Large matrices generallybigger th
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93Interval and ordinal scales for p
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99Representing mixed-motive games a
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101Representing mixed-motive games
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103Mixed-motive games without singl
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Page 120 and 121: 109Mixed-motive games without singl
Page 122 and 123: 111Mixed-motive games without singl
Page 124 and 125: 113Summary of features of mixed-mot
Page 126 and 127: 115The Cournot, von Stackelberg and
Page 140 and 141: 129Solving games without Nash equil
Page 146 and 147: 7 Repeated gamesLife is an offensiv
Page 148 and 149: 137Infinitely repeated gamesincenti
Page 152 and 153: 141Finitely repeated gamescontinuou
Page 154 and 155: 143Finitely repeated gamesBUPALarge
Page 156 and 157: 145Finitely repeated gamesGHGLarge
Page 158 and 159: 147Finitely repeated gamesNatureGHG
Page 160 and 161: 8 Multi-person games, coalitions an
Page 162 and 163: 151Mixed-motive multi-person gamesp
Page 164 and 165: 153Partially cooperative multi-pers
Page 166 and 167: 155Indices of power: measuring infl
Page 186 and 187: 175Rationalityexperimental evidence
Page 188 and 189: 177Indeterminacygot locked into a l
Page 190 and 191: 179Inconsistencythe pot of £49 at
Page 192 and 193: 181Conclusionpost-industrial econom
Page 194 and 195: 183Appendix APlayer 1 wants to maxi
Page 196 and 197: 185Appendix APlayer 1choosesrow 1Pl
Page 198 and 199: 187Appendix Aon the straight line b
Page 200 and 201: 189Appendix Amax p min q Σw ij p i
Page 202 and 203: 191Appendix Bandp(B/A 2 )·p(A 2 )p
Page 204 and 205: 193BibliographyBenoit, J.P. & Krish
Page 206 and 207: 195BibliographyJenkinson, T. (Ed.)
Page 208 and 209: 197BibliographyRobinson, M. (1975)
Page 210 and 211: Indexancestors and descendants 49ap
Page 212 and 213: 201Indexmaximin principle, in chanc
Page 214 and 215: 203Indexschool buses college cooper
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