Decision Making using Game Theory: An introduction for managers

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41Games of chance involving riskyu(x) µ E(x)[u≤(x) = 0 ]XFigure 3.3The von Neumann–Morgenstern utility function and the expected value have a linear relationship.value of 1 and £50 has a utility value of 2, then £500 has a utility value of3.)The results of Example 3.3 can be represented graphically and thethree categories of risk generalised to deWnitions, as follows:∑ If the von Neumann–Morgenstern utility function, which representsa player’s preferences among expected values, and the expected valueitself have a linear relationship, the player is said to be risk-neutral(see Figure 3.3). In such a linear von Neumann–Morgenstern utilityfunction, the player is essentially ranking the values of the game inthe same order as the expected values. This, by deWnition, is what itmeans to be risk-neutral. (The values of the game in Example 3.3 are:£300 for the diploma option; and £144 for the certiWcate option,being the average of £180 and £108.)Notice that, for a linear function,∑u"(x) 0If the von Neumann–Morgenstern utility function is proportional toany root of the expected value, the player is said to be risk-averse (seeFigure 3.4). Generally, risk-averse functions are of the form:u(x i ) n E(x i )Notice that the derivative of a concave function such as this is clearlydecreasing, so:u"(x) 0
42Games of chanceyu xuxnE xXFigure 3.4The von Neumann–Morgenstern utility function is proportional to a root of the expected value.yu(x) µ[ u ≤ (x)nE(x)
Page 2 and 3: Decision Making Using Game TheoryAn
Page 5 and 6: CAMBRIDGE UNIVERSITY PRESSCambridge
Page 7 and 8: viContents4 Sequential decision mak
Page 9 and 10: MMMM
Page 11 and 12: xPreface∑∑∑To Wnd new solutio
Page 13 and 14: 2Introductionvying for business fro
Page 16 and 17: 5TerminologyTable 1.1 The union’s
Page 18 and 19: 7Classifying gamesGAME THEORYGames
Page 20 and 21: 9A brief history of game theoryIn 1
Page 22 and 23: 11A brief history of game theoryano
Page 24 and 25: 13A brief history of game theoryIt
Page 26 and 27: 15Layoutexplained. Games involving
Page 28 and 29: 2 Games of skillIt is not from the
Page 30 and 31: 19Linear programming, optimisation
Page 38 and 39: 27The Lagrange method of partial de
Page 44 and 45: 33An introduction to basic probabil
Page 46 and 47: 35An introduction to basic probabil
Page 48 and 49: 37Games of chance involving riskV(X
Page 50 and 51: 39Games of chance involving riskthe
Page 54 and 55: 43Games of chance involving riskExa
Page 56 and 57: 45Games of chance involving uncerta
Page 58 and 59: 47Games of chance involving uncerta
Page 60 and 61: 49Representing sequential decision
Page 62 and 63: 51Representing sequential decision
Page 64 and 65: 53Sequential decision making in sin
Page 78 and 79: 67Sequential decision making in two
Page 84 and 85: 73Cooperative two-person gamesapply
Page 86 and 87: 75Cooperative two-person games∑it
Page 88 and 89: 5 Two-person zero-sum games ofstrat
Page 90 and 91: 79Representing zero-sum gamesPlayer
Page 92 and 93: 81Games with saddle pointsFigure 5.
Page 94 and 95: 83Games with saddle pointsSurgeonSt
Page 96 and 97: 85Games with saddle pointsPlayer 1a
Page 98 and 99: 87Games with no saddle pointsPlayer
Page 100 and 101: 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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113Summary of features of mixed-mot
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115The Cournot, von Stackelberg and
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129Solving games without Nash equil
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7 Repeated gamesLife is an offensiv
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137Infinitely repeated gamesincenti
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141Finitely repeated gamescontinuou
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143Finitely repeated gamesBUPALarge
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145Finitely repeated gamesGHGLarge
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147Finitely repeated gamesNatureGHG
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8 Multi-person games, coalitions an
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151Mixed-motive multi-person gamesp
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153Partially cooperative multi-pers
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155Indices of power: measuring infl
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175Rationalityexperimental evidence
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177Indeterminacygot locked into a l
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179Inconsistencythe pot of £49 at
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181Conclusionpost-industrial econom
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183Appendix APlayer 1 wants to maxi
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185Appendix APlayer 1choosesrow 1Pl
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187Appendix Aon the straight line b
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189Appendix Amax p min q Σw ij p i
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191Appendix Bandp(B/A 2 )·p(A 2 )p
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193BibliographyBenoit, J.P. & Krish
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195BibliographyJenkinson, T. (Ed.)
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197BibliographyRobinson, M. (1975)
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Indexancestors and descendants 49ap
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201Indexmaximin principle, in chanc
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203Indexschool buses college cooper
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