Table Of ContentDISCRETE MATHEMATICS AND GAMETHEORY
THEORY AND DECISION LIBRARY
General Editors: W. Leinfellner (Vienna) and G. Eberlein (Munich)
SeriesA: PhilosophyandMethodologyoftheSocialSciences
SeriesB: MathematicalandStatisticalMethods
SeriesC: GameTheory.MathematicalProgrammingandOperationsResearch
SeriesD: SystemTheory,KnowledgeEngineeringan ProblemSolving
SERIES C: GAMETHEORY, MATHEMATICALPROGRAMMING
ANDOPERATIONSRESEARCH
VOLUME22
Editor: S. H. Tijs (University ofTilburg); EditorialBoard: E.E.C. van Damme (Tilburg), H. Keiding
(Copenhagen), J.-F. Mertens (Louvain-la-Neuve), H. Moulin (Durham), S. Muto (Tokyo University),
T. Parthasarathy (New Delhi), B. Peleg (Jerusalem), H. Peters (Maastricht), T. E. S. Raghavan
(Chicago), J. Rosenmiiller (Bielefeld), A. Roth (Pittsburgh), D. Schmeidler (Tel-Aviv), R. Selten
(Bonn),W.Thomson(Rochester, NY).
Scope: Particularattention is paid in this series to game theory and operations research, their formal
aspectsand theirapplications to economic, political and social sciencesas wellas tosocio-biology. It
will encourage high standards in the application ofgame-theoretical methods to individualand social
decisionmaking.
The titlespublishedinthisseriesarelistedattheendofthisvolume.
DISCRETE MATHEMATICS
AND GAME THEORY
by
Guillermo Owen
Naval Postgraduate School,
Monterey, California, U.S.A.
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4613-7266-0 ISBN 978-1-4615-4991-8 (eBook)
DOI 10.1007/978-1-4615-4991-8
Printed on acid-free paper
AlI Rights Reserved
© 1999 Springer Science+Business Media Dordrecht
OriginaIly published by Kluwer Academic Publishers in 1999
Softcover reprint ofthe hardcover Ist edition 1999
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permis sion from the copyright owner.
TableofContents
ChapterI. VectorsandMatrices
1. AlgebraicOperations 1
2.RowOperationsandtheSolutionofSystemsofLinearEquations 7
3. SolutionofGeneral mxnSystemsofEquations 15
ChapterII.LinearProgramming 27
1. LinearPrograms 27
2. TheSimplexAlgorithm: SlackVariables 37
3. TheSimplexTableau 41
4. TheSimplexAlgorithm: Objectives 47
5. TheSimplexAlgorithm: ChoiceofPivots 48
6. TheSimplexAlgorithm: StageI. 55
7. TheSimplexAlgorithm: ProofofConvergence 58
8. EquationConstraints 66
9. DegeneracyProcedures 70
10. SomePracticalComments 74
11.Duality 77
12. TransportationProblems 92
13. AssignmentProblems 102
ChapterIII.TheTheoryofProbability 113
1. Probabilities 113
2. DiscreteProbabilitySpaces 115
3. ConditionalProbability. 123
4. CompoundExperiments 126
5. Bayes'Formula 131
6. RepetitionofSimpleExperiments;TheBinomialDistribution 138
7.DrawingswithandwithoutReplacement 141
8. RandomVariables 145
9. ExpectedValues. MeansandVariances 151
10.Rulesfor ComputingtheMeanandVariance 155
11.TwoImportantTheorems 158
12. MarkovChains 167
13. RegularandAbsorbingMarkovChains 172
ChapterIV. TheTheoryofGames 181
1. Games: ExtensiveandNormal Form 181
2. SaddlePoints 185
3. MixedStrategies 192
4. Solutionof2x2Games 199
5. 2xnandmx2Games 203
6. SolutionsbyLinearProgramming 209
7. SolutionofGamesbyFictitiousPlay 218
8. ThevonNeumannModelofanExpandingEconomy 222
9. ExistenceofanEquilibriumExpansionRate 228
10. Two-PersonNon-Zero-SumGames 236
11.EvolutionaryStableSystems 245
ChapterV. CooperativeGames 253
1. n-PersonGames 253
2. TheCore 256
3. TheShapleyValue 260
4. VotingStructures 266
ChapterVI.DynamicProgramming 271
1. ThePrincipleofMaximality 271
2. TheFixed-ChargeTransportationProblem 279
3. Inventories 287
4. StochasticInventorySystems 294
ChapterVIl. GraphsandNetworks 303
1.Introduction 303
2. CriticalPathAnalysis 304
3. TheShortestPaththroughaNetwork 314
4. Minimal SpanningTrees 319
5. TheMaximalFlowinaNetwork 324
vi
I. VECTORS AND MATRICES
1. ALGEBRAIC OPERATIONS
We assume thatthe reader is already familiar with the ordinary algebraofvectors
andmatrices. We givethe basics below.
Amatrix is arectangulararrayofnumbers, arrangedin rows and columns. Ifithas
mrows and ncolumns, we saythat it is an mxn matrix. Forexample,
4 1 2 0
A= 2 -4 5 I
3 0 -1 2
is a3x4 matrix, while
6 3
B= 7 4
8 -3
-2 5
is a4x2 matrix.
Thenumbers thatmake up the matrix are calledentries. We willfrequently use
letters with two subscriptsto representthese entries. Thus, would be the entry in
aij
the ithrow andjthcolumn ofmatrix A, b would be the entry inthe 3'd row and pI
31
columnofB, etc.
Matrices are in some sense ageneralizationofthe conceptofnumber. Thus itis
not surprisingthatcertainoperations canbe carriedoutonmatrices, similarto the
arithmetic operations. These include addition andmultiplicationofmatrices, and
scalarmultiplication. There is also the operationoftransposition.
Transposition. IfAis an mxn matrix, then the transpose ofAis an nxm matrix, B,
defined by
We will use AI to denote the transpose ofA. Forexample,
l I D
1.1.1 7 1 -4
If A= 1 3 then 3 2
-4 2
Addition ofmatrices. IfAand Bare two mxn matrices (notethe two matriceshave
the same mandthe same n) then theirsum, A+B, is also an mxn matrix, C, defmed
by
= b
Cij aij+ ij.
Thus,
3 4 -1 + 2 -6 3 5 -2 2
1 -3 0 I 5 2 2 6 -1 2
Scalar multiplication. IfAis an mxn matrix, ands is anumber. Thenthe product,
sA, is also an mxn matrix, D, definedby
1.1.2 dij =sajj
Thus
4 3 4 -1 12 16 -4
I
1 -3 0 I 4 -12 0
Note thatinthis context, the numbers (whichmultiplies the matrix) is frequently
calledascalar.
Multiplicationofmatrices. LetAbe an mxn matrix, and letBbe an nxpmatrix.
Thenthe productAB is an mxpmatrix, Q, defined by
1.1.3
2
Thus,
T I
l 1 5 6+4 30+8 10 38
2 4 4 8 2+16 10+32 18 42
-1 3 -1+12 -5+24 11 19
Note that AB is notnecessarilyequalto BA. Infact, itmay be thatone ofthese
products exists while the otherone does not; evenifthey bothexist, moreover, they
neednot be equal.
I.l.l. Rules for operations on matrices
Thefollowing hold for the operations:
1. IfAand Bare bothmxn matrices, then
2. Forany A,
3. IfAandBare both mxn matrices, then
A+B=B+A
4. IfA, B, andCare three mxn matrices, then
(A+B)+ C= A+(B +C)
5. IfAandBare mxn matrices, ands is ascalar, then
s(A+B) =sA+sB
6. IfAis amatrix and rands are scalars, then
(r+s)A = rA +sA.
3
and
(rs)A =r(sA)
7. Forany givenmand n, there exists an mxn matrix, 0, suchthat, ifAis also mxn,
then A+O = A.
8. IfAis mxn, Bisnxp, and Cispxq, then
(AB)C = A(BC)
9. IfAis mxn, Bisnxp, ands is ascalar, then
(sA)B =A(sB) =s(AB)
10. IfAand Bare both mxn n, and Cisnxp, then
(A+B) C= AC +BC
11. IfAis mxn, and BandCare bothnxp, then
A(B +C)= AB +AC
12. Foranyn, there exists andnxn matrix, I, suchthat, ifAis any mxn matrix, then
AI=A
and, ifBis any nxpmatrix, then
ID=B.
13. IfAis mxn, and Bisnxp, then
(AB)' = B'A'
Thematrix 0, referred to in (7) above, is simply an mxn matrix, all ofwhose entries
are o. Itis known as the additive identity, orzero matrix. Note thatfor any A, we
candefine the matrix -Aas the product(-I)A. Itis easyto checkthat
4
