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Two new computer based results in game theory related to combinatorial games and Nash equilibria

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Title
Two new computer based results in game theory related to combinatorial games and Nash equilibria
Name (type = personal)
NamePart (type = family)
Oudalov
NamePart (type = given)
Vladimir
NamePart (type = date)
1954-
DisplayForm
Vladimir Oudalov
Role
RoleTerm (authority = RULIB)
author
Name (type = personal)
NamePart (type = family)
Prekopa
NamePart (type = given)
Andras
DisplayForm
Andras Prekopa
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
chair
Name (type = personal)
NamePart (type = family)
Gurvich
NamePart (type = given)
Vladimir
DisplayForm
Vladimir Gurvich
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
co-chair
Name (type = personal)
NamePart (type = family)
Boros
NamePart (type = given)
Endre
DisplayForm
Endre Boros
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
internal member
Name (type = personal)
NamePart (type = family)
Beck
NamePart (type = given)
József
DisplayForm
József Beck
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
outside member
Name (type = personal)
NamePart (type = family)
Elbassioni
NamePart (type = given)
Khaled
DisplayForm
Khaled Elbassioni
Affiliation
Advisory Committee
Role
RoleTerm (authority = RULIB)
outside member
Name (type = corporate)
NamePart
Rutgers University
Role
RoleTerm (authority = RULIB)
degree grantor
Name (type = corporate)
NamePart
Graduate School - New Brunswick
Role
RoleTerm (authority = RULIB)
school
TypeOfResource
Text
Genre (authority = marcgt)
theses
Genre (authority = ExL-Esploro)
ETD doctoral
OriginInfo
DateCreated (qualifier = exact)
2013
DateOther (qualifier = exact); (type = degree)
2013-01
Place
PlaceTerm (type = code)
xx
Language
LanguageTerm (authority = ISO639-2b); (type = code)
eng
Subject (authority = RUETD)
Topic
Operations Research
Subject (authority = ETD-LCSH)
Topic
Game theory--Computer programs
Subject (authority = ETD-LCSH)
Topic
Combinatorial analysis
RelatedItem (type = host)
TitleInfo
Title
Rutgers University Electronic Theses and Dissertations
Identifier (type = RULIB)
ETD
RelatedItem (type = host)
TitleInfo
Title
Graduate School - New Brunswick Electronic Theses and Dissertations
Identifier (type = local)
rucore19991600001
Identifier
ETD_4395
Identifier (type = hdl)
http://hdl.rutgers.edu/1782.1/rucore10001600001.ETD.000067806
Identifier (type = doi)
doi:10.7282/T33R0RKB
PhysicalDescription
Form (authority = gmd)
electronic resource
InternetMediaType
application/pdf
InternetMediaType
text/xml
Extent
vi, 58 p. : ill.
Note (type = degree)
Ph.D.
Note (type = bibliography)
Includes bibliographical references
Note (type = vita)
Includes vita
Note (type = statement of responsibility)
by Vladimir Oudalov
Location
PhysicalLocation (authority = marcorg); (displayLabel = Rutgers, The State University of New Jersey)
NjNbRU
Abstract (type = abstract)
This thesis consists of two chapters.The first chapter is about the new version of NIM recently introduced by Gurvichtogether with a generalization of the minimal excludant function (mex). This gameNIM(a,b) is played with two piles of matches. Two players alternate turns. By onemove, each player can take either “almost the same” number of matches from each pile(the difference of the two numbers is strictly less thana) or any number of matches fromone pile and strictly less thanbfrom the other. This game further extends Fraenkel’sNIM = NIM(a,1), which, in its turn, is a generalization of the classic Wythoff NIM =NIM(1,1).Gurvich introduced a generalization mexbof the standard minimum excludant mex =mex1defining mexb(S⊂Z+) = min{n:∀s∈S s≤n⇒s+b≤n}. He also showedthat P-positions (the kernel) of NIM(a,b) are given by the following recursion:xn= mexb({xi,yi|0≤i < n}), yn=xn+an;n≥0,and conjectured that for alla,bthe limits`(a,b) =xn(a,b)/nexist and are irrationalalgebraic numbers. Here we prove it showing that`(a,b) =ar−1, wherer >1 is the Perron root of the polynomialP(z) =zb+1−z−1−a−1∑i=1zdib/ae,wheneveraandbare coprime; furthermore, it is known that`(ka,kb) =k`(a,b).In particular,`(a,1) =αa=12(2−a+√a2+ 4). In 1982, Fraenkel introducedthe game NIM(a) = NIM(a,1), obtained the above recursion and solved it explicitlygettingxn=bαanc, yn=xn+an=b(αa+a)nc. Here we provide a polynomial timealgorithm based on the Perron-Frobenius theory solving game NIM(a,b), although wehave no explicit formula for its kernel.The second chapter of the thesis is about the existence of Nash equilibria (NE) inpure stationary strategies inn-person positional games with no moves of chance, withperfect information, and with the mean or total effective cost function.We construct a NE-free three-person game with positive local costs, disproving theconjecture suggested by Boros and Gurvich in Math. Soc. Sci. 46 (2003) 207-241.Still, the following four problems remain open:Whether NE exist in alltwo-person games with total effective costs such that (I) alllocal costs are strictly positive or (II) without directed cycles of the cost zero?If NE exist in alln-person games with the terminal (transition-free) cost functions,provided all directed cycles form a unique outcomecand (III) assuming thatcis worsethan any terminal outcome or (IV) without this assumption?Forn= 3 cases (I) and (II) are answered in the negative, while forn= 2 cases(III) and (IV) are proven. We briefly survey other negative and positive results onNash-solvability in pure stationary strategies for the games under consideration.
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The author owns the copyright to this work.
RightsHolder (type = personal)
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Oudalov
GivenName
Vladimir
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DateTime (encoding = w3cdtf); (qualifier = exact); (point = start)
2012-11-25 17:10:17
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Vladimir Oudalov
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Rutgers University. Graduate School - New Brunswick
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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windows xp
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