Table Of ContentNewHandbookofMathematicalPsychology
The field of mathematical psychology began in the 1950s and includes
bothpsychologicaltheorizinginwhichmathematicsplaysakeyrole,and
applied mathematics motivated by substantive problems in psychology.
CentraltoitssuccesswasthepublicationofthefirstHandbookofMath-
ematicalPsychologyinthe1960s.Thepsychologicalscienceshavesince
expandedtoincludenewareasofresearch,andsignificantadvanceshave
been madein both traditional psychological domains and in the applica-
tionsofthecomputationalsciencestopsychology.Upholdingtherigorof
the original Handbook, the New Handbook of Mathematical Psychology
reflects the current state of the field by exploring the mathematical and
computational foundations of new developments over the last half cen-
tury.Thefirstvolumefocusesonselectmathematicalideas,theories,and
modeling approaches to form a foundational treatment of mathematical
psychology.
william h. batchelder is Professor of Cognitive Sciences at the
UniversityofCaliforniaIrvine.
hans colonius is Professor of Psychology at Oldenburg University,
Germany.
ehtibar n. dzhafarov is Professor of Psychological Sciences at
PurdueUniversity.
jay myung isProfessorofPsychologyatOhioStateUniversity.
New Handbook of
Mathematical Psychology
Volume 1. Foundations and Methodology
Editedby
William H. Batchelder
Hans Colonius
Ehtibar N. Dzhafarov
Jay Myung
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Contents
Listofcontributors pagevii
Preface
william h. batchelder, hans colonius,
ehtibar n. dzhafarov, and jay myung ix
1 Selectedconceptsfromprobability
hans colonius 1
2 Probability,randomvariables,andselectivity
ehtibar n. dzhafarov and janne kujala 85
3 Functionalequations
che tat ng 151
4 Networkanalysis
john p. boyd and william h. batchelder 194
5 Knowledgespacesandlearningspaces
jean-paul doignon and jean-claude falmagne 274
6 Evolutionarygametheory
j. mckenzie alexander 322
7 Choice,preference,andutility:probabilisticanddeterministic
representations
a. a. j. marley and michel regenwetter 374
8 Discretestatemodelsofcognition
william h. batchelder 454
9 Bayesianhierarchicalmodelsofcognition
jeffrey n. rouder, richard d. morey,
and michael s. pratte 504
10 Modelevaluationandselection
jay myung, daniel r. cavagnaro,
and mark a. pitt 552
Index 599
v
Contributors
j. mckenzie alexander, LondonSchoolofEconomics(UK)
william h. batchelder, UniversityofCaliforniaatIrvine(USA)
john p. boyd, Institute for Mathematical Behavioral Sciences, University of
CaliforniaatIrvine(USA)
daniel r. cavagnaro, MihayloCollegeofBusinessandEconomics,Califor-
niaStateUniversityatFullerton(USA)
hans colonius, OldenburgUniversity(Germany)
jean-paul doignon, Département de Mathématique, Université Libre de
Bruxelles(Belgium)
ehtibar n. dzhafarov, PurdueUniversity(USA)
jean-claude falmagne, Department of Cognitive Sciences, University of
CaliforniaatIrvine(USA)
janne v. kujala, UniversityofJyväskylä(Finland)
anthony a. j. marley, Department of Psychology, University of Victoria
(Canada)
richard d. morey, UniversityofGroningen(TheNetherlands)
jay myung, OhioStateUniversity(USA)
che tat ng, DepartmentofPureMathematics,UniversityofWaterloo(Canada)
mark a. pitt, DepartmentofPsychology,OhioStateUniversity(USA)
michael s. pratte, DepartmentofPsychology,VanderbiltUniversity(USA)
michel regenwetter, Department of Psychology, University of Illinois at
Urbana-Champaign(USA)
jeffrey n. rouder, Department of Psychological Sciences, University of
Missouri(USA)
vii
Preface
Aboutmathematicalpsychology
Therearethreefuzzyandinterrelatedunderstandingsofwhatmathemat-
icalpsychologyis:partofmathematics,partofpsychology,andanalyticmethod-
ology. We call them “fuzzy” because we do not offer a rigorous way of defining
them.Asarule,aworkinmathematicalpsychology,includingthechaptersofthis
handbook, can always be argued to conform to more than one if not all three of
these understandings (hence our calling them “interrelated”). Therefore, it seems
safer to think of them as three constituents of mathematical psychology that may
bedifferentlyexpressedinanygivenlineofwork.
1.Partofmathematics
Mathematicalpsychologycanbeunderstoodasacollectionofmathematicaldevel-
opmentsinspiredandmotivatedbyproblemsinpsychology(oratleastthosetra-
ditionallyrelatedtopsychology).Agoodexampleforthisisthealgebraictheory
ofsemiordersproposedbyR.DuncanLuce(1956).Inalgebraandunidimensional
topology there are many structures that can be called orders. The simplest one is
thetotal,orlinearorder (S,(cid:2)),characterizedbythefollowingproperties:forany
a,b,c∈S,
(O1) a(cid:2)borb(cid:2)a;
(O2) ifa(cid:2)bandb(cid:2)cthena(cid:2)c;
(O3) ifa(cid:2)bandb(cid:2)athena=b.
The ordering relation here has the intuitive meaning of “not greater than.” One
can,ofcourse,thinkofmanyotherkindsoforder.Forinstance,ifwereplacethe
property(O1)with
(O4) a(cid:2)a,
we obtain a weaker (less restrictive) structure, called a partial order. If we add
to the properties (O1–O3) the requirement that every nonempty subset X of S
possessesanelementa suchthata (cid:2)aforanya∈X,thenweobtainastronger
X X
(more restrictive) structure called a well-order. Clearly, one needs motivation for
ix
