Context dependent choice

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Context dependent choice

• Fuad Aleskerov
• State University Higher School of Economics
• and
• Institute of Control Sciences
• Russian Academy of Sciences

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The set of alternatives AA presentation A
binary relation preference Utility function u
A?Choice function C
s.t.Rational choice model

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xPy means that x is better than y When
holds? Answer Sröder (1890 - 1895), Cantor
(1985) P is a weak order Indifference relation
is transitive! ab, bc gt ac P a linear order
u
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• P weak order, i.e.
• Irreflexive
• Transitive
• Negatively transitive
• Indifference relation
• is transitive
• Examples a) Radio-carbon method
• Coffee and sugar
• Bicycle and pony
• Dibbets work

BR
P
B
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Strong intervality condition
Semitransitivity condition
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Theorem

• P is representable as (1) with econstgt0 iff is a
  semiorder (Luce (1956))
• P is representable as (1) with ee(x)0 iff P is
  an interval order (Wiener (1914))
• P is representable as (1) with ee(x) iff P is a
  biorder (Ducamp and Falmagne (1969))

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Context dependent binary relations (simplest
case)
xPy ? u(x) gt u(y) e(x,y) additive non-negative
e(x,y) d(x) d(y) gives interval order
multiplicative e e(x,y) d(x)d(y), d(x)
au(x)ß, a gt 0 gives semiorders if 0ltßlt1
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Context dependent binary relations
Example Renault Clio e(X1)0,5 max u(x) min
u(x) 1450
e(X2) 600
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Context dependent binary relations (formal
definition)

• Forms of threshold
• e e(x, y, X)
• e e(y, X)
• e e(x, X)
• e e(X)

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Example
Mercedeces
e(X3) 10,450
e(X4) 8,450
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Theorem
Every choice function is rationalizable as (4)
with e e(x, y, X). Such e is non-negative iff
Fixed Point condition holds.
Theorem
Choice function is rationalizable as (4) with e
e(X) iff WARP holds
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Over-lath choice
(5)
Theorem 7. (5) is equivalent to (4) with e
e(X)
Examples of L
xk being the menu alternative on X with respect
to the ordering derived from u()
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Context independent relations with context
dependent numerical representation
Characterization weak bi-orders
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Condorcet principle PC (pair-wise comparison)
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Other ConditionsHeredity H Chernoff (1954),
Sen (1974), Arrow (1959)
for allConcordance C
Chernoff (1954), Sen (1974), Arrow (1959)
Outcast O Chernoff (1954), Sen (1974)
Arrows Choice Axiom (ACA) Nash (1949),
Chernoff (1954), Arrow (1959)
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H
O
by partial order
Pareto optimal choice
ACA
by weak order
utility function
C
H-O-ACA
rationalizable by any acyclic relation
C
The case of single valued choice
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Choice of m superior alternatives on linear order
PFor m1 it is just the choice of undominated
alternative on PTheorem (Aizerman,Aleskerov
1995 )Choice of m superior alternatives
satisfies (and ) and
satisfies in two cases only for m1
and mA
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Joint- extremal choice (Aizerman, Malishevski
1981)
each
is a
choice function choosing undominated alternative
on linear order
Theorem
is rationalizable by joint- extremal choice
model iff C satisfies (or
path-independent c.f.)
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Example Joint- extremal choice
The very idea of rationality is violated!
dominates
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Hyper - relations
Unilateral hyper - relation
For
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Strong hyper acyclicity
there are no sets
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Domain H unilateral hyper-relation
Strongly dominant choice rule
Theorem
iff it is rationalizable by strongly
hyper-acyclic
unilateral
and strongly dominant rule.
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Weak hyper acyclicity there is no set
of elements s.t
and
where
means
for any
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Domain C unilateral
Weakly - dominant rule
Theorem
if C is rationalizable
by unilateral weak hyper-acyclic ? and weakly
dominant rule
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Domain O

The set of dominants of X wrt
is called correct if
Hyper dominant rule
Theorem
if it is rationalizable by
and hyper dominant rule
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Domain

is unilateral
is hyper-transitive if
and
s. t.
s. t.
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Theorem
iff
is rationazable by
and strong - dominant
hyper-transitive
rule
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Thank you