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Justifiable Choice
Yuval HellerTel-Aviv University
(Part of my Ph.D. thesis supervised by Eilon Solan)
http://www.tau.ac.il/~helleryu/
Bonn Summer School
July 2009
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Contents
Introduction Choice with incomplete preferences and justifications
Violating WARP and binariness
Convex axiom of revealed non-inferiority (CARNI)
Applications of the new axiom: Taste-justifications
Belief-justifications
Related literature & concluding remarks
Incomplete Preferences
Most existing models of rational choice assume
complete psychological preferences
Rationality does not imply completeness: DMs may be indecisive when comparing 2 alternatives
Complicated alternatives
Multiple objectives (multi-criteria decision making)
Group decision making (social choice)
Aumann (62), Bewely (86), Dubra et al. (04), Mandler (05)3
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Choice Correspondence - C
C specifies the choosable alternatives: C(A) A.
A - a closed and non-empty set of alternatives
Interpretation: When facing A, DM always chooses an act in C(A)
All acts in C(A) are sometimes chosen
The unique choice in C(A) is not modeled explicitly Interpretations: justifications, subjective randomization
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Weak Axiom of Revealed Preferences (WARP)
WARP is often violated when preferences are incomplete
Example: x, y are incomparable acts, x’ is a little bit better than x
xy BA xC(B)xC(A)
yC(B)
x, y AB
yC(A)
x’xyAC(A)={x,y}
C(B)={x’,y}
xC(A)
yC(B)xC(B)\
B=AU{x’}
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Insights from the Psychological literature
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Behavior depends on payoff-irrelevant information
DM has several ways to evaluate acts, each with a
different justification (rationale) Observable information determines which justification to use
The chosen act: the best according to this justification
Examples: Availability heuristics, Anchoring (Tversky &
Kahneman, 74), Framing effect (Tversky & Kahneman, 81),
Reason-based choice (Shafir, Simonson & Tversky, 93)
Taste Justifications
Influence tastes over consequences
Example (regret justification, Zeelenberg et al., 96) Choice between safe & risky lotteries of equal attractiveness
(when feedback is only on the chosen lottery)
Having feedback on the risky lottery caused people to
choose it more often
Similar phenomena in real-life: Dutch postcode lottery
Your lottery number = Your postcode / address
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Belief Justifications
Influence beliefs over state of nature
Example (mood justification, Wright and Bower, 92): Happy/sad moods were induced (by focusing on happy/sad
personal experiments)
Induced mood influenced evaluation of ambiguous events
Happy people are optimistic: higher probability for positive
ambiguous events
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Weak Axiom of Revealed Non-Inferiority (Eliaz & Ok, 05)
WARNI: alternative is chosen if it is not revealed
inferior to any chosen alternative (WARP WARNI)
x is revealed inferior to y if x is not chosen in any set that
includes y
WARNI binariness: Choice is binary if it maximizes a binary relation (x is
chosen in A iff it is chosen in any couple in A)
Justifications often induce non-binary choice9
Example for Violating Binariness (Taste Justifications)
Alice chooses a restaurant for lunch x - serves meat, y - serves chicken
z – randomly serves either meat, chicken or fish
Incomplete preferences: Indecisive between meat &
chicken (uses justifications), fish is a little bit worse
Plausible choice: zC(x,z), zC(y,z), zC(x,y,z)
Remark: z is dominated by alternatives in the convex
hull of x & y (mixtures )10
Convex axiom of revealed non-inferiority (CARNI)
CARNI: x is chosen in A if it is not inferior to any
alternative in the convex hull of C(A) x is revealed inferior to y, if: yconv(A) xC(A)
WARP + independence CARNI
Why comparing to conv(A) (= not choosing z): Choice between x & y according to a toss of a coin
Multiple choices of z are strictly worse then multiple
choices between y & x
No justification (linear ordering) supports z 11
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Applying CARNI in Different Models of Choice
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Model 1 - Taste Justifications
Von-Neumann-Morgenstern’s framework: X - Finite set of outcomes
Alternatives: lotteries over X (A(X))
3 Axioms imposed on choice: Continuity ( for all g :{ f | fC({f,g}) is closed ,
{ f | {f}=C({f,g}) is open )
Independence ( fC(A) g+(1-)f C(g+(1-)A) )
CARNI (instead of WARP in vN-M’s model)
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Theorem 1 - Taste Justifications(multiple utilities)
C satisfies continuity, independence & CARNI
C has a multi-utility representation: A unique (up to
positive-linear transformations) closed and convex set U
of vN-M utility function, such that a lottery is chosen
iff it is best w.r.t. to some utility in U
Interpretation: Justification triggers the DM to think
primarily about a particular “anchoring” utility in U
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Relation with Eliaz-Ok (04)
Eliaz & Ok assume WARNI instead of CARNI
Their representation: a lottery x is chosen iff for
each y in the set there is a utility uy in U such that x
is better than y w.r.t. to uy
Allows the choice of unjustified alternative, which
is not best w.r.t. to any of the utilities
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Model 2 – Belief Justifications
Anscombe-Aumann’s framework (1963) : S – finite set of states of nature, X - finite set of outcomes Alternatives (acts): functions that assign lottery for each state Notation: f(s) – the constant function that assigns in all states
the lottery that f assigns in s
3 new axioms: Non-triviality: there is A, s.t. C(A)A Monotonicity: For all sS, f(s)C(A(s)) fC(A) WARP over unambiguous (constant) alternatives
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Anscombe-Aumman’s Framework
A (X)S
Set of acts (alternatives)
f1
f2
f3
states of natureS (finite set)
DM
Lotteries over X
X
finite set of outcomes
0.7 + 0.3
0.4 + 0.6
0.5 + 0.5
0.5 + 0.5
0.1 + 0.9
0.8 + 0.2
Theorem 2 - Taste Justifications(multiple priors)
C satisfies continuity, independence, CARNI,
non-triviality, monotonicity and unambiguous WARP
C has a multi-prior representation: A unique
closed and convex set P of priors and a unique vN-M
utility u, such that an alternative is chosen iff it is best
w.r.t. to some prior in P
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Relation with Bewely (02) & Lehrer-Teper (09)
They axiomatize preference relation Implict assumption: choice is binary
Our axioms = their axioms + CARNI
Their representation: an act f is chosen iff for each g in the
set there is a prior pg in P such that f is better than g w.r.t.
to pg
Allows the choice of unjustified alternative, which is not
best w.r.t. to any of the utilities
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Models for Both kinds of Justifications (Tastes & Beliefs)
Ok, Ortoleva & Riella (08) present a few axiomatic
models for prefernces that generelize multiple utilities
and mutliple priors A model that is either multy-utility or multi-prior
3 models of different kinds of state-dependant utilities
One can add CARNI to all of these axiomatic models
get the analog justification representations
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Concluding Remarks & Related Literature
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Model’s Primitive & Binariness
Multiple-priors may be interpreted as models for
choice when ambiguity’s evaluation has different
justifications
Most existing models combine the different
justifications into binary preferences ()
We demonstrate why justifications should be combined
into a non-binary choice correspondence (C)
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Global Binariness
Our models have a “global-binariness” property: Preferences (=binary choices) over the couples in A do
not reveal the choice in A
The preferences over all the couples in the grand set (or
at-least in conv(A)) reveal the choice in A
A few examples for non-binary choice models: Social choice - Batra & Pattanaik (72), Deb (83)
Preferences of elements over sets - Nehring (97)
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Status-Quo Justification
Violating WARP in a dynamic environment may be
vulnerable to money pumps
This can be avoided by a status-quo justification: DM uses justifications that are consistent with past choices
Example: choosing the most recently chosen act in C(A)
A related formal construction in Bewley (2002)
Strong empirical psychological support
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Conjectural Equilibrium (Battigalli, 87)
Each player has partial information about the actions of the others. In equilibrium she plays a best response against one of the consistent action profiles Similar concepts in the learning literature: Fudenberg & Levine
(93), Kalai & Lehrer (93), Rubinstein & Wolinsky, (94)
Modeled by belief-justifications: Each player has a set of priors – P
A common set when information is symmetric
Justification triggers each player into a specific prior
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Attitude to Uncertainty (Belief-Justifications)
Example: |S|=2, X={x,y}, y=C(x,y), P includes a segment around 0.5
Let: g= (x,y), f=(0.5x+0.5y,0.5x+0.5y)
Minimax model (Gilboa-Schmeidler, 1989) predicts: f g Our model predicts that both acts are choosable
Heath & Tversky (1991) – people are: Uncertainty-averse – when DM feels ignorant or uninformed
Uncertainty-seeker – when DM feels knowledgeable26
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Summary
We present a new axiom, CARNI, which behaviorally
describes (non-binary) choice when there are
incomplete preferences and multiple justifications
A convex variation of Ok-Eliaz (04) WARNI axiom
We apply the new axiom in different choice models:
Taste justifications (multiple utilities)
Belief justifications (multiple priors)
Generalizations (a la Ok, Ortoleva & Riella, 08)
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Questions & Comments?
Y. Heller (2009), Justifiable choice, mimeo.
http://www.tau.ac.il/~helleryu/weaker.pdf