Anchored Preference Relations

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Anchored Preference Relations

• Jacob Sagi
• Berkeley-Stanford Spring 2003 Finance Workshop

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Outline of Talk

• Reference dependent choice
• Why?
• Primitives of choice
• Naïve anchoring traps
• What to rule out
• Axiom 1 the no anchor cycling property
• Consequences of Axiom 1
• No prospect theory
• Not kinky enough!!
• An alternative to prospect theory
• Risk attitudes and loss aversion
• Accommodating Allais

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Reference dependent choice- why?

• Loss Aversion (Markowitz (1952), Kahneman
  Tversky 1979)
• Group 1
• Youre given 1000 and
• a choice of 500 sure gain or a 50-50 chance at
  1000.
• Most choose 1000 500 for sure
• Group 2
• Youre given 2000 and
• a choice of 500 sure loss or a 50-50 chance at
  losing 2000.
• Most choose 2000 gamble
• Consequences are identical
• Subjects demonstrate higher sensitivity to
  losses then gains in arriving at same place.
• Try to avoid losses even if it means taking risks
• People prefer not to get something at all then to
  get it and then have it taken away tangibility
  of ownership.

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Reference dependent choice- why?

• Endowment Effect (Thaler (1980), Samuelson
  Zeckhauser (1988))
• Endowment/Status Quo receives preferential
  treatment
• TIAA CREF
• New plans where the default investment was safe
  vs. risky.
• WTA-WTP disparity
• Subjects consistently value something more when
  they own it.
• Status quo bias is ubiquitous.
• Branding
• First mover advantage

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Reference dependent choice- why?

• 1st-order risk aversion Epstein-Zin (1990),
  Thaler Benartzi (1995), Rabin (2000),
  Barberis-Huang-Santos (2001)
• Smooth utility implies 2nd-order RA
• When risk is small relative to endowment, agent
  must be nearly risk neutral or exhibit near CARA
  (i.e., be pathologically averse to highly
  attractive gambles).
• Individuals habitually exhibit risk aversion in
  the large as well as the small
• Kinks in the utility function around the
  endowment can cause 1st-order RA in the small
• To exhibit this everywhere, utility function must
  be pathological or reference dependent.

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Primitives

• X a set of outcomes
• Compact metric space
• P(X) the set of Borel measures on X
• With appropriate topology
• a set of transitive, complete
  and continuous binary relations on P(X)
• Represented by Ue(.)
• e is the reference point, also called
  the anchor
• q is preferred to p when anchored
  at e
• Reference points can be stochastic!

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The dangers of a naïve theory(highly stylized)
1500
A
2000
0.5
B
1000
reference point expected value (portfolio
wealth) B chosen. Wait until new reference point
is incorporated at eB 1500. Before B is
resolved offer A once more. Relative to endowment
face
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One way to avoid cycle is to make the setting
manifestly temporal and impose time consistency
(backward induction)
A
eA
A
B
A
e0
eB
B
B
eA eB implies A is always preferred at second
node Thus, no loss aversion!!
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Moral

• One has to be very careful in constructing a
  theory with reference dependence.
• Choice
• Ignore relationship between
• Impose ad-hoc inter-temporal assumptions about
  the rate of endowment absorption.
• Impose time-consistency through backward
  induction and risk losing some important effects
• Alternatively,
• Impose appealing structure relating
• In the spirit of transitivity

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What to rule out?

• Axiom 1
• If q is better than p when the reference point is
  at p, then it is always better than p
• Affinity for a prospect is greatest when it is
  the reference point
• Prevents simple choice cycle
• Prevents more complex cycles
• Set anchor at e, choose p over q p becomes the
  new anchor, pay extra to get back q.

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No-cycle Equivalent

• Anchored uc sets are nested within non-anchored
  uc sets

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Implications of Axiom 1

• If all the are smooth a.e. and at a.e.
  anchor, then there is no reference dependence.
• With non-additive , result requires mild
  continuity wrt e
• CPT with non-additive probabilities (Choquet
  integrals)
• Requires a certainty equivalent for the anchor
• Structure of the model is smooth a.e. and at
  a.e. reference point

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Intuition
If there is reference dependence, the
indifference surfaces of some two relations cross
a.e. in some non-zero measure region. The smooth
a.e. and at a.e. anchor assumption means that at
some crossing point the crossing relations have a
linear approximation and the crossing point
itself is a smooth anchor.
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Zooming in

• Indifference surface of at e is smooth by
  assumption.
• Can be approximated by flat surface near e.
• Axiom 1 surface has to be nested

PROBLEM cant be both nested and flat
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Implications of Axiom 1

• No linear prospect theory is consistent with
  Axiom 1
• A similar result applies to generalized CPT
• Prospect theory is simply not kinky enough!

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Alternatives?

• Explicitly impose Axiom 1
• Add more structure
• Technical
• Anchor continuity
• Convex continuity if uc set is strictly convex
  at the anchor, all uc sets in the neighborhood of
  the anchor are also convex.
• Behavioral
• Independence

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Interpreting Independence

• Editing principle of KT (1979)
• Subject is only interested in the relative
  properties of prospects
• Cancel out common attributes
• If there is reference dependence, must cancel out
  relative attributes
• Translation and scale invariance
• Subject only cares about q-e, and does not pay
  attention to the scale of the difference between
  the distributions.
• No anchor is singled out for special treatment

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Interpreting Independence

• Note that
• vNM Independence wrt to mixing with the anchor.
• Counterfactual
• The common consequence version of the Allais
  Paradox
• Will return to this later
• Independence is just a behavioral assumption
  can relax it after examining rep.

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Representation of Ue(p)

• Y is some set of utility functions
• Reduces to EUT if Y is a singleton.
• Subject evaluates the prospect, p, based on a
  worst case utility scenario relative to the
  anchor.
• Note Ue(e) 0
• Ue(p) Ue(e) iff EUT of p exceeds that of e for
  EVERY utility function in Y.
• Resistance for moving away from the anchor
  (status quo bias)
• No cycling!
• uc set at anchor is a cone
• Kink at every anchor!!
• Proof is quite technical.

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Risk Attitudes of Repn

• Attributes of Ue(e) correspond to attributes of
  the ?s.
• All ?s concave, agent will always be RA
• Loss aversion requires Y contain both a risk
  averse (concave) and risk seeking utility
  (convex) function.

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Examples Applications

• Loss aversion
• Samuelson-Zeckhauser (1988) Endowment Effect
• WTA-WTP disparity
• Affinity is greatest for anchor (Axiom 1)
• The ce of a prospect is greatest when it is the
  anchor
• Charge more to give anchor than to receive it

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Example Applications

• X L, I, H
• Y ?1, ?2
• Utility vectors
• ?1 (0, 1, 4) ? RL
• ?2 (0, 1, 4/3) ? RA
• p (pL, 1 - pL - pH, pH)
• Let A13, A21/3
• Ue(p)

pH
pL
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Limitations

• Still hard or impossible to get
• All Allais-type behavior
• All cases of Risk aversion in the large small
• Okay if prospect entails both gain and loss
• Not okay otherwise (pure gains/loss prospects)
• The missing attributes are generally present in
  non-additive theories (CEUT)
• U(p) E?(p)v, where ?(p) are probability
  weights
• ?(xxi) w( prby(x ? xi) ) - w( prby(x ? xi1)
  )
• w is a non-linear monotonic transformation
• How to incorporate them here?

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Conclusions

• Reference dependence no-cycling
• Restricts admissibility of representations
• CPT not admissible
• Representation in terms of worst case utility
  relative to anchor is okay
• Simple worst case EUT
• More comprehensive worst case CEUT

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Berk