PREFERENCE FOR FLEXIBILITY AND THE OPPORTUNITIES OF CHOICE

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PREFERENCE FOR FLEXIBILITYAND THEOPPORTUNITIES
OF CHOICE

• Salvador BARBERÀ
• Birgit GRODAL

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• WHO EXHIBITS PREFERENCES FOR FLEXIBILITY?
• Agents who rank sets of alternatives
• and
• always prefer any set to its subsets.
• WHY DO WE NEED A THEORY ABOUT SUCH PREFERENCES?
• Formally, because they (seem to) contradict the
  standard notion of indirect utility
• Substantially, because it is an observed feature
  of preferences, and we would like to explain its
  origins.

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• BACKGROUND
• PEOPLE RANK SETS WHY DO THEY?
• Non-utilitarian explanations. Even if they will
  choose the same, they value the chance to choose
  from larger/smaller sets
• Preferences for freedom (Sen, Pattanaik Xu)
• Embarrassment of choices
• Utility-based explanations.
• Sets as the basis for lotteries
• Sets as non-discarded elements for a future round
  of choice preferences for flexibility
• Marshak, Kreps
• Barberà Grodal
• Barberà, Bossert Pattanaik, Ranking sets of
  objects, in Barberà, Hammond Seidl (Eds.),
  Handbook of Utility Theory. Volume II Extensions,
  Kluwer Academic Publishers, Dordrecth. 893-977,
  2004.

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PREFERENCES FOR FLEXIBILITY IN TWO-STAGE DECISIONS

• KREPS (Econometrica 1979)
• SCREENING in the 1st period
• CHOICE in the 2nd WITH UNCERTAINTY ABOUT
  PREFERENCES
• Agents prefer larger sets because they offer the
  possibility of changing decisions if preferences
  change.
• BARBERÀ GRODAL
• SCREENING in the 1st period
• CHOICE in the 2nd WITH UNCERTAINTY ABOUT THE
  AVAILABILITY OF ALTERNATIVES
• Agents prefer larger sets because they offer the
  possibility of changing decisions if some
  alternative becomes unavailable
• (The restaurants example)

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• SET UP
• A, a finite set of alternatives
• 2A, its power set
• SURVIVAL PROBABILITIES
• Given by lotteries in the form
• l 2A??, with ?E?2A l(E)1.
• L is the set of such lotteries.
• For l, E, l(E) is the probability that exactly
  the alternatives in E survive in the second
  period.
• Now, for l, C, D, define
• lC(D)? E?2Al EnCD l(E)
• lC(D) is the probability that D will be the set
  of alternatives available, given that
• C was screened in period 1
• some E has survived, s.t. EnCD

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• PREFERENCES
• On alternatives
• uA?? ?? s.t. u(?)
• On sets (to be taken as primitive/observable)
• ?, a total order on 2A
• . complete . transitive .antisymmetric (to be
  extended)
• EXPECTED OPPORTUNITY OF SETS (for u, l)
• V2A??
• V(C)? max u(x) . lC(D)
• D?2A x ?D??
• EXPECTED OPPORTUNITY RANKINGS
• The order ? on 2A is an expected opportunity
  ranking iff
• ?l?L, uA?? ?? such that the corresponding V
  satisfies
• B ? C ? V(B)gtV(C)

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• The inclusion property
• An ordering ? satisfies the inclusion property
  iff, for all B,C?2A
• B?C ?C?B
• Theorem
• Let A be a finite set and ? a total ordering on
  2A.
• ? is an expected opportunity ranking iff it
  satisfies the inclusion property.
• Comments
• Same basic result as in KREPS
• Extension to preorders? Yes, if we impose
• positive probabilities for all sets
• positive utilities for all alternatives

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• Comments to thin 1 (ctd.), and further analysis
• The proof is constructive, relies on an
  appropriate assignment of survival proabilities
  while keeping all alternatives indifferent.
• The construction implies that we cannot derive
  any knowledge about the agents utilities on
  alternatives from knowledge about set rankings.
• Restricting the admissible form of uncertainty
  regarding survival may imply further
  restrictions.
• We now turn to the implications of assuming
  independently distributed survival probabilites,
  with lA? 0,1 and
• l(B)?b?B l(b) ?c?B (1- l(c))

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• RESERVALS
• Let E, F ? 2A
• A set B ? 2A, with B?(E?F)?
• reverses E?F iff F?B?E?B
• WHY REVERSALS?
• Adding z to x has two effects
• V(x,z)-V(x)max?u(z)-u(x))l(x)l(z),
  o(1-l(x))V(z)
• 1st term. Utility gain
• 2nd term. Insurance gain
• When added to sets (in more general terms), the
  gains that come from new sets can be of different
  intensity.

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• SOME RESTRICTIONS ON SET RANKINGS DERIVED FROM
  THE ASSUMPTION OF INDEPENDENT SURVIVAL
  PROBABILITIES
• If all elements in B are better than all those
  in C and D, adding B to C and to D cannot
  reverse.
• Corollary for three alternatives
• Since one of x,y and z must be best, it cannot be
  that (simultaneously)
• x reverses y and z
• y reverses x and z
• z reverses x and y
• Application the order
• xyz is not an expected opportunity
• zy
• xz order with independent
• xy
• x survival probabilities
• y
• z
• INDEPENDENCE HAS BITE!

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• THEOREM
• Let A3. The total ordering ? on 2A can be
  represented as an expected opportunity ranking
  with independent survival probabilities iff it
  satisfies
• The inclusion property, and
• for any labeling x,y,z of the three alternatives
• in A, if y reverses x?z, then z does not
  reverse x?y, nor y?x.
• FINAL COMMENTS
• We can learn some partial facts about utilities
  and probabilites from set ranking.
• Hard to extend, probably not impossible.