Memory and Anticipation

1
Memory and Anticipation

• B. Douglas Bernheim, Stanford University
• Raphael Thomadsen, Columbia University
• April, 2004

2
Introduction

• Combines two strands of literature
• Imperfect memory
• Piccione and Rubinstein (1997), Benabou and
  Tirole (2002), Mullainathan (2002)
• Anticipatory emotions
• Loewenstein (1987), Elster and Loewenstein
  (1992), Caplin and Leahy (2001,2003), Koszegi
  (2002)

3
Why?

• Imperfect memory creates a natural role for
  anticipatory emotions
• The combination has interesting implications

4
Preview of Results

• Reasonable to select dominated strategies
• Resolution of Newcombs Paradox (requires only a
  tiny weight on anticipatory emotions)
• Cooperation in the Prisoners Dilemma with
  probability arbitrarily close to unity

5

• When the individual can influence what is
  remembered (e.g. through reminders)
• Role of reminders in improving concurrent
  decisions
• Preference for coarse (categorical) information
• Preference for ignorance
• As-if overoptimism
• Behaviors associated with cognitive dissonance

6
Other related literature

• For some results, there are other theories
  explaining similar phenomena (involving different
  mechanisms)
• Preference for ignorance (Carillo and Mariotti
  (2000), Benabou and Tirole (2002), Koszegi
  (2002))
• Overoptimism or overconfidence (Rabin and Schrag
  (1999), Koszegi (2000), Hvide (2002), Benabou and
  Tirole (2002), Postelwaite and Compte (2003), Van
  den Steen (2004))
• Cognitive dissonance (Akerlof and Dickens (1982))
• Formally, Benabou-Tirole (2002) is the closest of
  these (will describe differences later)

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Modeling Imperfect Memory

• Assume people are sophisticated with respect to
  memory imperfections (in tradition of
  Piccione-Rubinstein or Benabou-Tirole, not
  Mullainathan)
• There is more than one way to model sophisticated
  behavior with imperfect memory, corresponding to
  different potential assumptions about what
  exactly is forgotten.

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• We use the Piccione-Rubinstein solution concept
• Treat the individual as a different player
  after each episode of forgetting
• Reduces problem to a game of incomplete
  information, to which we can apply standard
  solution techniques (sequential equilibrium)
• Interpretation the individual confronts many
  similar decision problems, and is used to follow
  some norm. Whenever his memory fails, he
  remembers the norm and forgets any plan he may
  have had to deviate from it.

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The Link Between Imperfect Memory and
Anticipatory Emotions

• Three period decision problem, t 0,1,2.
  Decision in t 0, payoff in t 2, and
  information changes in between
• In t 0 the DM cares about expected period 2
  utility, and the expected period 1 expectation of
  period 2 utility (anticipatory feelings)

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• With perfect memory, the law of iterated
  expectations holds. Period 0 expectation of
  period 2 utility is equal to the period 0
  expectation of the period 1 expectation of period
  2 utility.
• If period 0 utility is linear, anticipatory
  feelings are irrelevant
• With imperfect memory, the law of iterated
  expectations breaks down. So anticipatory
  feelings are always relevant, even if period 0
  utility is linear.

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• Simple example
• State of the world is either good or bad
• These states are equally likely
• When the state is good, the payoff is 10 when it
  is bad, the payoff is 0
• The DM knows the state in period 0, but forgets
  it by period 1
• In period 1 the expected outcome is 5
• The period 0 expectation of the period 1
  expectation is 5
• But in period 0 the expected outcome is either
  zero or 10

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Newcombs Paradox

• Popularized by the philosopher Robert Nozick.
  Intended to depict a hypothetical situation where
  standard decision making principles would come
  into conflict.
• There are two boxes
• Open box, contains 1,000
• Closed box, contains either 1,000,000 or nothing

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• The subject must either choose
• Closed box only, or
• Both boxes
• Twist before the subject makes his choice, a
  famous psychic predicts what the subject will do.
• If the psychic predicts closed box only, a
  referee puts 1,000,000 in the closed box
• If the psychic predicts both boxes, a referee
  puts nothing in the closed box

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• The subject knows that the psychic has played
  this games with hundreds of thousands of other
  subjects
• Many have chosen both boxes, and many have chosen
  only the closed box
• The psychics prediction has always been correct
• What should the subject do?
• Dominance obviously should choose both boxes

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• Many people seem to feel that closed box only
  is the right choice. Why?
• One prevalent idea seems to be the sense that the
  psychic knows something subtle about their
  predisposition
• But this still does not come to grips with the
  causality issue
• We try to make sense of this intuition while
  adhering to the principle that, while the psychic
  may be knowledgeable, she cant violate the rules
  of causality, and everyone understands this

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• The general approach
• Formulate the problem as a game where the psychic
  moves first (with the objective of predicting
  correctly), and the subject moves second
• Assume subjects differ in some aspect of
  preferences, and the psychic can observe this
• Exhibit an equilibrium in which some subjects
  choose closed box only, some subjects choose
  both boxes, and the psychic always predicts
  correctly (putting 1,000,000 in the closed box
  only when the subject subsequently picks closed
  box only)

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Psychic
Predict C (PC)
Predict B (PB)
Closed only (C)
v(a), 1
v(0), 0
Subject
Both (B)
v(b), 1
v(a b), 0
Discussed special case of a 1,000,000 and b
1,000
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The Extended Game

• Nature randomly selects a preference parameter ?
• Represents subjects weight on ultimate payoff
  relative to anticipatory feelings
• Observed both by subject and by psychic
• Psychic makes her prediction the referee puts a
  in the box if the prediction is C
• Subject chooses either the closed box only, or
  both boxes, placing weight ? on the ultimate
  outcome relative to anticipatory feelings

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• The subject forgets the value of ?, but remembers
  his action
• The subject waits to learn the outcome, forms an
  expectation of what it will be, and experiences
  anticipatory feelings
• Payoffs are realized

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There is always one equilibrium in which the
subject always chooses both boxes. But this
isnt the only equilibrium.

• Theorem 1 Assume v is linear. Let ? (a
  b)/b. There exists an equilibrium in which
• For ? ? ?, the DM chooses both boxes and the
  psychic correctly predicts this, and
• For ? lt ?, the DM chooses the closed box only
  and the psychic correctly predicts this

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Intuition

• In equilibrium, the parties create correlation
  between their choices by conditioning on
  something commonly observed.
• The thing that is commonly observed is
  subsequently forgotten by the subject
• Choosing closed box only doesnt change whats
  in the closed box. But, because of the
  equilibrium correlation, it does cause the
  subject to subsequently infer that the psychic
  has predicted closed box only, which means the
  money is in the box
• Creating this inference is valuable because it
  improves the subjects anticipatory feelings, and
  is more valuable when ? is smaller.

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• This does NOT require the subject to place much
  weight on anticipatory emotions
• With v linear, a 1,000,000, and b 1,000, we
  have ? 999. So the subject chooses closed
  box only unless actual payoff is nearly 1,000
  times as important as anticipatory feelings.
• Extends to v concave, with ? indeterminate over
  some range. For v(y) ? y-2, the largest ?
  consistent with equilibrium is 500,000,000. So
  the subject chooses closed box only unless the
  actual payoff is more than 500 million times as
  important as anticipatory feelings.

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The Prisoners Dilemma
Player B
N
C
a, a
b, c
C
Player A
c, b
d, d
N
Assumption P1 c gt a gt d gt b (standard)
Assumption P2 d ? b gt c ? a (larger incentive
to play N rather than C when opponent plays N
rather than C)
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Extended Game

• Nature randomly selects a signal, ?, distributed
  uniformly over the interval 0,1. The value is
  observed by both players.
• The players simultaneously choose their actions
  (C or N), placing weight ? on the ultimate
  outcome relative to subsequent anticipatory
  feelings
• The players forget the value of ? but remember
  their choices

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• The players wait to learn the outcome, form
  expectations of what it will be, and experience
  anticipatory feelings
• Payoffs are realized

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• Theorem 2 Suppose that ? lt (a ? d)/(c ? a).
  Then, for all ? gt 0, there exists an equilibrium
  for which the players cooperate (play C) with
  probability greater than 1 - ?.

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Intuition

• In equilibrium, the parties create correlation
  between their choices by conditioning on
  something commonly observed.
• The thing that is commonly observed is
  subsequently forgotten by both players
• While playing cooperatively doesnt cause the
  opponent to play cooperatively, it does cause the
  player to subsequently infer that the opponent
  has played cooperatively
• Creating this inference is valuable because the
  subject has anticipatory feelings

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but

• Remember that players play both N and C with
  positive probability
• If the previous argument is right, why are they
  also sometimes willing to play N?

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• Assumption P2 tells us that playing N rather than
  C has a bigger impact on a players ultimate
  payoff when the opponent plays N, rather than C
• That means that, for some intermediate values of
  ?, a player can be both
• Deterred from deviating to N when hes supposed
  to play C
• Not tempted to choose C when hes supposed to
  play N
• Getting this to work for small ? involves mixed
  strategies.

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Some Evidence

• Shafir and Tversky (1992)
• Standard treatments, subjects play C in 37 of
  trials
• When informed that opponent played N, 3 play C
• When informed that opponent played C, 16 play C
• Since both figures are less than 37, uncertainty
  about opponents choice plays a significant role
  in generating cooperative behavior

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Behavior with Reminders

• Possible mechanisms for endogenizing memory
• Internal mechanisms (rehearsal, repression)
• External mechanisms (reminders)
• We focus on external mechanisms, though could
  model some internal mechanisms similarly
• Two types of reminders considered
• Hard reminders (verifiable information)
• Soft reminders (cheap talk e.g. notes or a
  diary)

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The Framework

• A payoff-relevant state of nature ? ? 0,1 is
  realized.
• The DM observes information s. Usually, s ?
  (the state of nature is observed).
• The DM potentially makes two decisions, placing
  weight ? on the ultimate outcome relative to
  subsequent anticipatory feelings
• Chooses reminders (more later)
• Chooses an initial action x1 ? 0,1

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• The DM forgets the state of nature s, but recalls
  his initial action x1 and any reminder he left
  for himself.
• The DM potentially selects a delayed action x2
  ? 0,1.
• The DM waits to learn the outcome, and
  experiences anticipatory feelings based on his
  expectation of what it will be.
• The outcome is realized.

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Details Concerning Reminders

• Hard reminder
• Chooses h ? 0,1??, subject to the constraint
  that h ? 0,1 implies h ?.
• Involves utility cost c
• Soft reminder
• m ? 0,1 chosen without constraints (could allow
  for a richer message set)
• Costless

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Details Concerning Payoffs

• Without a hard reminder, ultimate payoff (in step
  7) is
• u1(x1, ?) u2(x2, ?)
• With a hard reminder, subtract c
• Assumptions on ui
• Differentiable and strictly increasing in ?
• xi 0 is better than xi 1 when ? 0
• xi 1 is better than xi 0 when ? 1
• ui(0, ?) ? ui(1, ?) is strictly decreasing in ?

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Delayed Action Only, No Reminders

• Decision is uninformed, and therefore not first
  best
• Obvious role for reminders

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Initial Action Only, No Reminders

• Theorem 3 Suppose there is an initial action, no
  delayed action, and no reminders. The DM selects
  x1 1 for a larger set of states of nature than
  when anticipatory feelings dont matter.
• Interpretation As-if overoptimism
• Intuition Self-signaling the DM distorts
  choices in the direction of the types he is
  trying to imitate (here, those with more
  favorable information)

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• Welfare The DM would be better off ex ante (in
  expectation) if he could commit to always making
  the choice that maximizes his final payoff
  (avoiding self-signaling)
• Self-signaling improves his anticipatory emotions
  in some states of the world, but worsens them in
  others.
• Since he cant systematically fool himself,
  self-signaling cant improve his ex ante expected
  anticipatory state
• Therefore, the only welfare-relevant consequence
  of self-signaling is the fact that he makes
  sub-optimal choice (from the perspective of
  ultimate payoffs) in some states

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• Remark Since the decision-making distortion
  results from memory imperfections, there is a
  potential beneficial role for reminders even
  though the decision is taken with full
  information.

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Soft Reminders Only, No Actions

• Soft reminders by themselves are useless

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Hard Reminders Only, No Actions

• Theorem 4 Suppose there are hard reminders and
  no actions. The DM leaves a hard reminder as
  long as the state of nature exceeds some
  threshold, which goes to zero as the cost of the
  reminder gets small. When the cost of reminders
  is positive but low enough so that he leaves
  reminders in some states, he is worse off than
  with no information at all.

42

• Interpretation
• Endogenously remembers positive information,
  forgets negative information
• Preference for ignorance.
• Intuition This is basically a disclosure problem
• The behavioral outcome reflects a well-known
  unraveling phenomenon.
• The welfare result reflects the same
  considerations as before he can improve his
  anticipatory emotions in some states only by
  worsening them in others, with no net gain
  overall because he cant systematically fool
  himself. From an ex ante perspective, all he
  ends up doing is burning some resources by
  leaving reminders.

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• Remark
• This may sound like an example of dynamically
  inconsistent preferences the DM would like to
  avoid leaving reminders in step 3, but is unable
  to follow through on this when step 3 arrives
• It is actually a different phenomenon. Given the
  equilibrium inferences he makes in step 5, he
  concurs ex ante with his choices in step 3.
• He wishes to constrain his future actions not
  merely to change the actions themselves (as would
  be the case with dynamically inconsistent
  preferences), but also to change inferences
  (which is what makes the change in actions
  desirable).

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• An extension
• Suppose there is a step 0 in which the DM chooses
  left or right. Left leads to the problem
  just described. Right inverts the scale for ?
  (low is good and high is bad)
• Solutions to the continuation problems are
  described by Theorem 4.
• When DM chooses left, endogenously forgets
  information when it tells him that ? was low and
  he made the wrong initial choice.
• Likewise, when DM chooses right, endogenously
  forgets information when it tells him that ? was
  high and he made the wrong initial choice.
• Phenomenon is associated with cognitive
  dissonance (Akerlof and Dickins (1982)).

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Initial Action, Soft Reminders

• Same as initial action only

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Initial Action, Hard Reminders

• Theorem 5 Suppose there is an initial action and
  costless hard reminders. Then the DM chooses the
  first-best action in every state, and is
  perfectly informed throughout.
• Interpretation Having hard reminders available
  improves concurrent decisions.
• Intuition Without reminders, the action x1
  doubles as a reminder. With reminders,
  unraveling assures that information is revealed,
  and the initial action no longer serves an
  informational role.

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Delayed Action, Soft Reminders

• Theorem 6 With a delayed action and soft
  reminders, the DM always prefers to receive
  categorical information over continuous
  information.
• Interpretation Use of categorical information
  widespread even when underlying information is
  continuous

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• Intuition
• The cheap-talk equilibrium involves giving one
  message to induce the choice of 0 in low states,
  and another message to induce the choice of 1 in
  high states.
• For the marginal state, the DM sends a message
  that distorts the decision from the first-best in
  order to improve anticipatory feelings.
• As usual, any improvement in anticipatory
  feelings in some states is offset by reductions
  in other states.
• The DM would be better off not knowing that he is
  close to the margin. We can accomplish this by
  making the information categorical.

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Delayed Action, Hard Reminders

• Clearly, with costless hard reminders we get full
  revelation and first-best decisions.

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Summary 1

• Soft reminders only irrelevant
• Hard reminders only if costly, recalls positive
  information and forgets negative information, and
  would prefer ignorance

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Summary 2

• Initial action only choice is distorted, as-if
  optimism
• Add soft reminders no change
• Add hard reminders if cheap, initial action is
  no longer distorted, and outcome is first-best,
  so the availability of hard reminders improves
  concurrent decisions.

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Summary 3

• Delayed action only choice is uninformed, not
  first best
• Add soft reminders can improve choice, but
  prefers categorical information to continuous
• Add hard reminders if cheap, the DM remains
  fully informed and actions are first-best

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Direction for Continuing Work

• Memory and re-experienced utility
• Relationship between memory and re-experienced
  utility has a long history in the psychological
  literature (e.g. repression of painful memories)
• Probability of recall is higher for unusual
  experiences and for good experiences
• Accounts for tendency to repeat mistakes
• Possible resolution for an old puzzle why people
  take unfair gambles with small probabilities of
  big payoffs, but are risk averse with respect to
  gambles with smaller stakes or large potential
  losses.