RISK, AMBIGUITY, GAINS, LOSSES

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RISK, AMBIGUITY, GAINS, LOSSES

• Sujoy Chakravarty
• Department of Humanities and Social Sciences
• Indian Institute of Technology,
• Hauz Khas, New Delhi 110019, India
• Jaideep Roy
• Department of Economics,
• Lancaster University,
• Lancaster LA1 4YW, United Kingdom.

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The Concept of Ambiguity
50 RED 50 YELLOW
100 RED YELLOW
URN A RISKY
URN B AMBIGUOUS

• Bet on a colour.
• Pick a bead.
• If the colour of the bead matches the colour you
  bet on, you receive 100.
• Otherwise you get nothing

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The Concept of ambiguity (2)

• Knight (1921)
• Risk measurable uncertainty which may be
  represented by precise odds or probabilities
• Ambiguity immeasurable uncertainty which cannot
  be readily represented by precise odds or
  probabilities
• Einhorn and Hogarth (1986)
• Casual empiricism suggests that people tend to
  impute numerical probabilities related to their
  beliefs regarding likelihood of outcomes
  (Ellsberg 1961). Thus ambiguity has been referred
  to as subjective risk about objective risk
• Camerer (1995)
• Ambiguity with respect to events as known to
  be missing information, or not knowing relevant
  information that could be known .

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Utility Theory

• Savage-Bayes Approach
• Expected Utility Theory (vNM 1944) and
• Subjective Expected Utility Theory (Savage 1954).
• Essential implication of this approach
• Probabilistic Sophistication requirement on part
  of a decision maker to posses a complete and
  exhaustive list of all the possible states of the
  world along with a subjective assessment of
  likelihood of uncertain events that can be
  represented by a unique and additive probability
  distribution.
• Experimental evidence since Ellsberg (1961)
  demonstrated the inability of SEU theory to
  describe behaviour under ambiguity.

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A theory of Smooth AmbiguityKlibanoff,
Marinacci, Mukerji 2006

• Three separate virtues of a decision maker in
  face of uncertainty
• Attitude towards risk utility function u for
  money
• Attitude towards ambiguity a smooth value f for
  expected utility for each given possibility of
  the urn
• Subjective beliefs additive beliefs regarding
  the probability of occurrence of a particular
  state.
• Thus, the Value of an ambiguous prospect V- is
  the expected value of f given subjective beliefs
  over nature of urns.

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A theory of Smooth Ambiguity (2)

• Let us represent a
  simplified Ellsberg Urn B given as,
• (10, 0, 9, 1,,1,
  9,0, 10), using KMM theory.

10 RED and 0 YELLOW 9 RED and 1 YELLOW 8 RED
and 2 YELLOW 7 RED and 3 YELLOW 6 RED and 4
YELLOW 5 RED and 5 YELLOW 4 RED and 6 YELLOW
3 RED and 7 YELLOW 2 RED and 8 YELLOW 1 RED
and 9 YELLOW 0 RED and 10 YELLOW

Possible Distributions of RED and YELLOW
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A theory of Smooth Ambiguity (3)

• Given that there are 11 potential distributions
  of R and Y, letting u(.) represent the decision
  makers utility function, s, the additive
  probabilistic belief about the likelihood of
  occurrence of each of these distributions, and
  the f function, the curvature of which gives a
  decision makers attitude to ambiguity, we can
  write the value of the expected value of the
  ambiguous urn as,

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A theory of Smooth Ambiguity (4)
Possible Distribution Win100 Win 0 EU f
1 1 0 u(100) f u(100) 1/10
2 0.9 0.1 0.9u(100)0.1u(0) F0.9u(100)0.1u(0) 1/10
3 0.8 0.2 0.8u(100)0.2u(0) F0.8u(100)0.2u(0) 1/10
4 0.7 0.3 0.7u(100)0.3u(0) F0.7u(100)0.3u(0) 1/10
5 0.6 0.4 0.6u(100)0.4u(0) F0.6u(100)0.4u(0) 1/10
6 0.5 0.5 0.5u(100)0.5u(0) F0.5u(100)0.5u(0) 1/10
7 0.4 0.6 0.4u(100)0.6u(0) F0.4u(100)0.6u(0) 1/10
8 0.3 0.7 0.3u(100)0.7u(0) F0.3u(100)0.7u(0) 1/10
9 0.2 0.8 0.2u(100)0.8u(0) F0.2u(100)0.8u(0) 1/10
10 0.1 0.9 0.1u(100)0.9u(0) F0.1u(100)0.9u(0) 1/10
11 0 1 u(0) f(u(0)) 1/10
Beliefs
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A theory of Smooth Ambiguity (5)

• In our experimental study we use a (1, 0, 0,
  1) version of the Ellsberg Urn

Possible Distributions of RED and YELLOW
10 RED and 0 YELLOW 0 RED and 10 YELLOW
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Experimental Questions

• Are our attitudes towards risk different across
  gains and losses?
• Has been asked, answered by Kahneman and Tversky
  (1979) using mostly unpaid questionnaires who
  claimed that we are risk averse in gains and risk
  seeking in losses.
• Results from recent paid experiments by Holt and
  Laury (2005) challenge the above claim.
• We check again and use it in our calibrations for
  ambiguity attitudes.
• Are our attitudes towards ambiguity different
  across gains and losses?
• never asked with the exception of Einhorn and
  Hogarth (1986) unpaid experiments
• but they used single binary betting decision
  problems a la Ellsberg (1961). Such
  experiments, even if paid, do not allow us to
  systematically measure the extent of ambiguity
  aversion as it differs from individual to
  individual. They merely allow us to gauge whether
  or not an individual is ambiguity averse, neutral
  or seeking.
• Are our attitudes in risk and ambiguity really
  independent virtues?
• never asked

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Design of the Experiment

• 85 subjects from Indian Institute of Management,
  Ahmedabad predominantly with engineering and
  computer science backgrounds.
• Multiple Price List procedure Holt and Laury
  (2002), Harrison et al. (2005), Chakravarty et al
  (2005)
• A total of 4 tasks, 2 for risk, 2 for ambiguity.
  Each subject made 40 paired decisions, total of
  3400 binary decisions.
• All subjects performed the risk tasks first.
• Half were given loss tasks first, the other half
  was given gains task first controlling for
  order effects two independent sessions
• As MPL method may suffer from anchoring Anderson
  et al. (2005), Bosch and Silvestre (2005), the
  reverse MPL scheme was administered to
  approximately half the subjects

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Design of the Experiment (2)

• Each subject could earn a maximum total of Rs.500
  (PPP USD 55). Could never owe anything to the
  experimenter, but could see that an act would
  certainly lead to losing money.
• They earned on average Rs.275 (PPP USD 30).
• Net payments were made at the very end when all
  tasks were performed. Hence, no possibility of
  updating current wealth.
• All tasks has independent randomization rules
  no possibility of hedging
• Subjects were educated in MPL tasks through
  questionnaires

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Risk Task (Gain)
Decision Option A Option B Your Choice (Circle A or B)
1 Die Roll 1-5 Rs 40 Die Roll 6-10 Rs 60 Die Roll 1-10 Rs. 0 A B
2 Die Roll 1-5 Rs 40 Die Roll 6-10 Rs 60 Die Roll 1-9 Rs 0 Die Roll 10 Rs 100 A B
3 Die Roll 1-5 Rs 40 Die Roll 6-10 Rs 60 Die Roll 1-8 Rs 0 Die Roll 9-10 Rs 100 A B
4 Die Roll 1-5 Rs 40 Die Roll 6-10 Rs 60 Die Roll 1-7 Rs 0 Die Roll 8-10 Rs 100 A B
5 Die Roll 1-5 Rs 40 Die Roll 6-10 Rs 60 Die Roll 1-6 Rs 0 Die Roll 7-10 Rs 100 A B
6 Die Roll 1-5 Rs 40 Die Roll 6-10 Rs 60 Die Roll 1-5 Rs 0 Die Roll 6-10 Rs 100 A B
7 Die Roll 1-5 Rs 40 Die Roll 6-10 Rs 60 Die Roll 1-4 Rs 0 Die Roll 5-10 Rs 100 A B
8 Die Roll 1-5 Rs 40 Die Roll 6-10 Rs 60 Die Roll 1-3 Rs 0 Die Roll 4-10 Rs 100 A B
9 Die Roll 1-5 Rs 40 Die Roll 6-10 Rs 60 Die Roll 1-2 Rs 0 Die Roll 3-10 Rs 100 A B
10 Die Roll 1-5 Rs 40 Die Roll 6-10 Rs 60 Die Roll 1 Rs 0 Die Roll 2-10 Rs 100 A B
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Risk Task (Loss)
Decision Option J Option K Your Choice (Circle J or K)
1 Die Roll 1-5 -Rs 60 Die Roll 6-10 -Rs 40 Die Roll 1-10 -Rs 100 J K
2 Die Roll 1-5 -Rs 60 Die Roll 6-10 -Rs 40 Die Roll 1-9 -Rs 100 Die Roll 10 Rs 0 J K
3 Die Roll 1-5 -Rs 60 Die Roll 6-10 -Rs 40 Die Roll 1-8 -Rs 100 Die Roll 9-10 Rs 0 J K
4 Die Roll 1-5 -Rs 60 Die Roll 6-10 -Rs 40 Die Roll 1-7 -Rs 100 Die Roll 8-10 Rs 0 J K
5 Die Roll 1-5 -Rs 60 Die Roll 6-10 -Rs 40 Die Roll 1-6 -Rs 100 Die Roll 7-10 Rs 0 J K
6 Die Roll 1-5 -Rs 60 Die Roll 6-10 -Rs 40 Die Roll 1-5 -Rs 100 Die Roll 6-10 Rs 0 J K
7 Die Roll 1-5 -Rs 60 Die Roll 6-10 -Rs 40 Die Roll 1-4 -Rs 100 Die Roll 5-10 Rs 0 J K
8 Die Roll 1-5 -Rs 60 Die Roll 6-10 -Rs 40 Die Roll 1-3 -Rs 100 Die Roll 4-10 Rs 0 J K
9 Die Roll 1-5 -Rs 60 Die Roll 6-10 -Rs 40 Die Roll 1-2 -Rs 100 Die Roll 3-10 Rs 0 J K
10 Die Roll 1-5 -Rs 60 Die Roll 6-10 -Rs 40 Die Roll 1 -Rs 100 Die Roll 2-10 Rs 0 J K
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Ambiguity Task (Gain)
Your choice of colour to bet on (circle your choice) GREEN BLUE
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Ambiguity Task (Loss)
Your choice of colour to bet on (circle your choice) RED YELLOW
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Results

• 1. Risk (Pooling sessions 1, 2, 3 and 4)
• Subjects are risk averse in both gains and
  losses, though they are more so in gains (as
  found in Holt and Laury, 2005). No reflection
  effect (contrary to what claimed by Kahneman and
  Tversky, 1979)

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Results (2)

• Observed Parameters, Risk Tasks over Gains and
  Losses
• Average r over gains 0.56 (risk averse)
• Average s over losses 0.65 (risk averse)
• Per subject number of safe choices (Gains) 6.14
• Per subject number of safe choices (Losses)
  5.62
• These are different using both parametric (paired
  t-test, p. value 0.0005) as well as
  non-parametric (Wilcoxon test, p. value 0.0004)
  
• Similar to Holt and Laury (2005)

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Results (3)

• Ambiguity (Pooling sessions 1, 2, 3 and 4)
• Subjects are risk averse in the domain of gains
  but mildly seeking in the domain of losses. So,
  there is a weak reflection effect.
• Average a over gains 0.99 (ambiguity averse)
• Average b over losses 0.99 (ambiguity seeking)
• We cannot use the number of non-ambiguous choices
  to compare behaviour as the neutral flip point is
  6 for gains and 7 for losses.

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Results (4)

• We use instead ambiguity preference scores
• Define the Ambiguity Preference Scores
• SG (6- Observed Switch Point) for gains and
• SL (7- Observed Switch Point) for losses,
• Where SG -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5
  and
• SL -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,
  6
• Average SG -0.48
• Average SL 0.67
• These are different over gains and losses using a
  parametric 2-sided paired t-test (0.0000) and a
  non-parametric 2-sided Wilcoxon test (0.0000)

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Results (5)

• The Ellsberg Decision
• We also compare the distribution of choices
  between the ambiguous and non-ambiguous prospects
  at the neutral point (decision 5 for gains and 6
  for losses) where the same prize amounts (0 and
  100 for gains and -100 and 0 for losses) result
  from losing or winning the bet, when drawing from
  either the non-ambiguous (0.5, 0.5) urn or the
  ambiguous (1, 0) (0, 1) urn.
• Pooling all four sessions,
• 71 out of 85 subjects (84) chose the
  non-ambiguous prospect at the neutral decision in
  gains.
• 40 out of 85 subjects (47 ) chose the
  non-ambiguous prospect in losses.
• Pooling all observations at this neutral point,
  the choice of the non ambiguous prospect over
  gains statistically and significantly exceeds
  that over losses at the one percent level using
  both parametric and non-parametric tests (p-value
  0.0000).
• Similar results to Einhorn and Hogarth (1986)

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Results (6)

• 4. Risk-Ambiguity connection
• Pooling all observations over all four sessions,
  we find a positive and significant relationship
  between the attitude to ambiguity and the
  attitude to risk over the domain of gains Cor
  (r, a) 0.36, p-value 0.0008 but no such
  significant relationship over the domain of
  losses.
• Risk and ambiguity attitudes have been in the
  past reported to be uncorrelated in studies by
  Cohen et al. (1985) and Einhorn and Hogarth
  (1990).

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Reflection Effects
Risk Tasks
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Reflection Effects (2)
GAINS
AVERSE NEUTRAL SEEKING
AVERSE 44.7 0 7.05
LOSSES NEUTRAL 0 0 0
SEEKING 22.35 0 25.88
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Reflection Effects (3)
Ambiguity
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Reflection Effects (4)
GAINS
AVERSE AVERSE NEUTRAL SEEKING
AVERSE 42.35 42.35 0 4.7
LOSSES NEUTRAL 0 0 0 0
SEEKING 28.23 28.23 0 24.71
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Conclusions

• RISK
• In the aggregate over risky prospects, subjects
  are risk averse over both losses and gains, so no
  reflection effect.
• However subjects are more risk averse over gains
  compared to losses.
• When individual behaviour is examined, almost 30
  of the subjects do display a reflection effect,
  majority of who are averse in gains and seeking
  in losses.

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Conclusions (2)

• AMBIGUITY
• In the aggregate, subjects are ambiguity averse
  over gains and ambiguity seeking over losses, so
  there is a weak reflection effect.
• This aggregate reflection effect is confirmed at
  the individual level with almost 30 displaying
  ambiguity aversion over gains and ambiguity
  seeking over losses.
• RISK/AMBIGUITY CONNECTION
• Attitudes towards risk and ambiguity are
  positively correlated over the domain of gains
  and almost unrelated over the domain of losses.

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Future Research

• Develop procedures that allow us to estimate
  subjects beliefs over the likelihood of
  occurrence of an event in a situation of
  ambiguity.
• Study the risk and ambiguity connection in a
  deeper way.