Lecture 24 Anomalies of EUT & Prospect Theory

Sairam

• Probabilistic Insurance
  • Situation
    • Possibility: insure property against damage (fire/theft)
    • After examining the risks \(\text{Purchase Insurance} \sim \text{Leave Uninsured}\)
    • Offering a new program: Probabilistic Insurance. Pay only 50% of the premium now
    • In case of damage, there is a 50-50 chance
      • Company will ask you to pay the rest of the premium and they will cover all the losses
      • Company will pay you back the premium and you have to cover all the losses
    • Will you take the insurance? 20% said "Yes", 80% (including Sudarshan) said "No"
  • More examples of PI
    • Installation of burglar alarm (You won't know whether it will be of use or not)
    • Replacement of old tires (You won't know which accident you avoided by changing this tire)
    • Stop smoking (You won't know whether you actually were going to get cancer)
  • Generally, PI is unattractive. Why? Let \(\hat{p}\) be probability of loss
    • "Reducing \(\hat{p}\) from \(p \to \dfrac{p}{2}\)" \(\prec\) "Reducing \(\hat{p}\) from \(\dfrac{p}{2}\to 0\)"
  • But EUT \(\implies\) PI >>> Regular/Contingent² Insurance, intuitively.
  • \(\implies\) Intuitive notion of risk is not adequately captured by assumed concavity of utility function of Wealth.
    • Crazy: because all insurance is, in a sense probabilistic.
  • \(\implies\) Two prospects that are equivalent in probabilities and outcomes could have different values, depending on formulation¹
• Intro to Prospect Theory
  • Empirical evidence \(\implies\) EUT might just be a descriptive model (fails in many situations)
  • Many new theories since 1970s, but PT dominates
  • Developed for simple prospects with monetary outcomes and stated probabilities. Can be extended!
  • Choice Process = Phase 1 + Phase 2
    • Phase 1: Editing \(\to\) Prelim analysis of offered prospects... simpler representation
    • Phase 2: Evaluation \(\to\) Edited (simplified) prospects are evaluated and the one with the highest value is chosen.

Prospect Theory - Editing

1. Coding:
   • Coding of Outcomes as gains/losses**
   • Perception of gains/losses (relative to a neutral reference point) rather than final outcome/state
   • Reference point \(\leftrightarrow\) Current asset position \(\implies\) Gains and losses = Actual amount gained/lost
     • can be affected by the formulation

2. Reference Point

   • Some attributes are more accessible than others (natural assessments: size, distance, loudness, temperature etc)
   • Natural assessment will usually be relative ("I know this is bigger" without actually knowing the exact dimensions)
   • To judge this relative magnitude .
   • Absolute vs relative
     • Carol: 4 mil \(\to\) 3 mil
     • Amanda: 1 mil \(\to\) 1.1 mil
     • SE says: \(u(3) > u(1.1)\) so Carol should be more happy
     • However, most will think that Carol lost \(1\) million but Amanda gained \(.1\) mil. So the latter will be happier

3. Combination

   • Simplify prospects by combining probabilities of identical outcomes. \((100, 0.30; 100, 0.30) \to (100,0.60)\)
   • Combining = Coalescing
4. Segregation
   • Any Prospect = riskless component + risky component
     • \((100, 0.7; 150, 0.3) \to (100) + (50,0.3)\)
     • \((-200, 0.8; -300, 0.2) \to (-200) + (-100,0.2)\) 4. Cancellation
5. Cancellation
   • Can be applied to a set of two+ (nested) prospects
   • Different prospects share identical components, which are discarded or ignored
   • Two-stage game
     • 0.75 end without winning
     • 0.25 move to second stage with a choice
       • \((4000, 0.80)\)
       • \((3000)\)
     • The choice must be made at the start of the game, respondents ignored the first stage of the sequential game
     • And evaluate based on the results of the second stage (based on the assumption that they have already got past the first stage)³
   • Another type of cancellation: Discarding of common constituents
     • Choose between \((200, 0.2; 100, 0.5; -50, 0.3)\) and \((200, 0.2; 150, 0.5 ;-100, 0.3)\)
     • Reduced to \((100, 0.5; -50, 0.3)\) and \((150, 0.5; -100, 0.3)\)
     • Thus, ignoring \(200,0.2\)
   • Another example: The bonus (1000 or 2000 that we were already given were being ignored while the question was posed, which presented the same expected values)
6. Simplification
   • Simplify prospects by rounding off outcomes or probabilities
   • \((99,0.51)\) or \((101, 0.49)\) can be coded as \((100,0.50)\)
   • Important form: Discarding of extremely unlikely outcomes
7. Detection of Dominance
   • Detect dominated alternatives (rejected without further evaluation)⁴
   • Dominating prospects may have elements in common with others, but other elements involve preferred outcomes and probabilities
   • \((200,0.3;99,0.51)\) and \((200,0.4, 101,0.49)\)
     • Rounding off the second component of each prospect, we get: \((200, 0.3; 100, 0.50)\) and \((200,0.4, 100, 0.50)\). Thus the first one dominates the second.
8. Prospect Theory - Editing
   • The Anomalies of preference result from the editing of prospects. E.g.
     • Inconsistency of Isolation Effect \(\Leftarrow\) Cancellation of common components
     • Intransitivity of choice \(\Leftarrow\) Simplification that eliminates small differences
   • Generally,
     • Preference order is not invariant across contexts
     • The same offered prospects could be EDITED in different ways.

---

1. Contingent insurance is generally more attractive than probabilistic insurance when the probabilities of unprotected loss are equated. ↩

2. Provides Coverage for a specific type of risk \(\implies\) Contingent. ↩

3. Something similar happens in life, when we are looking at ourselves in the future, doing some important work in some huge organization or research institution etc, but we completely ignore the first stage of the hard work that would be required to get ourselves to that stage. ↩

4. When I am in front of an ocean of jewels, why should I ask for glass beads? I asked for wisdom and peace... ↩