Lecture 28 Cumulative Prospect Theory I

  • Stochastic Dominance
    • \(A = (x_{1},p_{1};\dots,x_{n},p_{n})\)
    • \(B = (y_{1},p_{1};\dots;y_{n},p_{n})\)
    • \(B\) weakly stochastically dominates \(A\) if \(x_{i} \leq y_{i}\) for all \(i\)
    • strictly if the above definition PLUS \(x_{i} \lt y_{i}\) for some \(i\)
    • An example of 4 gambles
      • B dominates A but C was preferred to D
      • Though \(C \sim A\) and \(D \sim B\)
      • Due to ambiguity in presentation
      • But when \(G\) is introduced, people started to \(A \succ G \succ B\) \(\implies\) \(A \succ B\)
      • PT fails to explain this: "Why are dominated prospects eliminated in the editing phase?"

  • Cumulative Prospect Theory (CPT)

  • Framing effect ^89062f

    • Rational theory of choice assumes Description invariance
    • But there are variations due to framing options
  • Non-linear preferences ^1cf13b
    • Utility of a risky prospect is linear in outcome probabilities
    • Allais's famous example: probability difference 0.99 and 1.00 has more impact on preferences than 0.10 and 0.11.
    • \(\implies\) Non-linear preferences
    • Major difference between PT & CPT: allowance for different weights for gains and losses
  • Source Dependence - Ellsberg Paradox^9b4bb6
    • Willingness to be on an uncertain outcome depends on
      • degree of uncertainty
      • source of uncertainty
    • Ellsberg observed that people prefer betting on an urn contain equal number of red and green balls rather than unknown proportions
    • More recent evidence: people are willing to bet on event in their area of competence than on a matched chance event (where they have uncertainty)1
    • Ambiguity Aversion
      • Urn 1 and 2
      • Gamble A: unknown proportion
      • Gamble B: 50-50
      • Preference is inconsistent, with EUT
        • People generally choose the urn where they are more certain about the outcome, even if it contrasts their prior beliefs.
  • Risk Seeking ^58969d
    • These choices are consistently observed in:
      1. People prefer a small probability of winning a large price \(\gt\) expected value of that prospect
      2. Choosing between a sure loss \(\lt\) substantial probability of a larger loss
  • Loss aversion ^8dfee5
    • Losses loom larger than gains in case of risk and uncertainty
    • Slope of utility function is more steeper in the domain of losses than that of gains.
  • The New Decision Weights Function
    • Lecture 27__The Weighting Function#^2432a8
    • Solved by rank-dependent or cumulative functional proposed by Quiggin (decision under risk) and Schmeidler (decision under uncertainty3)
    • Revised model transforms the entire cumulative distribution function
    • Applies cumulative functional2 to losses and gains separately
    • Setup
      • \(S:\) set of events. Each state \(s \in S\) has the consequence \(x \in X\)
      • Uncertain prospect \(f\), sequence of pairs \(f(s) = f(x_{i},A_{i})\)
        • \(x_{i}\) yields if \(A_{i}\) occurs
        • Outcomes are Sorted in increasing order \(x_{i} \gt x_{j} \iff\) \(i \gt j\)
        • \(A_{i}\) is a partition of \(S\)4
      • If \(f(s)\) is given by a probability \(\implies\) "risky" prospect
        • \(p(A_{i})= p_{i}\)
        • \((x_{i},p_{i})\)
      • Value function has two components
        • \(V(f) = V(f^+)+V(f^-)\)
          • \(f^+\) and \(f^-\) are positive and negative parts of the prospects
          • \(f^+(s) = f(s)\) if \(f(s) \gt 0\) else \(f^+(s) = 0\)
          • \(f^-(s) = f(s)\) if \(f(s) \lt 0\) else \(f^-(s) = 0\)
    • For a mixed or regular prospect (\(* = +/-\))
      • \(V(f^*) = \sum_{i=0}^n \pi_{i}^* v(x_{i})\)
    • Decision weights:
      • For \(0,1,2\dots i\dots n-1\)
      • \(\pi_{i}^+ = w^+(p_{i}+\dots+p_{n})- w^+(p_{i+1}+\dots+p_{n})\)
      • For \(1-m\dots,i,\dots-3,-2, -1, 0\)
      • \(\pi_{i}^- = w^-(p_{-m}+\dots+ p_{i})- w^-(p_{-m}+\dots+p_{i-1})\)
      • And
        • \(\pi_{n}^+ = w^+(p_{n})\) and
        • \(\pi_{-m}^- = w^- (p_{-m})\)
      • \(w^+\) and \(w^-\) are strictly increasing functions which are \(w^*(0)=0\) and \(w^*(1) = 1\)
    • Example
      • Win even \(x\) or lose odd \(x\)
      • consequences of \(f\) = \((-5,-3,-1, 2,4, 6)\)
        • So, \(f^+ = \left(0, \dfrac{1}{2}; 2, \dfrac{1}{6}; 4, \dfrac{1}{6}; 6, \dfrac{1}{6}\right)\)
        • and, \(f^+ = \left(-5, \dfrac{1}{6}; -3, \dfrac{1}{6}; -1, \dfrac{1}{6}; 0, \dfrac{1}{2}\right)\)
      • What cumulative means...
      • Then evaluate the entire value function by adding the two components \(V(f) = V(f^+) + V(f^-)\)

  1. Even though the former probability is vague. ↩

  2. A functional is just a function that takes in a function as an argument (e.g. a curve, is represented by \(f(x) = 2x^2+4\)) and returns a single number as result (e.g. the arc length) ↩

  3. A #doubt what is the difference between risk and uncertainty - I think "uncertain" means, we don't know the probabilities - When probabilities are known, it is referred to as "risky" ↩

  4. Partitions are mutually exclusive and their unions form \(S\) ↩