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HandbookDecision-Making Under UncertaintyAdvanced Topics Prospect Theory - IntroductionPrinter FriendlyDecision-Making Under Uncertainty - Advanced TopicsAn Introduction to Prospect TheoryOver time, researchers have become all too aware of the limitations of expected utility theory, especially those raised by theSt. Petersburg,Allais, andEllsbergparadoxes. As a result, numerous alternative theories have been developed to overcome the limitations of expected utility theory without losing its explanatory power. Prospect theory, developed by Daniel Kahneman and Amos Tversky is perhaps the most well-known of these alternative theories. This section covers the following topics:1. Motivation - certainty, reflection, and isloation effects2. The editing and evaluation phases3. Properties of the value and weighting functions4. Cumulative Prospect TheoryMotivationIn 1979, Daniel Kahneman and Amos Tversky conducted a series of thought experiments testing theAllais Paradoxin Israel, at the University of Stockholm, and at the University of Michigan. Everywhere, the results followed the same pattern. The problem was even framed in many different ways, with prizes involving money, vacations, and so on. In each case, thesubstitution axiomwas violated in exactly the same pattern. Kahnemann and Tversky called this pattern thecertainty effect- meaning, people overweight outcomes that are certain, relative to outcomes which are merely probable.Using the term prospect to refer to what we have so far called lotteries or gambles, (i.e. a set of outcomes with a probability distribution over them), Kahnemann and Tversky also state that where winning is possible but not probable, i.e. when probabilities are low, most people choose the prospect that offers the larger gain. This is illustrated by the second decision stage in theAllais Paradox.More generally, if x and y are outcomes; 0 p,q,r 0 and p + q = 1.It is strictly negative if all its outcomes are negative.It is regular if it is neither strictly positive, nor strictly negative.So for a regular prospect, i.e. either p + q y 0, or x y 0, then:V(x, p; y, q) = v(y) + (p)v(x) - v(y),so the value of a strictly positive or strictly negative prospect equals the value of the riskless component plus the differences between the values of the two outcomes, multiplied by the weight associated with the more extreme outcome. Note that a decision weight is applied only to the risky component, not the riskless one.Example:V(400,0.25; 100,0.75) = v(100) + (0.25)v(400) - v(100)Note that a decision weight is applied to the value difference v(x) - v(y), but not to the riskless component, v(100).Also note that the equation above reduces to (p)v(x) + 1-(p)v(y). This reduces to (p)v(x) + (q)v(y), the equation for a regular prospect, if (p) + (1-p) = 1. However, this is not generally satisfied.While the prospect theory equations appear to resemble those of expected utility theory, the crucial differences are:1. Values are attached to changes, rather than final states, and2. The decision weights need not coincide with probabilities.Next:The value and weighting functionsMore on value functionsThe emphasis on changes as the carriers of value does not mean that the value of a particular change is independent of the initial position. Value functions are likely to become more linear with increases in assets. A change from $100 to $200 is likely to have a much higher value than a change from $1100 to $1200. The value function is then concave above the reference point (v(x) 0), and convex below it (v(x) 0 for x y0, then (y,0.5; -y,0.5) is preferred to (x,0.5; -x,0.5).This means that v(y) + v(-y) v(x) + v(-x). Setting y = 0 gives us v(x) -v(-x), and letting y approach x gives us v(x) p for very small p.Cumulative Prospect TheoryIn their 1992 paper, Kahneman and Tversky developed an updated form of prospect theory, which they termedCumulative Prospect Theory. The theory incorporates rank-dependent functionals which transform cumulative, rather than individual probabilities, in response to a growing literature, and satisfies stochastic dominance, which the original form of prospect theory does not. It retains the most crucial features, though, viz.:1. Gains and losses, i.e. income is the carriers of value, not final assets or wealth.2. The value of each outcome is multiplied by a decision weight, not an additive probability.The Cumulative FunctionalA risky prospect f is a function that maps states of the world sS into consequences xX. Outcomes of prospects are arranged in increasing order, so that a prospect f is represented by a sequence of pairs (xi, Ai), and yields xiif Aioccurs, where Aiis a partition of S.A prospect is calledstrictly positiveorstrictly negativeif all its outcomes are positive or negative respectively. It is labeledpositiveornegativeif all its outcomes are non-negative or non-positive, respectively. All other prospects are calledmixed. We have the positive part of f, f+(s) = f(s) if f(s) 0, and f+(s) = 0 if f(s)0. The negative part of f, f-(s) = f(s) if f(s) 0, and f-(s) = 0 if f(s)0.A number V is assigned to each prospect, so that fg if & only if V(f)V(g).AcapacityW is a function that assigns to each AS a number W(A) satisfying W() = 0, W(S) = 1, & W(A)A(B) when ever AB.There exists a strictly increasing value function v:X, satisfying v(x0) = v(0) = 0, and capacities W+& W-, such that for f = (xi, Ai), -min:V(f) = V(f+) + V(f-)V(f+) =nv=0+iV(f-) =nv=-m-iIf the prospect f is defined by a probability distribution p(Ai) = pi, then the decision weights+(f+) = (+0.+n) and-(f-) = (-m.-0) are defined by:+n= w+(pn)-m= w-(p-m)+i= w+(pi+ . + pn) - w+(pi+1+ . + pn), 0i1;-i= w-(p-m+ . + pi) - w-(p-m+ . + pi-1), 1-mi0.where w+& w-are strictly increasing functions from the unit interval into itself, & they satisfy w+(0) = w-(0) = 0, & w+(1) = w-(1) = 1.Explaining the formulationThe decision weight+iis the difference between the capacities of the events the outcome is at least as good as xi, and the outcome is strictly better than xi.The decision weight-iis the difference between the capacities of the events the outcome is at least as bad as xi, and the outcome is strictly worse than xi.In this manner, the decision weight associated with each outcome is the marginal contribution of the event defined in terms of the capacities. If each capacity W is additive, it is a probability measure & soiis simply the probability of Ai.For both positive and negative prospects, the decision weights add to 1, but the sum can be greater or less than 1 for mixed prospects. This is because the decision weights for gains & losses are defined by different capacities.Additional conceptsAs in the original version of prospect theory, v is assumed conc