1 BAMS 517 – 2011 Decision Analysis -IV Utility Failures and Prospect Theory Martin L. Puterman UBC Sauder School of Business Winter Term

2 Two lotteries – Lottery I Choice A Choice A* 10 Million 0 50 Million 10 Million Would you choose Lottery A or A* ? For real!

3 Two lotteries – Lottery II Choice B* Choice B Million 0 50 Million Would you choose Lottery B or B* ? For real!

Behavioral Results In experiments, most people choose A in I and B* in II.  Clemen and Reilly (p.580) report 82% preferred A to A* while 83% preferred B* to B.  Did you?  Argue why this might make sense? Unfortunately this behavior is inconsistent with using expected utility to evaluate gambles! 4

Utility calculation for previous example From B and B*.9u(0)+.1u(50 mill) >.89u(0)+.11u(10 mill).01u(0) +.89u(10 mill) +.1u(50 mill) > u(10 mill) This last inequality is inconsistent with preferring A to A*. So in practice expected utility is breaking down here. This is often referred to as Allais’ paradox So alternatives to expected utility may be necessary. 5

6 Lottery I rewritten Choice A* Choice A Million 50 Million 0 1/11 10/11

7 Lottery II rewritten Choice A* Choice b Million 0 50 Million 0 1/11 10/11

8 What is utility? Consistency Axioms for outcomes and preferences For any outcomes x,y,z,w and numbers p,q between 0 and 1 the following hold: 1. Weak ordering. (a) x >~ x. (Reflexivity) (b) x >~ y or y >~ x. (Connectivity) (c) x >~ y and y >~ z imply x >~ z. (Transitivity) 2. Reducibility. ((x,p,y),q,y) ~ (x,pq,y). 3. Independence. If (x,p,z) ~ (y,p,z), then (x,p,w) ~ (y,p,w). 4. Betweenness. If x > y, then x > (x,p,y) > y. 5. Solvability. If x > y > z, then there exists p such that y ~ (x,p,z).

Independence Axiom revisited Also called Independence of Irrelevant Alternatives Equivalent statements 1. If (x,p,z)~(y,p,z) then (x,p,w)~(y,p,w) 2. If (x,p,z)>~(y,p,z) then (x,p,w)>~(y,p,w) 3. If x~y then (x,p,z)~(y,p,z) for any p and z 4. If x>~y then (x,p,z)>~(y,p,z) for any p and z If the independence axiom holds (in forms 1. or 2.) then decision makers who follow these axioms should be indifferent between lotteries I and II.  To see this, take x to be 10 million for sure and y to be a lottery which yields 0 with probability 1/11 and 50 million with probability 10/11. Thus peoples choice between gambles may depend on their reference outcome.  Kahneman and Tversky call this framing. 9

Another problem Question: Imagine that the Canada is preparing for the outbreak of an unusual disease which is expected to kill 600 people. Two alternative programs to combat the disease have been proposed. Assume the exact scientific estimate of the consequences of the programs are as follows.“  Program A: "200 people will be saved"  Program B: "there is a one-third probability that 600 people will be saved, and a two-thirds probability that no one will be saved“ Or instead:  Program C: "400 people will die"  Program D: "there is a one-third probability that nobody will die, and a two-third probability that 600 people will die“ In studies 72% preferred A to B while 78% preferred D to C. Does this make sense? 10

New theories These are just two of the problems faced when trying to determine whether people’s behavior is consistent with expected utility theory. Alternatives  Prospect theory  Cumulative prospect theory Features  A “prospect” is a gamble; that is a series of outcomes with a probability associated with each.  A value function v() that is risk averse for gains, risk seeking for losses and anchored at 0 so that v(0)=0  A weighting of probabilities π() that overweights low probabilities and underweights high probabilities Decisions are evaluated in terms of π(p 1 )v(o 1 ) + …+ π(p n )v(o n ) when there are n outcomes. Prospect theory and cumulative prospect theory construct π() differently. 11

Sample Value functions and Weight Functions from Cumulative Prospect Theory 12 Question: “Should we model the way people behave and use it for decision making or develop a model of how they should behave and encourage them to use it?” A Value Function A Weighting Function

Concluding Comment 13 ‘Prospect theory has probably done more to bring psychology into the heart of economic analysis than any other approach. Many economists still reach for the expected utility theory paradigm when dealing with problems, however, prospect theory has gained much ground in recent years, and now certainly occupies second place on the research agenda for even some mainstream economists. Unlike much psychology, prospect theory has a solid mathematical basis — making it comfortable for economists to play with. However, unlike expected utility theory which concerns itself with how decisions under uncertainty should be made (a prescriptive approach), prospect theory concerns itself with how decisions are actually made (a descriptive approach). Prospect theory was created by two psychologists, Kahneman and Tversky, who wanted to build a parsimonious theory to fit a number of violations of classical rationality that they (and others) had uncovered in empirical work. Prospect theory bears more than a passing resemblance to expected utility theory.’ MONTIER, James, Darwin’s Mind: The Evolutionary Foundations of Heuristics and Biases. Dresdner KleinwortWasserstein – Global Equity Strategy.Darwin’s Mind: The Evolutionary Foundations of Heuristics and Biases