Uncertainty and Risk - DARP

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Microeconomics CHAPTER 8. UNCERTAINTY AND RISK Exercise 8.6 An example to illustrate regret. Let P := f(x ! ; ! ) : ! 2 g P 0 := f(x 0 !; ! ) : ! 2 g be two prospects available to an individual. De…ne the expected regret if the person chooses P rather than P 0 as X ! max fx 0 ! x ! ; 0g (8.1) !2 Now consider the choices amongst prospects presented in Exercise 8.4. Show that if a person is concerned to minimise expected regret as measured by (8.1), then it is reasonable that the person select P 2 when P 1 is also available and then also select P 4 when P 3 is available. Outline Answer: Denote the regret in (8.1) by r (P; P 0 ). If I choose P 2 when P 1 is also available then the regret is r (P 2 ; P 1 ) = 0:1 [0] + 0:89 [0] + :01 [1] = 10; 000 Whereas, had I chosen P 1 when P 2 was available, then the regret would have been r (P 1 ; P 2 ) = 0:1 [4; 000; 000] + 0:89 [0] + :01 [0] = 400; 000 If I choose P 4 when P 3 is also available then the regret is r (P 4 ; P 3 ) = 0:1 [0] + 0:89 [0] + :01 [0] = 0 Whereas, had I chosen P 3 when P 4 was available, then the regret would have been r (P 3 ; P 4 ) = 0:1 [0] + 0:89 [0] + :01 [5; 000; 000] = 50; 000 cFrank Cowell 2006 120
Microeconomics Exercise 8.7 An example of the Ellsberg paradox . There are two urns marked Left and Right each of which contains 100 balls. You know that in Urn L there exactly 49 white balls and the rest are black and that in Urn R there are black and white balls, but in unknown proportions. Consider the following two experiments: 1. One ball is to be drawn from each of L and R. The person must choose between L and R before the draw is made. If the ball drawn from the chosen urn is black there is a prize of $1000, otherwise nothing. 2. Again one ball is to be drawn from each of L and R; again the person must choose between L and R before the draw. Now if the ball drawn from the chosen urn is white there is a prize of $1000, otherwise nothing. You observe a person choose Urn L in both experiments. Show that this violates the Revealed Likelihood Axiom. Outline Answer The implication of the revealed likelihood axiom is that there exist subjective probabilities ! . The result is proved by showing that it the stated behaviour is inconsistent with the existence of subjective probabilities. In this case the revealed likelihood axiom implies that for each urn there is a given subjective probability of drawing a black ball L (left-hand urn) and R (right-hand urn) such that preferences can be represented as v black (x black ; x white ) + [1 ] v white (x black ; x white ) (8.2) where = L or R and x black and x white are the payo¤s if a black ball or a white ball are drawn respectively. Note that the representation (8.2) does not impose either the State Irrelevance Axiom (which would require that v black () and v white () be the same function) or the Independence axiom (which would require that v black () be a function only of x black etc.). Nor does it impose the common-sense requirement that L = 0:49. All we need below is the very weak assumption that preferences are not perverse: v black (1000; 0) > v white (1000; 0) (8.3) and v black (0; 1000) < v white (0; 1000) (8.4) Condition (8.3) simply says that if the $1000 prize is attached to a black ball then the utility to be derived from having selected a black ball is higher than selecting a white ball; condition (8.4) is the counterpart when the prize attaches to the white ball.. Experiment 1 suggests that L v black (1000; 0) + [1 L ] v white (1000; 0) > R v black (1000; 0) + [1 R ] v white (1000; 0) (8.5) cFrank Cowell 2006 121
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