Prospect Theory for Risk and Uncertainty PDF

Decision Theory–Part II Prospect Theory for Risk and Uncertainty Ferdinand M. Vieider Department of Economics, Ghent University Ferdinand M. Vieider Decision Theory–Part II Introduction Markowitz-EU is violated by probability variations; dual-EU is violated by...

Decision Theory–Part II Prospect Theory for Risk and Uncertainty Ferdinand M. Vieider Department of Economics, Ghent University Ferdinand M. Vieider Decision Theory–Part II Introduction Markowitz-EU is violated by probability variations; dual-EU is violated by stake variations (‘dual violation’) One can combine the two to obtain Prospect Theory: probability distortions and reference dependence Prospect theory for more than two outcomes: good news probabilities and rank-dependence Revisiting the paradoxes: framing, common ratio and common consequence effects, lottery and insurance, the equity premium puzzle, response mode effects, Ellsberg paradox Limitations of prospect theory: gain-loss separability; rank-dependence; what is the reference point? CASE STUDY: are preferences stable? Evidence of environmental adaptability. Ferdinand M. Vieider Decision Theory–Part II Index 1 Setting the stage: Violations of Dual-EU 2 Prospect theory 3 CASE STUDY: Utility around the world 4 CASE STUDY: Measuring PT parameters 5 Multi-outcome prospects and rank dependence 6 Revisiting the paradoxes 7 Decisions under uncertainty and ambiguity aversion 8 The standard model of time discounting 9 Paradoxes for risk and for time 10 Conclusion Ferdinand M. Vieider Decision Theory–Part II Dual expected utility: Stake effects w(p) 1 0.79 0.69 0.48 0.38 0.21 0.14 0 0.1 0.5 0.9 1 p Dual-EU weighting functions for £100 or 0 versus £10, 000 or 0 Ferdinand M. Vieider Decision Theory–Part II The Dual versus the Primal Just as Markowitz-EU was violated by changes in probability, dual-EU is violated by changes in outcomes Up to £10,000, relative risk aversion increases relatively slowly over stakes; the increase is much quicker over probabilities When representing preferences over moderate stakes, Markowitz is thus more severely violated than dual-EU This may not hold true when stakes become large (>10m), where the violations of the dual may become more important The trouble is always caused by the dimension being treated linearly. SOLUTION: treat both dimensions nonlinearly Ferdinand M. Vieider Decision Theory–Part II Index 1 Setting the stage: Violations of Dual-EU 2 Prospect theory 3 CASE STUDY: Utility around the world 4 CASE STUDY: Measuring PT parameters 5 Multi-outcome prospects and rank dependence 6 Revisiting the paradoxes 7 Decisions under uncertainty and ambiguity aversion 8 The standard model of time discounting 9 Paradoxes for risk and for time 10 Conclusion Ferdinand M. Vieider Decision Theory–Part II Motivation Both M-EU and Dual-EU could account for some of the paradoxes, but neither could account for all of them M-EU and Dual-Eu were violated in some simple settings (the first for probability variations, the second for stake variations) In 1979, Kahneman and Tversky published the first ‘prospect theory’ paper in Econometrica The idea of prospect theory is to combine elements from M-EU and Dual-EU into one integrated framework: 1 Likelihood-dependence: Probabilities are subjectively transformed into decision weights 2 Reflection: small probabilities are over-weighted and large probabilities underweighted for both gains and losses 3 Decreasing sensitivity: Utility is defined over changes in wealth, and is concave for gains, convex for losses 4 Loss aversion: ‘losses loom larger than gains’ However: theoretical apparatus requires some amendments Ferdinand M. Vieider Decision Theory–Part II Gain-loss separability Given that utility is defined over changes in wealth, gains, losses and mixed prospects may be treated differently Assume an indifference z ≥ (x , p; y ). For x > y Ø 0, this may be represented as: v (z) = w (p)v (x ) + [1 ≠ w (p)]v (y ) For losses, the representation is the same whenever x < y Æ 0 (loss aversion parameter drops out of the equation) For mixed prospects, the representation consists of the two parts for gains and losses. For x > 0 > y we can thus write: v (z) = w (p)v (x ) + [1 ≠ w (p)]v (0) ≠ ⁄[w (1 ≠ p)v (≠y ) + [1 ≠ w (1 ≠ p)]v (0)] Here ⁄ indicates loss aversion; this is known as gain-loss separability: preferences over mixed composed of gain and loss preferences Ferdinand M. Vieider Decision Theory–Part II PT stylised facts: probability weighting Small probabilities of gains are overweighted: risk seeking for small probability gains Large probabilities of gains are underweighted: risk aversion for moderate to large probability gains Small probabilities of losses are overweighted: risk aversion for small probability losses Large probability losses are underweighted: risk seeking for moderate to large probability losses This results in a four-fold pattern of risk preferences under v (x ) = x However, preferences now also depend on utility curvature Ferdinand M. Vieider Decision Theory–Part II PT stylised facts: utility curvature Utility is defined on changes in wealth, and is concave for gains, contributing to risk aversion for large gains Utility is defined on changes in wealth, and is convex for losses: contributes to risk seeking for large losses Utility for losses is steeper than utility for gains, resulting in loss aversion: losses weigh about twice as heavy as gains Utility curvature, loss aversion, and probability weighting all contribute to risk preferences depending on choice Risk seeking for small stakes traditionally ignored, so that utility presents a uniform pattern (‘peanut effect’) This can, however, be accommodated by a suitable utility function without additional degrees of freedom Ferdinand M. Vieider Decision Theory–Part II Typical utiity function Ferdinand M. Vieider Decision Theory–Part II Index 1 Setting the stage: Violations of Dual-EU 2 Prospect theory 3 CASE STUDY: Utility around the world 4 CASE STUDY: Measuring PT parameters 5 Multi-outcome prospects and rank dependence 6 Revisiting the paradoxes 7 Decisions under uncertainty and ambiguity aversion 8 The standard model of time discounting 9 Paradoxes for risk and for time 10 Conclusion Ferdinand M. Vieider Decision Theory–Part II Adding utility to worldwide risk preference patterns We have seen many commonalities as well as difference in aggregate risk preferences around the world Used dual M-EU: how would patterns change if we allowed for utility curvature in addition to probability weighting? Identification: the tasks employed have changes in outcomes as well as across probabilities, so this is possible Expectation: since the stake ranges are moderate, adding utility should change relatively little in the M-EU results Ferdinand M. Vieider Decision Theory–Part II Some typical utility function Ferdinand M. Vieider Decision Theory–Part II Changes in pessimism for gains Germany Australia United States Spain South Africa Czech Republic Poland India Belgium Chile Costa Rica China Russian Federation France Japan Thailand Colombia Guatemala Kyrgyz Republic Tunisia Brazil Malaysia Cambodia Vietnam United Kingdom Saudi Arabia Ethiopia Peru Nigeria Nicaragua 0.2.4.6.8 1 mean of beta mean of beta_pt Ferdinand M. Vieider Decision Theory–Part II Changes in sensitivity for gains United Kingdom Guatemala Thailand Spain United States Belgium Japan Czech Republic Brazil Germany Kyrgyz Republic China Russian Federation Malaysia Australia Poland Saudi Arabia Colombia France South Africa Costa Rica Vietnam Ethiopia Chile Nicaragua Peru Cambodia Tunisia India Nigeria 0.2.4.6.8 1 mean of alpha mean of alpha_pt Ferdinand M. Vieider Decision Theory–Part II Changes in optimism for losses Cambodia Guatemala India United Kingdom Russian Federation Thailand Kyrgyz Republic South Africa Nicaragua Ethiopia Costa Rica Colombia Poland Vietnam Tunisia Peru United States France Nigeria Japan Germany Brazil Spain Czech Republic China Saudi Arabia Belgium Malaysia Chile Australia 0.5 1 1.5 mean of delta mean of delta_pt Ferdinand M. Vieider Decision Theory–Part II Changes in sensitivity for losses Guatemala Thailand Japan Czech Republic United States France Brazil Germany Spain Poland Colombia Russian Federation Belgium Australia Vietnam China Costa Rica South Africa Chile Saudi Arabia Malaysia United Kingdom Kyrgyz Republic Peru India Tunisia Ethiopia Nicaragua Cambodia Nigeria 0.2.4.6.8 1 mean of gamma mean of gamma_pt Ferdinand M. Vieider Decision Theory–Part II Index 1 Setting the stage: Violations of Dual-EU 2 Prospect theory 3 CASE STUDY: Utility around the world 4 CASE STUDY: Measuring PT parameters 5 Multi-outcome prospects and rank dependence 6 Revisiting the paradoxes 7 Decisions under uncertainty and ambiguity aversion 8 The standard model of time discounting 9 Paradoxes for risk and for time 10 Conclusion Ferdinand M. Vieider Decision Theory–Part II Measuring PT parameters Results just shown require parametric assumptions and structural equation models to separately identify v and w This is a general drawback of prospect theory, or in general complex models with multiple preference parameters However, it is possible to identify the parameters non-parametrically using multiple stages and careful design This has advantages, but also drawbacks: cumbersome; difficult to incentivize; error propagation However; very instructive to look at; we will study the method of Abdellaoui (2000) in some detail Ferdinand M. Vieider Decision Theory–Part II Outcome intervals with equal utility We start from obtaining 6 outcome intervals that are equally spaced in terms of utility; for 0 Æ r < R < x0 , obtain x1 : p p x0 x1 ≥ R r 1≠p 1≠p Substitute x1 for x0 , and obtain x2 : p p x1 x2 ≥ R r 1≠p 1≠p The continue up to obtaining x6 : p p x5 x6 ≥ R r 1≠p 1≠p Ferdinand M. Vieider Decision Theory–Part II Standard outcome sequence Advantage of this method: probabilities drop out, so that it does not matter if they are subjectively distorted From first indifference, one obtains w (p)u(x0 ) + (1 ≠ w (p))u(R) = w (p)u(x1 ) + (1 ≠ w (p))u(r ) u(x1 )≠u(x0 ) 1≠w (p) After rearranging: u(R)≠u(r ) = w (p) u(xi )≠u(xi≠1 ) 1≠w (p) And in general: u(R)≠u(r ) = w (p) Substituting into each other, we get u(x6 ) ≠ u(x5 ) = u(x5 ) ≠ u(x4 ) =... = u(x1 ) ≠ u(x0 ) i Normalizing u(x0 ) = 0 and u(x6 ) = 1 we get: u(xi ) = 6 This gives us a fully non-parametric utility function up to x6 Ferdinand M. Vieider Decision Theory–Part II Probability equivalents The second step consists in obtaining five probability equivalents for the following tasks using a careful procedure: pi x6 ≥ xi x0 1 ≠ pi Assume p1 is the probability equivalent using x1 ; then w (p1 )u(x6 ) + (1 ≠ w (p1 ))u(x0 ) = u(x1 ) Since all utilities are known it is straightforward to obtain w (p1 ) = 16 , and so forth for x2 to x6 (i.e. p2 to p6 ) This gives us a non-parametric probability weighting function by plotting the known w (pi ) against the elicited pi This can be repeated for losses (loss aversion is more complex and not treated in this paper; the procedure is similar) Ferdinand M. Vieider Decision Theory–Part II Index 1 Setting the stage: Violations of Dual-EU 2 Prospect theory 3 CASE STUDY: Utility around the world 4 CASE STUDY: Measuring PT parameters 5 Multi-outcome prospects and rank dependence 6 Revisiting the paradoxes 7 Decisions under uncertainty and ambiguity aversion 8 The standard model of time discounting 9 Paradoxes for risk and for time 10 Conclusion Ferdinand M. Vieider Decision Theory–Part II OPT and stochastic dominance violations Take a prospect with three non-zero outcomes (x , p; y , q; z). Under original prospect theory this could be written as w (p)u(x ) + w (q)u(y ) + (1 ≠ w (p) ≠ w (q))u(z) This can violate stochastic dominance: Under SD, (x , p) º (y , q) implies (x Õ , p Õ ) º (y , q) if x Õ Ø x or p Õ Ø p; SD violations are undesirable, violating some fundamental rationality principles and threatening mathematical tractability Quiggin (1982): instead of transforming raw probabilities, transform ‘good news probabilities’ This solution is incorporated in all modern version of prospect theory and rank dependent utility (dual-EU with utility curvature) Ferdinand M. Vieider Decision Theory–Part II OPT and multi-outcome prospects x1 x2 x3 xn p1 p2 p3...... pn 1 Ferdinand M. Vieider Decision Theory–Part II Stochastic dominance violations In general, we will have w (p1 + p2 ) ”= w (p1 ) + w (p2 ) In the case of w (p1 + p2 ) > w (p1 ) + w (p2 ): w (pn ) w (pn ) w (pn ) w (p3 ) w (p3 ) w (p3 ) w (p2 ) w (p2 ) w (p2 ) w (p1 + p2 ) w (p1 ) w (p1 ) w (p1 ) xn x3 x2 x1 xn x3 x2x1 xn x3x1 = x2 (a) (b) (c) a) depicts a dual-EU case with transformed probabilities b) x1 lowered to approach x2 ; nothing happens c) x1 coincides with x2. The utility of the prospect increases Ferdinand M. Vieider Decision Theory–Part II Rank-dependence Under modern PT, we transform good-news probabilities of obtaining an outcome at least as good as a given value For (x , p; y , q; z) and x > y > z: w (p)v (x ) + [w (p + q) ≠ w (p)]v (y ) + [1 ≠ w (p + q)]v (z) [w (p + q) ≠ w (p)] indicates the probability of obtaining an outcome at least as good as y , minus the probability of getting something even better, i.e. x The formula for losses is perfectly equivalent, except that the convention is to weigh losses using ‘bad-news probabilities’ Ordering is crucial: from best outcome to worse (gains), or from worst outcome to best (losses) Ferdinand M. Vieider Decision Theory–Part II Index 1 Setting the stage: Violations of Dual-EU 2 Prospect theory 3 CASE STUDY: Utility around the world 4 CASE STUDY: Measuring PT parameters 5 Multi-outcome prospects and rank dependence 6 Revisiting the paradoxes 7 Decisions under uncertainty and ambiguity aversion 8 The standard model of time discounting 9 Paradoxes for risk and for time 10 Conclusion Ferdinand M. Vieider Decision Theory–Part II The framing effect Problem 1: You are given a cash gift of £200. Those £200 are yours to dispose of. Additionally, you are given a choice between obtaining £50 for sure and playing a prospect with a 25% probability of winning another £200 or else nothing. What would you rather do? O I would rather take the additional £50. O I would rather take the prospect giving me a 25% chance to win an additional £200 and a 75% chance of winning nothing. Problem 2: You are given a cash gift of £400. Additionally,you are given a choice between losing £150 for sure and playing a prospect with a 75% probability of losing £200 and a 25% probability of losing nothing. What would you rather do? O I would rather give up £150 for sure. O I would rather take the prospect with a 75% probability of losing £200 and a 25% probability of losing nothing. Ferdinand M. Vieider Decision Theory–Part II Explaining framing effects Which of the different theories we have seen can explain the framing effect? Which element in the theory is crucial? Markowitz: can explain the effect through convex utility for losses and concave for gains; assumption: the EV falls into the respective sections (doubtful for small stakes) Dual M-EU: can account for the effect through underweighting of moderate to large probabilities; PT assumes isolation of choices (defined over changes in wealth) and incorporates probability weighting CONCLUSION: all three theories can account for framing under some circumstances and parameter values; however, PT does so most generally; dual-EU cannot account for it Ferdinand M. Vieider Decision Theory–Part II The common ratio effect (example) Think about your choice between the following two prospects: 0.5 0.25 1000 2500 º 0 0 0.5 0.75 Now think about your choice between the following two: 0.05 0.025 1000 2500 ª 0 0 0.95 0.975 The choice pattern shown above (or any inversion between the two preference pairs) constitutes a violation of EUT (and M-EU, since it relies on the same linearity principle) Ferdinand M. Vieider Decision Theory–Part II Explaining the common ratio effect Markowitz can clearly not explain the common ratio effect, as it directly contradicts probabilities being treated linearly Dual-EU, dual M-EU, and PT can all explain the common ratio effect through non-linear probability weighting Under EUT, only the ratio of probabilities counts, so that –p/–q = p/q Under probability transformation, in general w (–p)/w (–q) ”= w (p)/w (q) (see exercises for details) Ferdinand M. Vieider Decision Theory–Part II The common consequence effect Take a look at the following choice tasks: 11% 1m 100% 1m 89% 0 10% 10% 5m 5m 89% 1m 1% 90% 0 0 (d) choice A (e) choice B The preference pattern (UP, DOWN) displayed above contradicts EUT; consistent: (UP,UP), (DOWN,DOWN) Ferdinand M. Vieider Decision Theory–Part II Explaining the common consequence effect Dual-EU and PT can both explain the common consequence effect through probability distortions Markowitz-EU in its original formulation cannot account for the paradox, since probabilities are linear HOWEVER: Markowitz can account for it if we allow the reference point to coincide with sure outcomes different from 0 Such endogenous reference points have been integrated into modern versions of PT Ferdinand M. Vieider Decision Theory–Part II Explaining lottery and insurance Markowitz-EU was originally developed to account for lottery and insurance, but has specific limitations Dual M-EU can account for lottery and insurance through the overweighting of small probabilities Dual M-EU cannot account for lottery and insurance play changing with the outcomes at stake PT accounts for lottery and insurance through nonlinear probabilities in conjunction with decreasing sensitivity in utility However: extent of deductible choices may require a combination of probability overweighting and loss aversion (Sydnor, 2010) Ferdinand M. Vieider Decision Theory–Part II The equity premium puzzle The equity premium puzzle consists in historically consistent higher returns to shares than to options Dual-EU cannot account for this phenomenon M-EU and PT can explain the equity premium puzzle, but need an extension: myopic loss aversion Myopia consists in decisions being examined in isolation; loss aversion then leads to avoiding potential losses in short term The issue is the same as for Samuelson’s colleague, who refuses (200, 0.5; ≠100) but would accept 100 such prospects The fallacy is failing to understand that hundreds of such choices do indeed present themselves over a lifetime Similar phenomena are known as ‘narrow bracketing’, ‘narrow framing’, the ‘isolation principle’, ‘mental accounting’, etc. Ferdinand M. Vieider Decision Theory–Part II Index 1 Setting the stage: Violations of Dual-EU 2 Prospect theory 3 CASE STUDY: Utility around the world 4 CASE STUDY: Measuring PT parameters 5 Multi-outcome prospects and rank dependence 6 Revisiting the paradoxes 7 Decisions under uncertainty and ambiguity aversion 8 The standard model of time discounting 9 Paradoxes for risk and for time 10 Conclusion Ferdinand M. Vieider Decision Theory–Part II One more paradox There are two urns: 1) 50 black and 50 red balls; 2) 100 red or black balls in unknown proportion: Choose a colour on which to bet, and an urn. Then extract one ball to win e100 if of the declared colour, and else 0 Which colour do you pick? Which urn do you pick? Now imagine you are offered the same bet, but you can win e100 from the known and e105 from the unknown urn Does this affect your choice? Ferdinand M. Vieider Decision Theory–Part II Ambiguity aversion Most people prefer to bet on the urn with the known proportion of colours; they are usually willing to leave some money on the table for this preference This contradicts (subjective) expected utility theory: you should pick the likelier ball if you think the urn is unfair This pattern can be generalized to different probability levels by using more colours of know or unknwon proportion For 50-50: probability aversion; for small known probabilities, this is less the case and may even be opposite What do you think is driven these choice patterns? Can you explain the reasoning underlying your own choice? Ferdinand M. Vieider Decision Theory–Part II The Ellsberg paradox The contradiction of (S)EU is known as the Ellsberg paradox Most people prefer to bet on the known urn when red wins: pk (R) > pu (R) Most people also prefer to bet on the known urn when black wins: pk (B) > pu (B) Together these two revealed probabilities imply that pu (R) + pu (B) < pk (R) + pk (B) = 1 This contradicts the principle that probabilities must sum to 1, i.e. the subjective probabilities are subadditive, which contradicts SEU Ferdinand M. Vieider Decision Theory–Part II Prospect theory for uncertainty Prospect theory can account for ambiguity aversion through source preferences (Abdellaoui et al. 2011) The solution is that pu (R) + pu (B) © 1, but w (pu (R)) + w (pu (B)) ”= 1; this accommodates subadditivity, as well as superadditivity People’s willingness to bet on an event depends on their perceived expertise (i.e., their confidence) This can be represented through source preferences—weighting functions tend to be different depending on the source of uncertainty We can then directly think about these weighting functions as incorporating people’s willingness to bet and confidence The explanation works only if wk (pr ) Ø wu (P(R)) for pr = P(R), and the same for black Ferdinand M. Vieider Decision Theory–Part II Index 1 Setting the stage: Violations of Dual-EU 2 Prospect theory 3 CASE STUDY: Utility around the world 4 CASE STUDY: Measuring PT parameters 5 Multi-outcome prospects and rank dependence 6 Revisiting the paradoxes 7 Decisions under uncertainty and ambiguity aversion 8 The standard model of time discounting 9 Paradoxes for risk and for time 10 Conclusion Ferdinand M. Vieider Decision Theory–Part II Discounted Utility Samuelson (1937) proposed the standard model Take a temporal prospect (ys , xt ); we can write DU(ys , xt ) = ”s u(y ) + ”t u(x ) (1) For (x1 , t1 ;...; xn , tn ) using a discount function D(t): T DU(x1 , t1 ;...; xT , tT ) = i=1 D(ti )u(xi ) (2) Additively separable function, like EU Similar assumptions on outcomes: consumption integration Assumption: D(t) is exponential, i.e. constant discounting Functional forms (continuous and discrete time): 3 4t 1 D(t) = exp(≠rt) , D(t) = (3) 1+r Implication: only absolute differences in time matter Parallels to linearity of probability under risk Ferdinand M. Vieider Decision Theory–Part II Index 1 Setting the stage: Violations of Dual-EU 2 Prospect theory 3 CASE STUDY: Utility around the world 4 CASE STUDY: Measuring PT parameters 5 Multi-outcome prospects and rank dependence 6 Revisiting the paradoxes 7 Decisions under uncertainty and ambiguity aversion 8 The standard model of time discounting 9 Paradoxes for risk and for time 10 Conclusion Ferdinand M. Vieider Decision Theory–Part II