Common ratio using delay

Theory Dec. (2010) 68:149 158 DOI 10.1007/s18-008-9130-2 Common ratio using delay Manel Baucells Franz H. Heukamp Published online: 4 January 2009 Springer Science+Business Media, LLC. 2009 Abstract We present an experiment in which we add a common delay in a choice between two risky prospects. The results show that delay produces the same change in preferences as in the well-documented common ratio effect in risky lotteries. The added common delay acts as if the probabilities were divided by some common ratio. Moreover, we show that there is a strong magnitude effect, in the sense that the effect of delay depends on the magnitude of the outcome. The results are consistent with the recently introduced probability time trade-off (PTT) model by Baucells and Heukamp. We present a parameterization of the model based on the experimental results, showing that the value function exhibits increasing relative risk aversion, the weighting function is s-shaped, and the intrinsic discount rate is decreasing. Keywords Risk and time preferences Common ratio effect 1 Introduction Several authors have recognized the parallelism between the role of uncertainty and time in decision making (Rotter 1954; Mischel and Grusec 1967; Prelec and Loewenstein 1991; Quiggin and Horowitz 1995). However, most research on decision making over time and decision making under uncertainty has evolved independently. In the domain of decisions with delays, the model of discounted utility has been developed as the standard normative model albeit M. Baucells F. H. Heukamp (B) Department of Managerial Decision Sciences, IESE Business School, Avenida Pearson 21, 08034, Barcelona, Spain e-mail: fheukamp@iese.edu M. Baucells e-mail: mbaucells@iese.edu

150 M. Baucells, F. H. Heukamp with many documented deviations from it (Cruz and Muñoz 2004). Discounted utility has its equivalent normative model in the domain of risk, namely, expected utility and also many deviations from it (Wu et al. 2004). A well-known deviation from discounted utility is the common difference effect: When subjects are indifferent between a higher amount to be received later and the smaller amount to be received sooner then adding a common delay will make most subjects prefer the larger later amount (Bostic et al. 1990; Keren and Roelofsma 1995; Weber and Chapman 2005). The analogous effect in the domain of risk is the common ratio effect. When subjects are indifferent between a higher amount with lower probability and a lower amount with higher probability, then multiplying the probabilities with a common ratio will make most subjects prefer the higher amount (Kahneman and Tversky 1979). In the context of the common difference effect, Keren and Roelofsma (1995) studied the relation between delay and probability. They showed that the preference switch associated with the common difference effect can equally be observed when a common probability is applied to the delayed payoffs. In this article, we focus on the the common ratio effect and study what happens if a delay is added in a choice between two risky prospects. If the delay acts as an additional uncertainty, then we expect to observe the same results as in the common ratio effect. The experiment adds to the literature related to the time probability tradeoff. While our experiment shares some features with the experiments of Weber and Chapman (2005), it differs in important aspects of the experimental design. The results of our experiment establish a baseline of evidence that empirically relevant models about probability and time trade-off need to adhere to. The experiment involved 221 subjects and was carried out with an incentive compatible random device. Based on the experimental results, we fit a stochastic choice model through the parameterization of the probability time trade-off (PTT) model by Baucells and Heukamp (2007). 2 The common ratio using delay experiment 2.1 The common ratio effect The common ratio effect is a robust behavioral finding in decision-making under risk. Most subjects prefer, say, a sure e 300 gain to an 80% chance at e 400. However, this preference is reversed if the probabilities in both prospects are divided by a common ratio of, say, ten. If chance to gain the e 300 is 10% and the chance to gain the e 400 is 8%, then most subjects prefer the second prospect (Kahneman and Tversky 1979). In the experiment we test whether the same change in preferences can be observed if a common delay is added to the prospects. That is, instead of multiplying probabilities by a common factor of 1/10, we add a common delay of, say, 6 months. The resulting option is a sure e 300 gain to be received in 6 months, or an 80% chance at e 400 to be received in 6 months. We call this the common ratio using delay effect. If time is treated as if it were uncertain, then the delay of 6 months acts as a common probability factor that induces some subjects to switch preference towards the second option.

Common ratio using delay 151 2.2 Design, incentives, and subjects Subjects were 221 participants of the MBA and EMBA programs of IESE Business School in Barcelona and Madrid. Participation in the experiment was voluntary and the experiment was self-administered through a spreadsheet. This spreadsheet was designed so that some tasks were answered by all subjects and others were randomly assigned to only some students. The goal was to test the common ratio effect and the common ratio using delay effect for different monetary amounts and delays. The total number of tasks was 25. To each student 14 tasks were presented in a first round and three additional ones in a second round 3 weeks later. Each task consisted of a choice between two prospects. A prospect was characterized by an amount x to be received at time t with probability p and otherwise zero. The stakes ranged from e 0.03 to e 300. The presentation of the tasks was randomized: The order of the tasks, within each task the presentation of alternatives 1 and 2, and the order of the attributes (x, t, p) were random. The choices used real incentives: three participants were chosen at random, and one of their preferred choices, also chosen at random, was played out with real delays and real incentives. Table 5 shows the design of the 25 tasks. 2.3 Results and discussion Table 1 shows the results for six of the tasks. In the high certainty treatment, Prospect 1involvese 9 for sure and Prospect 2 e 12 with an 80% chance. This choice is being made for delays of 0, 1, and 3 months. Similarly, in the low certainty treatment, Prospect 1 involves e 9 with a 10% chance and Prospect 2 offers e 12 with an 8% chance. Again, the delays are 0, 1, and 3 months. In the absence of delay (column t=now), we reproduce the common ratio effect: Most subjects prefer e 9 for sure in the high certainty treatment (58%) and, in violation of the independence axiom, most prefer the e 12 with an 8% chance in the low certainty treatment (78%). Consider now the high certainty treatment when the delay is added. The preference for Prospect 1 is reduced to the point that for a delay of 3 months, Prospect 2 is preferred by most of the subjects (57%). Therefore, we are able to reproduce the common Table 1 Proportion of subjects preferring the corresponding outcome for three different levels of delay. Boldface indicates results above 50% at p = 5% Delay of monetary reward t = now (%) t = 1 month (%) t = L3 months (%) High certainty 1. e 9forsure 58 55 43 2. e 12 with 80 42 45 57 Low certainty 1. e 9 with 10 22 26 21 2. e 12 with 8 78 74 79

152 M. Baucells, F. H. Heukamp Table 2 Proportion of subjects preferring the corresponding sure outcome for two levels of delay and varying stakes Amount for sure Delay of monetary reward t = now (%) t = 3 months (%) e 300 80 75 e 25 62 60 e 9 58 43 e 3 54 47 ratio effect by adding a delay. The low certainty treatment with delay produces no surprises, as it reinforces the preference for the larger outcome. While Table 1 extracts the results for stakes of e 9 and e 12, respectively, a similar pattern holds for the other stakes (see Table 5). If a common delay is added to the two prospects of a task, then the preference for the sure prospect is always reduced. Indeed, with the exception of a e 300 prospect with a delay of 6 months, the preference for the risky prospect increases as the delay is increased. The delay seems to be perceived as additional uncertainty and therefore the sure prospect is not considered for sure anymore. As shown in Table 2, the reduction in preference for the sure prospect when adding delay ranges from a few percentage points (sure gains of e 300 or e 25) to a preference reversal (sure gains of e 9ore 3). Diverging from Weber and Chapman (2005), we show that added delay leads to similar results as dividing the probability by a common ratio. One explanation for this difference in results is our use of a simple design. We simply divide the probabilities in the high certainty case by the common ratio of 10. In contrast, Weber and Chapman (2005) use a more complicated Allais Paradox design. Also, we use relatively short delays of up to 12 months Weber and Chapman (2005) go up to 25 years that correspond to a realistic time of continuing interaction with the subjects as program participants at the business school and which excludes potential doubts about payments etc. Finally, in contrast with Weber and Chapman (2005), we offer real incentives. 3 A Model-based analysis 3.1 The PTT model Simple prospects of the form (x, t, p) are in the domain of applicability of the PTT Model axiomatized in Baucells and Heukamp (2007). The PTT model rests on the condition that additive changes in delays are perceived as multiplicative changes in probabilities, with a magnitude-dependent exchange rate between them. Specifically, it stipulates that an (x, t, p) prospect is evaluated as V (x, t, p) = w(pe r(x)t )v(x). (1) Here r(x) is an intrinsic discount rate function that can depend on the stakes, w is a probability weighting function and v(x) is a value function. The PTT model addresses

Common ratio using delay 153 Table 3 Proportion of subjects preferring the corresponding outcome for three different levels of probability (Keren and Roelofsma 1995, Table 1) Probability of monetary reward p = 1.0 (%) p = 0.9(%) p = 0.5(%) Imminent future A. 100 now 82 54 39 B. 110 in 4 weeks 18 46 61 Remote future C. 100 in 26 weeks 37 25 33 D. 110 in 30 weeks 63 75 67 All amounts are in Dutch Guldens. In 1995, 1 Gulden was equivalent to roughly $$0.6 only the trade-off between delay and probability. The shapes of r, w, and v are taken as given. From Eq. 1 we see that for a decision maker who follows the PTT model, a delay creates an additional subjective uncertainty in addition to the objective probability p. This also implies that in the PTT framework, the common ratio effect, i.e., dividing the probability p by some factor, is equivalent to the common delay effect, i.e., adding a delay to the current time t. This analogy has been suggested before by Prelec and Loewenstein (1991). The PTT model accommodates the experimental findings reported in this article and the ones by Keren and Roelofsma (1995) (see Table 3). Keren and Roelofsma (1995) conducted an experiment with simple prospects of the type (x, t, p) in which a common probability was added to delayed prospects. Their results demonstrated that the preference reversal observed in the common difference effect can equally be obtained by adding risk (Table 3). The PTT model is not the only way to account for preferences over triples (x, t, p). However, we now show that other forms are not possible to reconcile with the evidence we have presented here. For instance, following the logic of discounted expected utility, an intuitive formulation for valuation of a risky prospect with delay would be V (x, t, p) = f (t)w(p)v(x) (2) for given functions f, w, and v. Evaluating pairs of prospects, say A and B in Table 3, leads to the following evaluation (indices A and B refer to the attributes of the respective prospect): V (x A, t A, p A ) V (x B, t B, p B ) = f (t A )w(p A )v(x A ) f (t B )w(p B )v(x B ) For the experiments by Keren and Roelofsma (1995) (Table 3) the probability is the same for the two prospects, p = p A = p B. This yields V (x A, t A, p A ) V (x B, t B, p B ) = w(p)[ f (t A )v(x A ) f (t B )v(x B )] Thus, if prospect A is preferred to prospect B for p = 1 then this should also be the case for any other p. This is in contradiction with the change of preferences

154 M. Baucells, F. H. Heukamp in the experiments by Keren and Roelofsma (1995). This falsifies the formulation given in (2) as behaviorally plausible. More generally, any formulation of the type V (x, t, p) = w(p)v(x, t) is incompatible with the evidence of Table 3. Of course, the PTT model given in Eq. 1 not only is compatible with the results by Keren and Roelofsma (1995) but specifically predicts that an added common probability p will do the same as a common delay. In the case of a common delay for the two prospects (Table 1), the evaluation of V (x, t, p) = f (t)w(p)v(x) leads to V (x A, t A, p A ) V (x B, t B, p B ) = f (t)[w(p A )v(x A ) w(p B )v(x B )] Because the preference for prospect A or B should not depend on t, we again confirm that the formulation given in (2) is incompatible with the experimental results of Table 1. The PTT model is again not only compatible with the results reported in this article but predicts that an added common delay causes the same changes in preferences as reducing the probability by a common ratio. Together, the results of Keren and Roelofsma (1995) on common delay with risk and the ones reported here on common ratio with delay show that the probability and time dimensions cannot be separated in an empirically relevant representation of V (x, t, p). 3.2 Magnitude effects An additional view on the effects of the stakes can be obtained from Table 4. The table displays the percentage of subjects who prefer the sure gain for all the tasks in the high certainty treatment (14 out of the total 25 tasks shown in Table 5). The stakes of the risky prospect (Prospect 2) are always 4/3 of the stakes of the sure prospect (Prospect 1) and its probability (p 2 ) is always 80%. By way of example, the e 300 column shows the percentage of subjects that prefer e 300 for sure to e 400 with an 80% chance, for the different delays. In this two-way table, we observe the effect of delay (rows) and magnitude (columns). Let us consider these two effects separately. As discussed before, for any fixed amount (or column), the preference for the sure prospect decreases with the delay. In addition, the preference for the sure prospect decreases as the stakes are reduced. This magnitude effect can be explained in two Table 4 Percentage of subjects who prefer the sure prospect (Prospect 1) as a function of x 1 and t t\x 1 e300 (%) e25 (%) e9(%) e3 (%) e0.3 (%) e0.03 (%) 0 80 62 58 54 46 39 3 75 60 43 47 6 79 54 12 75 51 The stakes of the risky prospect (Prospect 2) are always 4/3 of the stakes of the sure prospect and its probability (p 2 )isalways0.8

Common ratio using delay 155 Table 5 Task design and results (last two columns) of the experiment of common ratio with delay # x 1 [e] t 1 [months] p 1 [1] x 2 [e] t 2 [months] p 2 [1] N 1[%] 2[%] Task design 1 300 0 1 400 0 0.8 41 80 20 2 300 3 1 400 3 0.8 221 75 25 3 300 6 1 400 6 0.8 183 79 21 4 300 12 1 400 12 0.8 167 75 25 5 300 0 0.1 400 0 0.08 41 24 76 6 300 3 0.1 400 3 0.08 221 23 77 7 300 6 0.1 400 6 0.08 41 34 66 8 25 0 1 33 0 0.8 207 62 38 9 25 3 1 33 3 0.8 221 60 40 10 25 6 1 33 6 0.8 115 54 46 11 25 12 1 33 12 0.8 166 51 49 12 25 0 0.1 33 0 0.08 115 23 77 13 25 3 0.1 33 3 0.08 221 28 72 14 25 6 0.1 33 6 0.08 115 15 85 15 9 0 1 12 0 0.8 142 58 42 16 9 1 1 12 1 0.8 65 55 45 17 9 3 1 12 3 0.8 221 43 57 18 9 0 0.1 12 0 0.08 65 22 78 19 9 1 0.1 12 1 0.08 65 26 74 20 9 3 0.1 12 3 0.08 221 21 79 21 3 0 1 4 0 0.8 144 54 46 22 3 3 1 4 3 0.8 221 47 53 23 3 3 0.1 4 3 0.08 221 24 76 24 0.3 0 1 0.4 0 0.8 108 46 54 25 0.03 0 1 0.04 0 0.8 113 39 61 Boldface indicates results above 50% at p = 5%. Columns 2 4 show the attributes (x 1, t 1, p 1 )ofprospect 1 and columns 5 7 the same attributes for Prospect 2. Columns 8 10 show the number of subjects who were presented with the task (N) and the percentages that preferred each prospect ways. It could be related to the shape of the value function of a subject. But if the value function were to exhibit constant relative risk aversion, then a multiplicative scaling of the payoffs would not alter the preferences (Pratt 1964) and the percentage of subjects choosing x for sure would not depend on x. This is not the pattern observed in the first row of Table 4. Hence, this first row alone rules out a power value function. The observed results rather indicate that subjects value function is consistent with increasing relative risk aversion. This same result was obtained in an experiment with real and similarly large stakes as ours by Holt and Laury (2002). Table 4 also shows the effect of time (row). Fixing a column, let us consider the effect of adding a delay. If discount rates were constant, then the decay rates of preference for the sure option should be the same, independent of the magnitude of outcomes.

156 M. Baucells, F. H. Heukamp However, what we observe is that the temporal discount rate is an increasing function of the magnitude of the payoffs. As the stakes increase, subjects may be more patient (Thaler 1981). In Table 4, this effect can be observed in the interaction between the effect of the delay, t, and the outcome size. The preference for the sure outcome is more stable with respect to a delay for high stakes (e.g. e 300) than for lower stakes (e.g. e 25 or even more for e 9). 3.3 A stochastic choice model Qualitatively, we have argued that the results of Tables 1 and 3 are consistent with the PTT model. This suggests that a parametric specification of the model can have some predictive value and provide additional insight. In the following, we estimate such a parametric model. For the three functions of the PTT model, we choose the following specifications. For the value function, we choose the expo-power form used in Holt and Laury (2002): v(x) = 1 exp( αx1 β ) α (3) As explained before, both Holt and Laury (2002) and our results indicate that a simple power form would not do justice to the data, which exhibits increasing relative risk aversion. With respect to the probability weighting function, we choose the form proposed by Prelec (1998). In Baucells and Heukamp (2007) we show that this function is particularly suited for our application, as its associated time discount function has a direct time acceleration interpretation. Hence, w(p) = exp( ( lnp) γ ) (4) Finally, regarding the intrinsic discount rate function, the only constraint we impose is that it should be non-increasing in order to capture the well-established absolute magnitude effect. We suggest a very simple one-parameter form: r(x) = R/ x (5) Of course, if x tends to zero, then the intrinsic discount rate tends to infinity. In summary, our parametric specification is: ( V (x, t, p) = w pe r(x)t) ( ( ) Rt γ ) 1 exp( αx 1 β v(x) = exp x lnp ) α which implies the estimation of four parameters: R, α, β, and γ. To estimate the parameters, we use the results of Task 1 20 presented in Table 5 (for which at each level of the stakes high and low certainty treatments are available). We assume that the logarithm of the odds of preferring Prospect 1 over Prospect 2 is linearly related to the relative difference in valuation: (6)

Common ratio using delay 157 Ln [ q / (1-q) ] 2 1.5 1 0.5 0-0.6-0.4-0.2 0 0.2 0.4 0.6-0.5 ( V1 - V2 ) / (V1+V2) -1-1.5 Fig. 1 Best fit line for the stochastic choice model based on the results from Tasks 1 20-2 ( ) q ln = V (x 1, t 1, p 1 ) V (x 2, t 2, p 2 ) 1 q V (x 1, t 1, p 1 ) + V (x 2, t 2, p 2 ) (7) where q is the probability of choosing Prospect 1. Minimizing the sum of the squared errors, the best fit is obtained for the following values of parameters: R = 1.52,α = 0.0018, β = 0.03, and γ = 0.7. Figure 1 shows the fit of the model. The γ of the weighting function matches closely values in the literature. The α of the expo-power value function corresponds to a comparatively small risk tolerance of around e 550. Our model is richer than that of Holt and Laury (2002), as the risk attitude of our decision maker is given by the combined effect of both w and v (Holt and Laury (2002) do not include probability weighting). Also, the PTT model captures two possible effects of payoff magnitude, one associated with v(x) and a second associated with r(x). Finally, a sub-proportional weighting function would explain simultaneously the common ratio effect and the common difference effect. 4 Summary and concluding remarks The experiment presented in this article showed that a common delay in simple prospects leads to the same changes in preferences as the common ratio effect. This adds to the experimental literature and proves that common ratio effect and common delay effect are intimately related. The PTT model predicts the results presented in this article and is also compatible with earlier work by Keren and Roelofsma (1995). The parameterization of the PTT model that we obtain leads to reasonable specifications of the weighting function and value function. We observe magnitude effects that can be explained by the intrinsic discount rate function of the PTT model.

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