Estimating the Relationship between Economic Preferences: A Testing Ground for Unified Theories of
Behavior: Appendix
Not for Publication August 13, 2012
1 Appendix A: Details of Measures Collected
Discount rate and Present Bias. We measure each subject’s discount rate using three ques- tions in which we elicit indi↵erences of the form (x, t) ⇠ (y, s) where (x, t) is the amount x received in t weeks. We used the triples (5, 6, 6), (6, 8, 7) and (5, 10, 7) for (t, y, s), and in each case we identify x using the multiple price list method. For each triple, we calculate the implied annual discount rate. For this section of the experiment, payment was made by cheque that was mailed to the subject. The date of payment was the date that the cheque was mailed.
Present bias refers to the phenomena by which subjects tend to exhibit higher discount rates when the soonest available payment is available immediately. In order to measure present bias, we repeat the above analysis, but ‘shift’ everything forward so the date of the sooner payment is immediate: in other words, we use the values (0, 6, 1), (0, 8, 1) and (0, 10, 2) for (t, y, s). We should emphasize that in case of payment in ‘0’ days, cheques were mailed on that day (rather than the subject receiving the cash immediately): this is done to reduce any e↵ect due to transaction costs.1
We call the discount rate measured using these questions ‘Discount Rate (Present)’ to discriminate from the degree of discounting elicited when all payments were in the future.
Present bias for each pair of questions is measured as the change in indi↵erence points due to the shift to immediate sooner payment as a proportion of the later amount (so, if a subject was indi↵erent between $8 in 5 weeks and $10 in 7 weeks, and $7 in 0 weeks and $10 in 2 weeks, their present bias would be measured as 10%).
Notice that here we use the term ‘discount rate’ purely behaviorally. We make no claim
1Indeed this implies that today’s payment are actually received on the following day, potentially elimi- nating present bias. This, however, does not seem to be the case in our data: we do find significant present bias similar in degree to that found in previous studies.
that this procedure is identifying the discount rate parameter derived from any particular model of behavior.
Risk Aversion in the Gain and Loss Domain. In the gain domain we measure risk aversion by eliciting the certainty equivalence of three 50/50 lotteries: between $6 and $0, $8 and $2 and $10 and $0. We measure risk aversion as the di↵erence between the certainty equivalence and expected value of the lottery as a proportion of the expected value.
In order to measure risk aversion in the loss domain, subjects are endowed with an extra
$10.2 Certainly equivalence is then extracted for three 50/50 lotteries, between -$10 and $0, -$8 and $0 and -$6 and $0.
Violations of Expected Utility Under Risk: Common Ratio and Common Consequence E↵ects. In our experimental design, we measure two classic violations of expected utility under risk: The common ratio and common consequence e↵ects.
The common consequence e↵ect is e↵ectively the standard Allais paradox. We measure it using two pairs of lotteries. First, we measure the value of x that makes subjects indi↵erent between lotteries A and B
- A: 100% chance of y
- B: 89% chance of y, 1% chance of $0, 10% chance of x then between the lotteries A0 and B0
- A0: 11% chance of y and 89% chance of $0 - B0: 10% chance of x0 and 90% chance of $0
We do this for two values of y: $4 and $8. Notice that, for an expected utility maximizer, it has to be the case that x = x0. The standard common consequence e↵ect is that x > x0, which is usually interpreted as implying that the subject needs more compensation in order to choose lottery A over B as A provides a prize with certainty. We estimate the size of the common consequence e↵ect as x xy 0.
The common ratio e↵ect also involves the comparison between two pairs of lotteries.
First, we find the z that makes the subject indi↵erent between C and D - C: 100% chance of w
- D: 80% chance of z and 20% chance of $0 Then between C0 and D0
2That is, subjects are told: “You are given an additional $10 for questions in this block. i.e. if a question in this block is selected for payment you will receive $10 on top of your show up fee.”
- C0: 25% chance o↵ w and 75% chance of $0 - D0: 80% chance of z0 and 80% chance of $0
Again, we use $4 and $8 as two values for w, and again, expected utility maximization implies that z0 = z. The standard common ratio e↵ect finds that z > z0. We measure the common ratio e↵ect as z zw0.
Ambiguity Aversion and Compound Lottery Aversion. In order to measure aversion to ambiguity and compound lotteries, we use a technique similar to that used in Halevy [2007]:
subjects are presented with bags filled with 40 poker chips that are either red or black. They can select a color to bet on, creating a gamble in which they will win a prize of value x if the poker chip that is drawn is of their selected color, and $0 otherwise. Their certainty equivalence of this gamble is then elicited. For each prize level x, the certainty equivalence is extracted for three di↵erent type of bags:
- Risk: subjects are told there are 20 red chips and 20 black chips.
- Ambiguity: subjects are told nothing about the composition of the bag.3
- Compound lottery: subjects are told the following “The number of red chips was determined as follows: a computer randomly chose a number between 0 and 40 with equal probabilities. The number chosen is the number of red chips in the bag. The remainder of the chips are black.”
Note that a subject who reduces the uncertainty of compound lotteries in the standard way should treat the risk and compound lottery bags in the same way, while a subject that makes choices according to subjective expected utility maximization must like to gamble on the ambiguous bag at least as much as on the risky bag.4 We therefore measure ambiguity aversion for each prize level as the certainty equivalence of the gamble on the risky bag minus the certainty equivalence of the gamble on the ambiguous bag, divided by the expected value of the gamble on the risky bag. Compound lottery aversion is measured in the same way, but using the value of the gamble on the compound lottery bag rather than that of the ambiguous bag. We measure each of these behaviors at three prize levels: $6, $8 and $10.
Loss Aversion and the Endowment E↵ect. Loss aversion in risky choice is usually defined in the context of a specific model of decision making: the utility function in the loss domain has a steeper slope that that in the gain domain. The behavioral implication of loss aversion in risk choices is essentially that risk aversion for lotteries that involve both gains and losses is higher than in those that contain gains alone and those that contain losses alone (see for example Thaler (1997)). As non-parametric ways of measuring loss aversion require a lot of
3Specifically, they are told “The bag contains 40 chips. The number or red and black chips is unknown.
It could be any number between 0 red chips (and 40 black chips) and 40 red chips (and 0 black chips).”
4The reason is, for any subjective belief r about the probability of a red ball being drawn, max(r, (1 r)) 0.5. Thus, as the subject gets to choose which color to gamble on, the ambiguous bag has to have at least as high a probability of winning as the risky bag.
choices to be observed, in what follows we use the parametric methodology of Abdellaoui et al. (2008). The answer to the questions on risky bets are used to estimate constant relative risk aversion utility functions for the gain and loss domain. The value of x that makes the subject indi↵erent between $0 for sure and a 50/50 gamble between $8 and -$x is then elicited. Loss aversion is estimated as the additional slope of the utility function in the loss domain relative to the gain domain that is necessary to match this choice conditional on the slopes estimated separately in the two domains.
The endowment e↵ect refers to the phenomena by which subjects tend to require more money to relinquish an item that they already have than they are prepared to pay when they do not own the item (the willingness to pay/willingness to accept gap). In order to measure this in our subjects, we use the certainty equivalence of the lotteries in the gain domain used to elicit risk aversion (described above) as an estimate of the ‘willingness to accept’ for these lotteries.5 The willingness to pay for the same lotteries was then extracted, by endowing subjects with an additional $10, then telling them: “...you will be o↵ered the opportunity to buy a lottery ticket. That is, you will be o↵ered the opportunity to use some of this additional $10 in order to buy a lottery ticket. If you choose to do so (and that question is selected as one that will be rewarded), then you will pay the specified cost for the lottery, and you would keep the remaining amount of money and the lottery.” The endowment e↵ect for each lottery is measured as the willingness to accept minus the willingness to pay for that lottery, as a proportion of the lotteries expected value.
Sender and Receiver Behavior in the Trust Game. The trust game is a standard tool in experimental economics used to estimate social preferences. The first mover in the trust game is endowed with a certain amount of money ($5 in our experiment). They then have to decide how much of this to keep, and how much to send to Player 2.6 Any amount they send is tripled. Player 2 then has to decide how much of this money to keep, and how much to return to Player 1.
In our experiment, we use the strategy method to elicit each subject’s play in each possible decision node in the game.7 Subjects are asked to report how much they would send if they were Player 1, and how much they would return as Player 2, conditional on each possible received amount. If this question was selected as one to be actualized, then their responses were paired with those of another subject to determine payment.
The unique subgame perfect equilibrium of this game is that Player 1 sends nothing, and Player 2 never returns anything. Subjects often do not conform to this behavior. We measure sender behavior as the amount that they choose to send as Player 1, and returner behavior as the average fraction of the amount that Player 1 sends that they choose to return.
Cognitive Ability, Overconfidence and Overplacement. We measure cognitive ability using
5These questions were phrased as follows: after a description of the lottery, the subjects were told “This lottery is yours to keep (if this is one of the questions that is selected at the end of the experiment). However, you will be o↵ered the opportunity to exchange this lottery for certain amounts of money (for example $5).”
6Our subjects were constrained to choose from 50c increments.
7This approach is standard, though may bias downward subject’s degree of trust (see Casari and Cason (2009)).
Raven’s matricies, a standard, non verbal measure of perceptual reasoning. We use a 12 questions subset from Raven’s Advanced Progressive Matricies test developed by Arthur and Day (1994),8 as well as a subset of 5 matrix questions from the set used by Putterman et al. (2010), giving 17 questions in total. We also measure intelligence using self reported SAT mathematics scores.
Experiemental subjects are often found to exhibit ‘overconfidence’ in their abilities. In this study we use two methods to estimate this. First, we ask subjects to report their expected performance in the cognitive test – i.e. how many of the 17 questions they think they got right. We measure overconfidence as the di↵erence between the predicted score and the actual score. We also ask them to estimate the average number of correct responses in the session. We then call ‘overplacement’ the di↵erence between their own predicted score an their predicted average for the room.
2 Appendix B: Details of Theoretical Models Consid- ered
2.1 Models of Ambiguity Aversion
There are several theories that make predictions about how individual behavior in risky and ambiguous settings should be related. We consider five of these: Subjective Expected Utility (SEU), the MaxMin Expected Utility (MMEU) model of Gilboa and Schmeidler (1989), the Multiple Priors Multiple Distortions (MPMD) and Joint Multiple Priors Multiple Distortions model of Dean and Ortoleva (2012) and the Recursive Non-Expected Utility (RNEU) model of Segal (1987).
Subjective Expected Utility (SEU). In the Subjective Expected Utility models of Savage (1954) and Anscombe and Aumann (1963) the decision maker forms a subjective belief on the likelihood of the di↵erent states of the world, and maximizes expected utility relative to this subjective belief. She follows Expected Utility also when lotteries are objective. According to the model, therefore, we should not observe either the common consequence or common ratio e↵ect. SEU is also incompatible with a positive ambiguity aversion. Furthermore, assuming that the agent treats the two color symmetrically (sometimes called the principle of insufficient reason), then SEU predicts an ambiguity aversion equal to zero.9 Moreover, while not originally a dynamic theory, the reduction of compound lotteries is a necessary part of expected utility theory in a dynamic setting (Segal (1990), Anscombe and Aumann (1963)). Thus, SEU theory predicts that the compound lottery urn should have the same value as the risky urn.
8Items 1, 4, 8, 11, 15, 18, 21, 23, 25, 30, 31 and 35
9If we allowed the decision maker to treat the two colors in a di↵erent way, then we could have a negative ambiguity aversion measure with SEU in our data: if the decision maker assigns a high probability to a Red ball being extracted, she would choose to bet on Red and value this bet more than the objective one.
Maxmin Expected Utility (MMEU). The Maxmin Expected Utility model was introduced by Gilboa and Schmeidler (1989) in order to extend SEU to allow for ambiguity aversion.
This is done by allowing the decision maker to have a set of subjective probability distribu- tions ⇧ over states of the world and assuming that any act f is evaluated using the worst of these probability distributions.10 While this model allows for ambiguity aversion, it assumes that the decision maker follows Expected Utility in the risk domain: one of the defining ax- ioms of the MMEU model is c-independence, which implies standard independence in the risk domain, thus ruling out the Common Consequence and Common Ratio e↵ects. Following the same argument used for SEU, this in turn rules out Compound Lottery aversion.
Multiple Priors Multiple Distortions (MPMD) and Joint Multiple Priors Multiple Dis- tortions (JMPMD) Models. While the MMEU model forces the decision maker to follow Expected Utility for risky choices, the Multiple Priors Multiple Distortions (MPMD) and the Joint Multiple Priors Multiple Distortions (JMPMD) models of Dean and Ortoleva (2012) allow for subjects to be pessimistic over risky outcomes, as well as being ambiguity averse over ambiguous outcomes, allowing for violations of independence in both domains.11 Both models prescribe that the decision maker evaluates risky prospects following a generaliza- tion of the Rank Dependent Expected Utility model of Quiggin (1982), in which the decision maker has a set of concave probability distortions , and evaluate risky prospects that re- turn $x with probability p and $y < $x with probability (1 p) by
min 2 (0.5)u(x) + (1 (0.5))u(y) where (0.5) 0.5.
Where the MPMD and the JMPMD model di↵er is in how agents evaluate ambiguous bets. In the former, these are evaluated in a way very similar to MMEU: the decision maker has a set of priors ⇧, and evaluates each act using the most pessimistic prior in the set, i.e. by min⇡2⇧P
⇡(s)U (f (s)) (where U is the utility that she uses to evaluate objective lotteries).
The JMPMW mode is a special case of the MPMD model in which the agent also has a set of priors ⇧0 over states of the world, but she evaluates ambiguous bets by first mapping them to objective lotteries using the worst prior in ⇧0, and then evaluating the resulting objective lotteries as she usually does, by using the most pessimistic distortion of probabilities in .12 That is, she evaluates the act f that returns x (for sure) in state s and y (for sure) in state s0 following
min⇡2⇧0 min
2 (⇡(s))u(x) + (1 (⇡(s0)))u(y).
Both of these models allow for violations of expected utility in the risk domain. If
10In particular, an act f mapping state space S to lotteries outcome space X is evaluated according to min⇡2⇧
X
s2S
⇡(s)U (f (s))
where U (f (s)) is the expected utility of the lottery over X that act f generates in state s.
11Similar insights can be gained from considering a decision maker who behaves in accordance with a concave rank dependent utility model in both the risky and ambiguous domains – see Wakker (2001).
12The fact that the JMPMD is a special case of the MPMD model is proved in Dean and Ortoleva (2012).
contains anything other than the linear function, than the decision maker will exhibit the common consequence e↵ect, and (under most commonly assumed probability weighting functions) the common ratio e↵ect. In terms of ambiguity attitudes, however, the two models di↵er. The MPMPD model is compatible both with ambiguity aversion and its opposite, even under the principle of insufficient reason. Assuming that red and black balls are treated equivalently (i.e. the set ⇧ is symmetric), the degree of ambiguity aversion13 in our experiment is given by
✓
min2 (0.5) min
⇡2⇧0 ⇡(r)
◆ u(x).
Thus, the agent’s ambiguity attitude is determined by their pessimism in the ambiguous domain (measured by min⇡2⇧⇡(r)) relative to their pessimism in the risky domain (measured by min 2 (0.5)).14 By contrast, the JMPMW model predicts that the agent is always (weakly) ambiguity averse: it is easy to see that in this model subjective bets are always distorted weakly more than objective ones. Ambiguity aversion is given by
✓
min 2 (0.5) min
2 min
⇡2⇧0 (⇡(r))
◆ u(x) which must be weakly negative, as ⇡(r) 0.5.
In terms of the correlation between ambiguity aversion and violations of expected utility, without further assumptions neither models make a prediction. However, notice that in the MPMD model we can see the set of priors ⇧ as representing the amount of agents ‘absolute’
pessimism towards ambiguity (see Section 4.1 of the main body of the paper), and the set as representing the pessimism towards risk. As we have discussed, if we assume that these two features are distributed independently in the population, then we should expect a negative correlation between ambiguity aversion and the size of the common ratio and common consequence e↵ects: the reasons is, ceteris paribus as we increase the distortion of objective bets, ambiguity aversion will fall. By contrast, in the JMPMD model the set ⇧0 can be seen as representing the agent’s ‘relative’ pessimism towards ambiguity, i.e., the additional pessimism in evaluating ambiguous gamble due to the fact that they are ambiguous and not only risky. Under this model, we should expect the relation between ambiguity aversion and the common ratio and common consequences e↵ect to reflect directly the relation between this form of relative pessimism towards ambiguity and pessimism towards risk (captured by ).
Finally, we discuss the implications of the two models for attitudes to compound lotter- ies. As we argued above, following Segal (1990) we know that (under basic assumptions) violations of Expected Utility imply a non-neutral attitude towards compound lotteries. In order to extend the MPMD and JMPMD to dynamic settings, we can use the recursive
13Measures by the di↵erence in utilities between the gamble with prize x on the ambiguous and on risky urns.
14In fact, as pointed out in Wakker (2001), ambiguity aversion is a measure of relative pessimism. Its sign and magnitude therefore depend on how big is the distortion on objective bets relative to the one on ambiguous ones.
approach of Segal (1987): the set of probability distortions is first used to determine the certainty equivalents of second stage lotteries. The same formula is then again used to evaluate first stage lotteries in which we have replaced the second stage lotteries with their certainty equivalents.15 Segal (1987) describes the conditions under which a compound lottery will be considered worse that then single stage lottery with the same distribution over final outcomes in the case in which is a singleton, and shows that they are tightly tied to those that lead to the common consequence and common ratio e↵ect under risky choice. It is easy to see that parallel conditions hold for the MPMD and JMPMW models.
Thus, we should expect to see a strong positive correlation between violations of expected utility and reduction of compound lotteries, as both stem from pessimism in the risk domain.
Ambiguity aversion should only be correlated with compound lottery aversion to the extent that it is also correlated with violations of EU.
Recursive Non-Expected Utility (RNEU). Starting from Segal (1987, 1990), a di↵erent channel has been suggested to connect violations of Expected Utility in the risk domain, compound lottery aversion, and ambiguity aversion. In particular, Segal (1990) shows how standard reduction of compound lotteries (along with compound independence) implies that subjects satisfy Expected Utility in risky choices, and derives the Recursive Non-Expected Utility (RNEU) model from studying the behavior of non-EU decision makers whan faced with compound lotteries.16 Segal (1987) then argues that the Recursive Model of Non- Expected Utility could be used to explain ambiguity aversion if ambiguous bets are seen as compound lotteries. For example, the decision maker may think of various ways in which the uncertain urn was filled, then assign probabilities to these ‘states,’ creating a two stage lottery. If a decision maker evaluates two stage lotteries recursively, and if she has non- expected utility preferences, then she may exhibit ambiguity aversion. For an agent that has rank-dependent utility preferences, the conditions under which a reduced lottery will be preferred to its extensive form are closely tied to the conditions that guarantee the common ratio and common conseqence e↵ect. We would therefore expect to see agents who exhibit these e↵ects to also exhibit aversion to compound lotteries and ambiguity aversion. Fur- thermore, this model predicts a strong positive correlation between attitudes to compound lotteries and ambiguity aversion.
15That is, the certainty equivalence of binary second stage lotteries are calculated using UM P M D(p) = min
2 (1 (p)) u(x)
where p is the probability of obtaining the prize in the second stage lottery. In the case of our compound urns, there would therefore be 41 second stage lotteries, with p ranging from 0 to 1. The utility of the first stage lottery is then calculated by reapplying this formula, so in the case of our compound urns, we have
UM P M D(Lc(x)) = min
2
X40 i=0
✓ ✓41 i 41
◆
(40 i 41 )
◆
UM P M D( i
40) (1)
16Dillenberger (2010) links preference for one shot resolution of uncertainty with an axiom called Negative Certainty Independence, which is linked to violation of Expected Utility in risky environments.
2.2 Models of Loss Aversion and the Endowment E↵ect
The concept of loss aversion, introduced by Kahneman and Tversky (1979), refers to the idea that ‘losses loom larger than gains’ in a↵ecting decision making. This concept has been used to explain two very di↵erent behavioral phenomena. On the one hand, loss aversion in risky choice has been used to explain why people are more risk averse for lotteries that involve both losses and gains than they are for lotteries that involve only one or the other.
On the other hand, loss aversion has also been widely used to explain the endowment e↵ect, i.e. the tendency of decision makers to assign a higher value to a good when they own it, as opposed to when they do not – the well-known Willingness to Pay/Willingness to Accept gap.
In the prospect theory model of Kahneman and Tversky (1979), the assessment of a binary lotteries over final wealth levels of the form (p, w2; 1 p, w1) for w2 w1 depends on the reference point ¯w. The utility of each prize is assessed according to whether it is a gain or loss from ¯w, with gains being assessed according to some ug(x) for x 0, and losses being assessed according to ul(x) = ug( x) for x < 0. The parameter is termed
‘loss aversion,’ as it captures the degree to which losses are weight more heavily than the equivalent gains. For ¯w2 [w1, w2] we therefore have
u(p, w2; 1 p, w1) = pug(w2 w)¯ (1 p)ug( ¯w w1).
Thus, while standard model would predict that the certainty equivalent of this lottery would be the same if the decision maker is initially endowed with w1, w2 or w1+w2 2, prospect theory does not. If the reference point is w2, the lottery is assessed using
0.5 ug(w1 w2) while for reference point w1 it is assessed using
0.5ug(w1 w2) and for w1+w2 2
0.5ug
✓w2 w1
2
◆
0.5ug
✓w2 w1
2
◆ .
In the case of linear utility, it is clear that for reference point w2 (where the lottery only involves losses) and for reference point w1 (where the lottery only involves gains), the certainty equivalent of the gamble is w1+w2 2. However, when the reference point is w1+w2 2, and so the lottery involves both gains and losses, the certainty equivalence is
w1+ w2
2 ( 1)(w2 w1)
4 .
This means that, for > 1, the subject will be risk averse.
More generally, loss aversion in risky choice is identified by estimating the utility function in the loss domain, the utility function in the gain domain, and then the ‘additional’ risk aversion in choices that involve gains and losses. This is the approach taken in this paper, following the methodology of Abdellaoui et al. (2008).
Loss aversion has also been widely used to explain the endowment e↵ect. Tversky and Kahneman (1991) consider the case where, for any object m, the utility of purchasing that object relative to the reference point of not having that object is given by v(m). However the utility for selling the same object relative to not having the same object is given by v(m). Therefore, assuming that the money side of the transaction is seen as neither a gain or loss, and that v(m) represents the value of the lottery to the decision maker17 a subject who is loss averse will also exhibit the endowment e↵ect – i.e. the typical Willingness to Pay/Willingness to Accept gap.
Another appraoch that indicates a link between loss aversion and the endowment e↵ect for lotteries is that of Koszegi and Rabin (2007). In their setup, the case in which the DM has to sell the lottery is one in which they have a stochastic reference point. In the simplified case of linear utility considered in Koszegi and Rabin (2007), the utility of receiving an amount x when the reference point is y is given by
x + µ(x y) if x > y x + µ(x y) if x y
where is the loss aversion parameter. For stochastic prospects and reference points, expectations are taken over these utilities. Thus, the utility of keeping a lottery (0.5, x; 0.5, 0) lottery that one is endowed with it given by
x 2 + 1
4(1 )µx
while the utility of selling the same lottery at price p is given by p + 1
2(µ (p) µ(x p))
Similarly the utility of buying the lottery for price p (when the reference point is not having the lottery) is given by
x 2 +1
2(µ (x p) µ(p)) , while the utility of not buying the lottery is given by p.
Koszegi and Rabin (2007) show that a DM who has > 1 (i.e. exhibits loss aversion) will exhibit an endowment e↵ect for risk. The break even selling price p⇤ of the lottery, such
that x
2 + 1
4(1 )µx = p⇤+ 1
2(µ (p⇤) µ(x p⇤))
17Admittedly a strong assumption, but one that Tversky and Kahneman (1991) claim to have circumstan- tial evidience for.
will be strictly greater that the break even purchace price - i.e. the p⇤⇤ such that x
2 +1
2(µ (x p⇤⇤) µ(p⇤⇤)) = p⇤⇤.
2.3 Models of Present Bias
We consider three theoretical models that predict links between risk and time preferences:
the standard model of exponential discounting, the hyperbolic discounting model, and a model in which the future is seen as inherently uncertain. These predictions are couched in terms of the relationship between discounting and the curvature of the utility function on the one hand, and the degree of probability weighting on the other. It is important to note that none of the measures that we describe in Section 2.1 of the paper relate precisely to these concepts.
Exponential Discounting. The standard model of time separable exponential discounting (used the vast majority of economics) predicts a link between intertemporal choice and risk aversion through the curvature of the utility function: the higher the curvature, the higher the risk aversion, and the higher the discount. As an illustration, consider an agent who has a constant relative risk aversion (CRRA) utility function in each period
u(x) = x1 ↵
1 ↵
for 0 < ↵ < 1, where higher ↵ means a higher risk aversion. Now consider the monetary value c1 such that the agent is indi↵erent between receiving c1 today an $10 in one period’s time. This will be the case if u(c1) = u(10), where is the discount factor, which leads to
c1 = 1 ↵1 10.
Thus, an increase in the curvature of the utility function (i.e. an increase in ↵) leads to a decrease in x (as < 1). Notice that a decision maker comparing c2 in one period’s time to $10 in two period’s time would also exhibit such a relation, as in this case we would have u(c2) = 2u(10), and so we would have again that c2 = 1 ↵1 10. Thus the correlation between curvature of the utility function and the discount rate to the present should be exactly the same as its correlation with the discount rate to the future.
It is well known that the exponential discounting model does not allow for present bias.
Furthermore, the standard model does not predict any relationship between probability weighting and discounting behavior.18
18We should note that the presence of a link between risk aversion and time preferences predicted by the standard economic model is often considered problematic, as it follows from the implicit assumption that a unique parameter, the curvature of the utility function, a↵ects both intertemporal tradeo↵s and risky choice.
Other models, most notably Kreps and Porteus (1978), propose instead to use two distinct parameters, therefore eliminating the connection suggested in the standard model. Indeed the lack of the predicted connection would provide an even stronger support for the use of these general models.
Hyperbolic Discounting. One way that the standard model has been generalized in order to allow for present bias is by allowing for discount functions that are not exponential. Here, we consider the quasi-hyperbolic discount function popularized by Laibson (1997).19 Thus, income in one period’s time is discounted by , and in two period’s time by 2. Repeating the thought experiments above, we see that
c1 = ( )
1 ↵1
10, c2 = ( )
1 ↵1
10.
Thus, as < 0, we have that c1 < c2 and so we have present bias. Present Bias is measured as
c2 c1 10 = ( )
1 ↵1
( )
1 ↵1
,
which is increasing in the curvature of the utility function.20 As a given change in ↵ a↵ects present discounting more than future discounting, ceteris paribus it should also be the case that curvature of the utility function is more tightly correlated to discounting to the present that to discounting to the future.
This model predicts no link between present bias and probability weighting.
The Future as a Risky Prospect. A second potential channel between risk/uncertainty attitudes and intertemporal choice is that the future can be seen as risky or uncertain. In particular, we can assume that the decision maker treats today’s payments as certain, while she assigns a strictly positive probability of not being able to enjoy future payments, be it for some risk of not being able to receive or enjoy the prizes (e.g. she doesn’t fully trust the experimenter), or more simply the mortality rate. Typically, models that see the future as risky assume a constant per period hazard rate. Thus, we have that
u(c1) = ((1 )) u(10) u(c2) = ((1 )2)
((1 )) u(10)
19Our focus on quazi-hyperbolic discounting is without loss of generality since we only consider 3 periods.
20While this is true for all utility functions such that u 1( u(10)) = u 1( )10, it may not be true in general. For a general utility function:
c2 c1
10 = u 1( u(10)) u 1( u(10)) 10
=
R u(10) 0
R @2u 1
@x2 (x)d(x)d(x) R u(10) 0
R @2u 1
@x2 (x)d(x)d(
10 x)
=
R u(10) u(10)
R @2u 1
@x2 (x)d(x)d(x) 10
An increase in risk aversion decreases the second derivative of u, but increases the second derivative of u 1 for all x. However, the range over which the integral will be evaluated also changes, due to the changes in u(10).Thus the direction of change in present bias is undetermined
where is the per period probability of payment not occuring, and is the probability weighting function. This model implies that present bias will be given by
c2 c1
10 = ( ((1 )) )1 ↵1
✓ ((1 )2) ((1 ))
◆1 ↵1
(2) As shown by Halevy (2008) and Saito (2012) there is now a tight link between non-expected utility attitudes towards risk (e.g. probability weighting) and present bias. Note that an agent who exhibits no probability weighting will exhibit no present bias, as
((1 )2)
((1 )) = (1 )2
(1 ) = (1 ) = ((1 )) .
Thus, a subject should exhibit present bias only if they exhibit probability weighting. More- over, the type of probability weighting required for present bias is the same as that required for the common ratio e↵ect. To see why, notice that present bias requires that, for some prize x and k = (1 )
(1, c1; 0, 0) ⇠ (1 , x; , 0)
(k, c1; 1 k, 0) (k(1 ), x; (1 k(1 ), 0)
where (p, x; (1 p), y) is the prospect that gives prize x with probability p and y otherwise.
This is precisely a violation of common ratio invariance, of which the common ratio e↵ect is another example.21 Diecidue et al. (2009) show that a probability weighting function exhibits common ratio invariance if and only if it is a power function.
Thus, this theory predicts a strong relationship between the common ratio e↵ect and present bias. Most weighting functions used in practice would also imply a correlation between present bias and the common consequence e↵ect. Unless is extremely large, we would also expect probability weighting to be positively weighted with discounting to the present. While probability weighting to the future could in fact be positive or negatively correlated to discount rate to the future, for reasonable parameter values the former is more likely than the latter.22
As with the quasi-hyperbolic discounting model, we expect the curvature of the utility function to be correlated with discounting to the future, more correlated with discounting to the present, and positively correlated with present bias.
21Recall that the common ratio e↵ect requires (1, x; 0, 0)⇠ (0.8, y; 0.2, 0) and (k, x; (1 k), 0) (0.8k, y; 1 0.8k, 0) for k = 0.25.
22Assuming, for example, Prelec (1998)’s one parameter weighting function and = 0.01, we have negative correlation for extremely high degrees of probability weighting but positive correlation for more reasonable values.
Appendix C: Experimental Instructions
Introduction:Welcome!
This experiment is designed to study decision making. The main part of the experiment will include 1 practice section and 8 short experimental sections. In each section you will be asked to answer a number of questions. Specific instructions will be given at the start of each section.
At the end of the experiment, one question will be selected at random from those you answered from the 14 experimental sections. The amount of money that you get at the end of the experiment will depend n your answers to these questions. Anything you earn will be added to your show-‐up fee of $10.
Unless otherwise stated, you will be paid with cash at the end of the experiment.
Please turn of cellular phones now.
We will start with a brief instruction period. During this instruction period, you will be given a
description of the main features of the experiment and will be shown how to use the program. If you have any questions during this period, please raise your hand.
After you have completed the experiment, pleas remain quietly seated until everyone has completed the experiment.
Most questions in the experiment will take the form of lists of choices. For example, Question A might ask you to choose between receiving some amount of money (say $12) in one week’s time, or different amounts of money now. In such a case, Question A would look like this:
For each line in the list, you must choose between the option on the left or the option on the right.
Note that on each line, the option on the left stays the same in each row, while the option on the right gets better as one goes down the list.
You can select the option you like by clicking on the button next to that option.
If question A was then selected as the one that will be paid at the end of the experiment, then ONE line will be selected at random from those in Question A, and you will be paid according to your choice on that line. That is, if Question A was selected, then a line would be randomly chosen between the first line (choice between $12 in 1 week's time and $1.00 today) and the last line (choice between $12 in 1 week's time and $7.00 today) with equal probability. If, for example, the first line was chosen, then your payment for the experiment would depend on your choice on the first line. If you had chosen ‘$12 in 1 week’s time’, then that is what you would receive-‐ $12 in one week’s time. If you have chosen $1.00 today, then that is what you would receive.
At the start of each round, all the buttons will be unselected. At the bottom of the screen there will be two buttons: ‘Auto complete left’ and ‘Auto complete right’. Clicking the `Auto Complete Left’ button at any time will select the left option on each line of the list for which you have not already made a choice.
Similarly, ‘Auto Complete Right’ will select the right option for each line in which you have not made a selection. Clicking the 'Clear Selection' button will reset all of the buttons.
You are free to change your selections at any time, whether or not you have used an Auto complete button. Once you have made a selection on every line, you may press the ‘Next’ button to move on to
the next question. Once you click 'Next', you proceed to the next question. You cannot go back and modify your answer to previous questions.
Practice Questions:
The experiment will begin with three practice questions, designed to familiarize you with the program.
The questions asked in this section will not be selected for payment. They have been included to give you an idea of what the questions will be like.
In these questions, you will be asked to choose between receiving a certain amount of money for sure, or playing a lottery. These lotteries will be represented in the following way:
This lottery ticket has a 60% chance of winning $5 and a 40% chance of winning $3.
Remember, this is a practice round, and these questions cannot be selected for payment.
There are 3 practice questions.
You have now completed all three practice questions, and are ready to begin the experiment.
Please note that the following questions are no longer practice, and may be selected as the questions that determine your payment.
Section 1:
In this section of the experiment you will be asked questions about amounts of money that you may receive IN THE FUTURE. In particular, these questions will concern amounts of money that you may receive after some delay – for example $5 received in 10 days’ time.
For the questions in this section only, payment will not be made in cash. Instead, we will prepare a check that you will receive after the relevant delay. This check will be made out today and placed in an
envelope which you will be asked to address. This envelope will then be mailed at the relevant time. For example, if the payment was to be made today then it will be sent today. If payment is in a week’s time, then the envelope will be mailed in seven days’ time.
There are 3 questions in this section.
Section 2:
In this section of the experiment, you will be given various lottery tickets. These lotteries will be represented in the following way:
This lottery ticket has a 60% chance of winning $5 and a 40% chance of winning $3.
This lottery is yours to keep (if this is one of the questions that is selected at the end of the experiment).
However, you will be offered the opportunity to exchange this lottery for certain amounts of money (for example $5).
There are 3 questions in this section.
Section 3:
In this section you will be asked to make choices between different lotteries. These questions will also be presented in the form of a list. Here is an example:
On the first line of the list you are asked to choose between the 'lottery' and the 'alternative lottery with a value of x=$0.50'. Thus, on this line, you have to choose either a 50% chance of winning $6 and 50%
chance of winning $0, or a 40% chance of winning $6, a 40% chance of winning $0 and a 20% chance of winning $0.50.
On the second line you are asked to choose between the 'lottery' and the 'alternative lottery with a value of x=$1.00'. Note that the option on the left (the lottery) stays the same one each line, with the option on the right (the alternative lottery with a value of x equal to some number) gets better as one goes down the list.
There are 15 questions in this section.
Section 4:
In this section of the experiment you will be asked to make choices based on the bags you can see at the front of the room. Notice that each question refers to a different bag.
These bags contain poker chips that are either red or black in color. You may be given some information about the number of red or black chips in the bag.
At the end of the experiment, a chip will be drawn from each bag by the research assistant. You will be asked to bet on the color of the chip that will be drawn, red or black. If the chip extracted is of the color you have bet on, then you win the bet (you will be told the amount you will win in each question).
Otherwise, you lose the bet, and get nothing. After the end of the experiment, you will be free to inspect the contents of each bag, if you so wish.
Once you have made your bet, you will be asked to choose between this gamble and different amounts of money. If you choose the gamble, then you 'play that gamble', and the amount of money you will win will depend on the color of the chip that will be extracted. If you choose to take the money, then you will receive that amount of money regardless of the color of the chip drawn.
For example, imagine that you have been told that a bag has 5 red and 5 black chips and that if you correctly predict the color of the chip that is extracted, you will win $3. Imagine that (again, for example) you choose to bet that a red chip will be drawn. Therefore, if you keep this gamble you will get $3 if a red chip is drawn and $0 otherwise. You are then asked if you would prefer to keep this gamble, or exchange it for $1. If you choose to make the exchange, you will get $1, regardless of the color of the chip drawn. If you choose to keep the gamble, then you will receive $3 if a red chip is drawn and $0 otherwise. In other words, you have to choose between the gamble and the amount of money before you discover which chip will be drawn from the bag.
There are 9 questions in this section.
Section 5:
For questions in this section of the experiment, you will be given an extra $10. That is, if a question from this section of the experiment is chosen as one that will be rewarded at the end of the experiment you will be given an extra $10 on top of your show up fee.
In this section of the experiment, you will be given various lottery tickets. This lottery is yours to keep (if this is one of the questions that is selected at the end of the experiment). However, you will be offered the opportunity to exchange this lottery for other alternatives. Both the lottery and the alternative may involve LOSING money. These losses will be taken out of the $10 you have been given for these
questions.
There are 3 questions in this section.
Section 6:
For questions in this section of the experiment, you will be given an extra $10. That is, if a question from this section of the experiment is chosen as one that will be rewarded at the end of the experiment you will be given an extra $10 on top of your show up fee. In this section you will be asked to make choices between different lotteries, which may involve LOSING money. These losses will be taken out of the $10 you have been given for these questions. These questions will also be presented in the form of a list. Here is an example:
On the first line of the list you are asked to choose between the 'lottery' and the 'alternative lottery with a value of x=-‐$5.50'. Thus, on this line, you have to choose either a 50% chance of losing $6 and 50%
chance of losing $0, or a 40% chance of losing $6, a 40% chance of losing $0 and a 20% chance of losing
$5.50 from the $10 that you were given.
On the second line you are asked to choose between the 'lottery' and the 'alternative lottery with a value of x=-‐$5.00'. Note that the option on the left (the lottery) stays the same one each line, with the option on the right (the alternative lottery with a value of x equal to some number) gets better as one goes down the list.
There are 3 questions in this section.
Section 7:
In this part of the experiment you will be paired with another person in this room. At no point will you know who you are paired with, nor will the person you are paired with know who you are.
You will either be player 1 or player 2. If you are player 1, then the person you are paired with is player 2, and vice versa.
Player 1 receives $5. She can keep this money if she so wishes, or send a proportion of it to player 2. Any amount sent to player 2 will be TRIPLED by the experimenter. That is, if player 1 sends $0, player 2 receives $0. If player 1 sends $5, player 2 receives $15.
Player 2 receives any amount that player 1 decides to send (after it has been tripled by the
experimenter). Player 2 can keep this money if she so wishes, or send a proportion of it BACK to player 1. Any amount sent to player 1 will NOT be tripled by the experimenter. That is, if player 2 sends back
$0, player 1 will receive $0. If she sends back $15, then player 2 will receive $15.
The payment for player 1 is the amount of money that she did not send to player 2 PLUS the amount of money that player 2 sends back to her. The payment for player 2 is how much money she keeps (i.e.
does not send back to player 1).
You will not learn whether you are player one or player two until after you have made your decisions.
Thus, you will be asked how much money you would send to player two, IF YOU WERE PLAYER 1. Then, you will also be asked how you would behave IF YOU WERE PLAYER 2, that is, how much you would choose to send back to player 1 depending on how much player 1 has sent you.
There are 2 questions in this section.
Section 8:
In this section of the experiment, you will be given the opportunity to purchase various lottery tickets.
These lotteries will be represented in the following way:
This lottery ticket has a 60% chance of winning $5 and a 40% chance of winning $3.
For questions in this section of the experiment, you will be given an extra $10. That is, if a question from this section of the experiment is selected as one that will be rewarded at the end of the experiment you will be given an extra $10 on top of your show up fee.
In each question you will be offered the opportunity to buy a lottery ticket. That is, you will be offered the opportunity to use some of this additional $10 in order to buy a lottery ticket. If you choose to do so (and that question is selected as one that will be rewarded), then you will pay the lottery the specified cost, and you would keep the remaining amount of money and the lottery.
For example, if you decide to pay 50c for the lottery, you get to keep $9.50 and you get the outcome of the lottery.
There are 3 questions in this section.
Survey and Ravens:
You have now completed the main section of the experiment. We will now ask you to complete a post-‐
survey questionnaire. This questionnaire asks some questions about your background. It also includes some questions designed to test various aspects of your personality. You do not have to answer any question that you do not wish to. Not answering a question will not affect the amount you are paid for the experiment.
You will now be presented with 17 questions. Each question will present you with a 3 x 3 matrix of images, with one missing. Please identify the missing element that completes the pattern. For example, examine the following image:
The correct answer would be 8, as the 8th image completes the pattern.
Appendix D: Typical Screenshot
Figure 1: Typical Screenshot
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