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A Bayesian examination of information and uncertainty in contingent valuation

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Abstract

A theoretical framework is presented to explain how agents respond to information under uncertainty in contingent valuation surveys. Agents are provided with information signals and referendum prices as part of the elicitation process. Agents use Bayesian updating to revise prior distributions. An information prompt is presented to reduce hypothetical bias. However, we show the interaction between anchoring and the information prompt creates a systematic bias in willingness to pay. We test our hypotheses in an experimental setting where agents are asked to make a hypothetical, voluntary contribution to a public good. Experimental results are consistent with the model.

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Notes

  1. Bayesian updating has been used to describe decision making under uncertainty in a variety of economic contexts such as learning about workplace risk (Viscusi and O’Connor 1984), herding behavior and information cascades (Anderson and Holt 1997), and global climate change (Cameron 2005).

  2. This method has been a popular method for eliciting the value of non-market goods. For an overview of this literature see Brookshire et al. (1982), Hausman (1993), Mitchell and Carson (1989), and Cummings et al. (1986).

  3. See Murphy et al. (2005a) for a review of the hypothetical bias literature in contingent valuation.

  4. We later extend our theory to address the double-bounded dichotomous choice (DBDC) format, where the issue of incentive incompatibility is discussed.

  5. Herriges and Shogren (1996) and McLeod and Bergland (1999) use a Bayesian approach to examine the issues of anchoring bias (where agents are induced by the question format itself to anchor their responses to an opening referendum price) and incentive incompatibility (where agents are induced by the question format to provide untruthful responses) in CV surveys. Unlike their studies, however, we aim to provide a more formal and general theory of the origins of hypothetical bias and the Bayesian updating process during the value elicitation process.

  6. The empirical evidence is mixed on whether cheap talk is, in general, an effective means of eliminating hypothetical bias in CV and field experiments. Cummings and Taylor (1999) find that a long cheap-talk script is effective in eliminating hypothetical bias. List (2001) and Lusk (2003) use a script similar to that of Cummings and Taylor and find that cheap talk only works for inexperienced consumers. Poe et al. (2002) report that a shorter cheap-talk script is ineffective in eliminating hypothetical bias; Loomis, Gonzales-Caban and Gregory (1994) and Neil (1995) find that reminders about budget constraints and substitutes also are ineffective. Aadland and Caplan (2003) find that, although cheap talk is ineffective overall, it successfully reduces hypothetical bias for certain groups of respondents. However in other work, Cummings, Harrison and Taylor (1995b) and Aadland and Caplan (2006) use a shorter script and find that cheap talk may even exacerbate the hypothetical bias. We offer a theory that is independent of script length and has the potential to explain some of the results that script length cannot. We note, however, that it would be fairly straightforward to incorporate a script-length effect into our Bayesian framework whereby long scripts evoke a larger WTP revision than short scripts.

  7. See Hogg and Craig (1978) for a discussion of Bayesian estimation.

  8. Throughout the paper, we refer to τ i as the “announced price”. In the contingent valuation literature, it is common to refer to τ i as the “referendum bid” or “bid”. We avoid using the term bid in this paper so as not to create any confusion associated with its use in other areas of economics such as auction theory. In the experiment described below subjects are presented with “investment” levels in the public good.

  9. In practice, not all cheap-talk scripts directly inform agents of the magnitude of the hypothetical bias. Instead they often report differences in actual and hypothetical participation rates for public programs or goods (e.g., Cummings and Taylor 1999; Lusk 2003), percentage difference in stated and revealed WTP e.g., (e.g., List 2001), or make a general statement that WTP tends to be misstated in hypothetical scenarios (e.g., Carlsson, Frykblom and Lagerkvist 2005; Aadland and Caplan 2006). It would be fairly straightforward to modify our theory so that, rather than being directly informed of μ and knowing it with certainty, the agent received an indirect signal about μ and was required to infer its value.

  10. The weighted-average form of the updating function in Eq. 11 results if g i (s i |δ i ) is a normal distribution and \(\alpha=\sigma_{h}^{2}/(\sigma_{g}^{2}+\sigma_{h}^{2})\), where \(\sigma_{g}^{2}\) is the variance of g i (s i |δ i ) and \(\sigma_{h}^{2}\) is the variance of the prior distribution h i (δ i ). The formal derivation of Eq. 11 is shown in Appendix 1.

  11. See Laibson and Zeckhauser (1998) for a discussion that relates Kahneman and Tversky’s work to the burgeoning field of “behavioral economics”.

  12. Some may argue that agents are unlikely to adjust their WTP perfectly to the signal c i  = μ. For simplicity, we assume perfect adjustments; however it is important to recognize that the subsequent results are robust to partial adjustments where 0 < E i (δ i |c i  = μ) < μ.

  13. An interesting implication of this result is that samples with a substantial number of nay-sayers (i.e., low WTP individuals (Carson 2000)) will appear to be more often associated with effective cheap-talk scripts.

  14. For further details on public good experiments, see chapter two in the Handbook on Experimental Economics by Ledyard (1995).

  15. Subjects were informed that not everyone in the group was receiving the same price but were not informed of the distribution of prices across players. In standard CV surveys, agents are given a randomized announced price but generally do not inquire about (and thus are not made aware of) the prices other respondents receive. This is because the cooperative nature of the public good game is not made explicit in field surveys, and because respondents complete the survey independently of one another, thus precluding the need to provide additional knowledge to respondents. The provision of this information represents a deviation from CV surveys in practice, but we do not feel that this alters the fundamental behavioral motives associated with hypothetical bias, cheap talk, and anchoring bias in our experiments.

  16. Recall that the cheap-talk meaure Δ i in Eq. 13 varies across all agents. Here, we are interested in specifying an estimable equation with a constant cheap-talk coefficient, Δ, that is similar to that commonly estimated in the literature and that will enable us to highlight the biases associated with failing to recognize the interaction between cheap talk and anchoring. Also, note that although i in Eq. 13 is defined as the difference between expected values (with and without cheap talk) for the same agent, the econometric analysis will contrast the expected WTP of one set of agents that receive cheap talk (treatment group) with a different set of agents that do not receive cheap talk (control group), holding all other observable factors constant.

  17. As pointed out by an anonymous reviewer, responses to open-ended WTP questions are governed by different incentive compatibility properties than responses to referendum questions. As such, one should exercise caution when using open-ended questions to guide responses to referendum questions. The extent to which the initial open-ended question might alter the agent’s response to cheap talk and the subsequent referendum question is an open and interesting research question.

  18. We also estimated WTP controlling for the demographic variables elicited on the last page of the experiment (see Appendix 2). The control variables include age, gender, income, college GPA, college rank and degree of risk aversion. The observed heterogeneity associated with these variables was not capable of explaining the willingness to invest in the public good. Most of the coefficient estimates associated with the demographic variables were statistically insignificant and did not qualitatively change the estimates of the anchoring and cheap-talk parameters.

  19. The early literature on incentive incompatibility in CV studies (Cummings et al. 1995a, 1997) appears to characterize incentive incompatibility more broadly than some of the more recent studies. For example, Cummings et al. (1995a) on page 260 state that incentive compatibility “implies that subjects will answer the CVM’s hypothetical question in the same way as they would answer an identical question asking for a real commitment.” Whitehead (2002) in a more recent study states on page 287 that in DBDC formats “if the follow-up questions are not incentive compatible, stated willingness to pay will be based on true willingness to pay with a shift parameter.”

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Author information

Authors and Affiliations

  1. Department of Economics and Finance, University of Wyoming, P.O. Box 3985, Laramie, WY, 82071-3985, USA

    David M. Aadland & Owen R. Phillips

  2. Department of Economics, Utah State University, 3530 Old Main Hill, Logan, UT, 84341, USA

    Arthur J. Caplan

Authors
  1. David M. Aadland
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  2. Arthur J. Caplan
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  3. Owen R. Phillips
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Corresponding author

Correspondence to David M. Aadland.

Additional information

Support from the Paul Lowham Research Fund is gratefully appreciated. We thank Chris McIntosh and NeilWilmot for their assistance with the experiments. Helpful comments on earlier drafts were received from Don McLeod and participants at the American Agricultural Economic Association meetings, Western Agricultural Economic Association annual meetings, the joint U.S. Forest Service / Colorado State University seminar series, University of Kentucky, and the CU Environmental and Resource Economics Workshop.

Appendix

Appendix

1.1 Appendix 1: Derivation of the Bayesian weighted-average updating function

Start by considering Bayes’ formula

$$ k(\textrm{WTP}|\tau)\propto g(\tau|\textrm{WTP})h(\textrm{WTP}) $$

where ∝ stands for “proportional to” as the marginal distribution for τ is dropped. This is standard in Bayesian analysis. Now let the conditional and prior distributions be

$$\begin{array}{*{20}l} g(\tau|\textrm{WTP}) & \sim N\left(\textrm{WTP},\sigma_{g}^{2}\right)=\frac{1}{\sqrt{2\pi}\sigma_{g}} \exp\left[ \frac{-(\tau-\textrm{WTP})^{2}}{2\sigma_{g}^{2}}\right] \\ h(\textrm{WTP}) & \sim N\left(\textrm{WTP}_{0},\sigma_{h}^{2}\right)=\frac{1}{\sqrt{2\pi}\sigma_{h}} \exp\left[ \frac{-(\textrm{WTP}-\textrm{WTP}_{0})^{2}}{2\sigma_{h}^{2}}\right] . \end{array}$$

The posterior distribution is then

$$ k(\textrm{WTP}|\tau)\propto\frac{1}{2\pi\sigma_{g}\sigma_{h}}\exp\left[ -\frac {(\tau-\textrm{WTP})^{2}}{2\sigma_{g}^{2}}-\frac{(\textrm{WTP}-\textrm{WTP}_{0})^{2}}{2\sigma_{h}^{2} }\right] , $$

which after expanding the squared terms and dropping constants gives

$$ k(\textrm{WTP}|\tau)\propto\exp\left[ -\frac{\textrm{WTP}^{2}\left(\sigma_{g}^{2}+\sigma_{h} ^{2}\right)+\textrm{WTP}\left(-2\tau\sigma_{h}^{2}-2\textrm{WTP}_{0}\sigma_{g}^{2}\right)}{2\sigma_{g}^{2} \sigma_{h}^{2}}\right] . $$

Finally, we complete the square in WTP to get

$$ k(\textrm{WTP}|\tau)\propto\exp\left[ -\frac{\left( \textrm{WTP}-\frac{\tau\sigma_{h} ^{2}+\textrm{WTP}_{0}\sigma_{g}^{2}}{\sigma_{g}^{2}+\sigma_{h}^{2}}\right) ^{2}} {\frac{2\sigma_{g}^{2}\sigma_{h}^{2}}{\sigma_{g}^{2}+\sigma_{h}^{2}}}\right] . $$

This implies that the mean of the posterior distribution (or the updated WTP value) is

$$ E(\textrm{WTP}|\tau)=\frac{\tau\sigma_{h}^{2}+\textrm{WTP}_{0}\sigma_{g}^{2}}{\sigma_{g} ^{2}+\sigma_{h}^{2}}=\left[ \frac{\sigma_{h}^{2}}{\sigma_{g}^{2}+\sigma _{h}^{2}}\right] \tau+\left[ \frac{\sigma_{g}^{2}}{\sigma_{g}^{2}+\sigma _{h}^{2}}\right] \textrm{WTP}_{0} $$

which if we define \(\alpha=\sigma_{h}^{2}/\left(\sigma_{g}^{2}+\sigma_{h}^{2}\right)\), can be written as

$$ \textrm{WTP}_{1}=\alpha\tau+(1-\alpha)\textrm{WTP}_{0}. $$

1.2 Appendix 2: Experimental instructions for NCT and CT treatments

1.2.1 Instructions

This is an experiment on how people make investment decisions. There are no right or wrong decisions. You have been given $10 to participate. This is yours to keep. You will not be paid anything more. Before the experiment begins, an example of how the experiment works is described. The actual experiment will be conducted after going through this example.

Suppose there are five people, each of whom is given $2 that he or she can invest. The individuals have made the following decisions:

  • Person #1—Invests nothing.

  • Persons #2 and #3—Invest $1 each.

  • Persons #4 and #5—Invest $2 each.

This results in a total of $6 invested from the five people, for an average investment of $6/5 individuals = $1.20. Using the table below, we can now calculate the return on the investment for each person.

Payout Chart—This is only an example

Average group investment

Range of payouts based on your investment choice

“YES, I’ll invest”

“NO, I won’t invest”

Min payout

Mid payout

Max payout

Min payout

Mid payout

Max payout

Greater than $0; less than or equal to $1

$0

$1

$2

$1

$2

$3

Greater than $1; less than or equal to $2

$1

$2

$3

$2

$3

$4

Begin by noting that each person’s payout range is determined in part by his or her investment choice and the average investment of the group. The average investment of $1.20 falls between $1 and $2 so we can focus on the second row of numbers in the table. The exact payout is then determined by the roll of a die. The roll of the die gives equal chances to the Min, Mid and Max payouts. For both the “YES” and “NO” columns, if a 1 or 2 is rolled the Min is paid; if a 3 or 4 is rolled the Mid is paid; and if a 5 or 6 is rolled the Max is paid.

For example, assume a “3” is rolled, so the Mid payout occurs. Person #1 invested nothing. The average group investment was $1.20. Therefore, the person receives a final payout of $3 ($3 payout less $0 invested). Persons #2 and #3 each invested $1. They receive a payout of $2, and their net return is $1 ($2 payout less $1 invested). Persons #4 and #5 each invested $2 and also receive a final payout of $2. Their net return is zero.

Are there any questions before we begin?

1.2.2 Experiment

1.2.2.1 Directions

Use the payout chart below to decide whether to hypothetically invest all, part, or none of your $10. If this experiment were for real, your payout range would be determined by your investment choice and the average investment of the group. (Note that if the total group investment is zero, the payout is zero to everyone.) The exact payout would be determined by the roll of a die. For both the YES and NO columns, if a 1 or 2 is rolled the Min is paid; if a 3 or 4 is rolled the Mid is paid; and if a 5 or 6 is rolled the Max is paid.

Payout Chart

Average group investment

Range of payouts based on your investment choice

“YES, I’ll invest”

“NO, I won’t invest”

Min payout

Mid payout

Max payout

Min payout

Mid payout

Max payout

Greater than $0; less than or equal to $2

$0

$1

$2

$1

$2

$3

Greater than $2; less than or equal to $4

$3

$4

$5

$4

$5

$6

Greater than $4; less than or equal to $6

$6

$7

$8

$7

$8

$9

Greater than $6; less than or equal to $8

$9

$10

$11

$10

$11

$12

Greater than $8; less than or equal to $10

$12

$13

$14

$13

$14

$15

(No cheap talk) Experiment (page 2)

The payout chart below is reproduced from the previous page in order to help you answer the following question.

Payout Chart

Average group investment

Range of payouts based on your investment choice

“YES, I’ll invest”

“NO, I won’t invest”

Min payout

Mid payout

Max payout

Min payout

Mid payout

Max payout

Greater than $0; less than or equal to $2

$0

$1

$2

$1

$2

$3

Greater than $2; less than or equal to $4

$3

$4

$5

$4

$5

$6

Greater than $4; less than or equal to $6

$6

$7

$8

$7

$8

$9

Greater than $6; less than or equal to $8

$9

$10

$11

$10

$11

$12

Greater than $8; less than or equal to $10

$12

$13

$14

$13

$14

$15

(Cheap talk) Experiment (page 2)

The payout chart below is reproduced from the previous page in order to help you answer the following question.

Payout Chart

Average group investment

Range of payouts based on your investment choice

“YES, I’ll invest”

“NO, I won’t invest”

Min payout

Mid payout

Max payout

Min payout

Mid payout

Max payout

Greater than $0; less than or equal to $2

$0

$1

$2

$1

$2

$3

Greater than $2; less than or equal to $4

$3

$4

$5

$4

$5

$6

Greater than $4; less than or equal to $6

$6

$7

$8

$7

$8

$9

Greater than $6; less than or equal to $8

$9

$10

$11

$10

$11

$12

Greater than $8; less than or equal to $10

$12

$13

$14

$13

$14

$15

Before answering the next question please note that in previous runs of this experiment we found that people typically overstate their true willingness to invest by approximately $2.00 when asked to do so in a hypothetical setting like this. Please keep this in mind when answering the next question.

1.3 Demographic questions

Please answer the following questions to the best of your ability. These questions are very important to us. Remember that all information is completely anonymous and confidential.

  1. 1.

    Gender:                      Male     Female   

     

  2. 2.

    Age_____________

     

  3. 3.

    Class:                         Freshman     

     

                                      Sophomore  

     

                                      Junior           

     

                                      Senior          

     

                                      Graduate      

     

  4. 4.

    Cumulative GPA _____________

     

  5. 5.

    Have you declared a major?

     

    Yes             No   

     

    If yes, what is your major? __________________________

     

  6. 6.

    In which range do you think your before-tax annual income falls (income includes wages, salary, and money from parents but excludes student loans)?

     

        Less than $10,000.

     

        Greater than $10,000 but less than $20,000.

     

        Greater than $20,000 but less than $30,000.

     

        Greater than $30,000.

     

  7. 7.

    Which would you choose?

     

        $10 with certainty.

     

        50% chance of $0; 50% chance of $20.

     

        I’m indifferent between the two choices above.

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Aadland, D.M., Caplan, A.J. & Phillips, O.R. A Bayesian examination of information and uncertainty in contingent valuation. J Risk Uncertainty 35, 149–178 (2007). https://doi.org/10.1007/s11166-007-9022-9

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