Willingness to Pay for Imperfect Information: Evidence from a Newsvendor Problem

Abstract

We present an experimental study on the willingness to pay (WTP) for imperfect information in a newsvendor context. Like a newsvendor who has to decide on the amount of papers to order for a given day, subjects have to order a quantity of a good before market demand is realized. Furthermore, subjects may commission an expert who correctly forecasts demand with a probability of 0.9. In the real world, this expert might be represented by a market research company. The WTP is measured with the Becker DeGroot Marschak-mechanism, and subject’s risk aversion is evaluated through their order behavior. We investigate the potential effects of conservatism and base rate fallacy, two well-known biases in Bayesian updating, on decision behavior by varying the skew of the demand distribution. Subjects in our experimental setting show a tendency to generally overvalue information. Interestingly, subjects seem to take the attractiveness of the decision task into account when evaluating their WTP for the information.

Appendix

1.1 Appendix A: Calculation of VOI Under Risk Neutrality

The basic structure of the problem of calculating the VOI is presented in Fig. A.1 for the example of the negatively skewed demand distribution.

Fig. A.1

Decision tree for the negatively skewed demand distribution

The a priori probabilities of a certain state of nature are updated according to the information provided by the expert. Remembering that I _i is the information that the demand quantity is i, i = 2, 4, 6, the probability of the demand being D given I _i is calculated using Bayes’ theorem:

$$ w(\left. D \right|{{I}_{i}})=\frac{w(D )\cdot w(\left. {{I}_{i}}\right|D )}{w({{I}_{i}})} $$

The a posteriori probabilities for the three demand distribution used are given in Table A.1.

Table A.1 A posteriori probabilities for the three demand distributions

If, e.g., the negatively skewed demand distribution is present and information I ₂ is given by the expert, the DM’s decision matrix looks as demonstrated in Table A.2.

Table A.2 Decision matrix under negative skew if the information I ₂ is present

$E[\pi (q )\left| {{I}_{i}}\right. ]$ is the expected profit of the order quantity q when information I _i is observed. In the case of i = 2, the risk neutral DM would consequently choose an order quantity of q = 2. Let

$$ E[{{I}_{i}}]=\underset{q}{\mathop{\max }}\,(E[\pi (q )\left| {{I}_{i}}\right. ] ) $$

then the expected profit with information is calculated as ($\overline{D}$ represents the set of all possible demand quantities):

$$ EI=\sum_{i\in \overline{D}}^{{}}{w({{I}_{i}})\cdot E[{{I}_{i}}]} $$

Subtracting the expected value without information, we obtain the VOI: VOI = EI–E[$\pi (q )$].

1.2 Appendix B: Individual Risk Preferences and VOIs for the Three Demand Distributions

Table A.3 Individual risk preferences and VOIs for the three demand distributions

1.3 Appendix C: Difference Between VOI_reg and VOI for Varying Degrees of Regret for Ex Post Inventory Errors (Fig A.2)

Fig. A.2

Difference between VOI_reg and VOI for the different demand distributions