Researching Preferences, Valuation and Hypothetical Bias

Abstract

A number of recent papers in environmental economics have focused on the process of researching preferences – agents are uncertain about preferences but with effort may narrow their uncertainty. This issue has arisen in formulating bids in contingent valuation (CV) as well as the debate over the divergence between WTP and WTA. In the context of CV, it has been suggested that the hypothetical nature of the preference elicitation process biases responses. This paper provides both a theoretical model and experimental evidence to contribute to this debate. The model is a model of competitive bidding for a private good with two components that are particularly relevant to the debate. The first component is that bidders are unsure of their own value for the private good but may purchase information about their own value (researching preferences). The second component is that there is a probability that the auction is hypothetical – that the winning bidder will not get the private good and will not pay the winning bid. The experiment tests this theoretical model of bidding equilibrium and analyzes the effects of variations in the parameters (hypotheticalness, information costs and number of agents) on the endogenous variables (such as the proportion of bidders who become informed and the winning bid). Experimental results suggest that an increase in the hypotheticalness of an auction tends to decrease the likelihood that bidders pay for information on their valuation with an ambiguous effect on the winning bid.

Appendix

Proof of Proposition 2

Consider the equilibrium condition (2) as an implicit function in p and N, and – just for the sake of exposition – assume that the first inequality holds as equality. Then, the equilibrium condition is:

$$H(p,N)=0$$

(A.1)

where the function H is defined as

$$H(p,N)\equiv\Omega(\Pi(p,N))-p$$

(A.2)

Therefore,

$$ \frac{{\hbox{d}}p}{{\hbox{d}}N}=-\frac{\partial H/\partial N}{\partial H/\partial p} =-\frac{(\partial\Omega/\partial\Pi)(\partial\Pi/\partial N)} {(\partial\Omega/\partial\Pi)(\partial\Pi/\partial p)-1}\ge 0 $$

(A.3)

Now, since the equilibrium p, p ^e, increases with N (provided Ω is not “flat”), it follows from the expressions (A.1)–(A.2) and the monotonicity of Ω that the equilibrium value of Π(p ^e, N) should be smaller the greater the value of N. Since the density function f has not changed, this is only possible through an increase of the exponent [(1−p)N−1] in expression (1).

Proof of Proposition 3

Let us define:

$$N^{\ast}\equiv (1-p)N$$

(A.4)

and

$$B(z, N^{\ast})=z-\int\frac{z}{V} \{\frac{F(\Theta)}{F(z)}\}^{[N^{\ast}-1]}{\hbox{d}}\Theta \quad \hbox{if}\,z\ge \overline{V\,} $$

(A.5a)

$$ =\overline{V\,}\quad \hbox{if}\,z\ge\overline{V\,} $$

(A.5b)

clearly,

$$ (i)\quad \frac{\partial B(z,N^{\ast})}{\partial z}\ge 0 $$

(A.6a)

$$ (ii)\quad \frac{\partial B(z,N^{\ast})}{\partial N^{\ast}}\ge 0 $$

(A.6b)

and that the expectation of the winner bid can be written simply as:

$$ (iii)\quad E(b)=\mathop{\int}\nolimits^{V_{\rm u}}_{V_{\rm l}}B(z,N^{\ast}) \gamma_{N^{\ast}}(z){\hbox{d}}z $$

(A.6c)

Now,

$$ \frac{{\hbox{d}}E(b)}{{\hbox{d}}N}=\frac{{\hbox{d}}E(b)}{{\hbox{d}}N^{\ast}}\cdot\frac{{\hbox{d}}N^{*}}{{\hbox{d}}N} $$

(A.7)

From the proof of Proposition 2, we already know that the second factor is positive. Thus, it suffices to prove that the first is non-negative. Differentiating (A.6c), we can say that:

$$\frac{{\hbox{d}}E(b)}{{\hbox{d}}N^{*}}=\int\nolimits_{V_{\rm l}}^{V_{\rm u}}\frac{\partial B(z,N^{*})}{\partial N} \gamma(z){\hbox{d}}z+\int\nolimits^{V_{\rm u}}_{V_{\rm l}} B(z,N^{*})\frac{\partial \gamma(z)}{\partial N^{*}}{\hbox{d}}z $$

(A.8)

where, for notation convenience, the subscript of the density function γ _N* has been dropped. It follows from (A.6b) that the first term in this expression is nonnegative. In turn, the second term can be written as:

$$ \mathop{\int}\nolimits_{0}^{V_{\rm u}}B(z,N^{*})\frac{\partial \gamma(z)}{\partial N^{*}}{\hbox{d}}z \equiv \mathop{\int}\nolimits_{V_{\rm l}}^{V_{\rm u}}B(z,N^{*})\frac{\partial}{\partial N^{*}}\left(\frac{\partial \Gamma(z)}{\partial z}\right){\hbox{d}}z $$

$$ \equiv \mathop{\int}\nolimits_{V_{\rm l}}^{V_{\rm u}}B(z,N^{*})\frac{\partial}{\partial z} \left(\frac{\partial \Gamma(z)}{\partial N^{*}}\right){\hbox{d}}z $$

(A.9)

which, by straightforward integration by part, yields:

$$ \equiv B(z,N^{*})\left(\frac{\partial \Gamma(Z)}{\partial N^{*}}\right)|_{V_{\rm l}}^{V_{\rm u}}-\mathop{\int}\nolimits_{V_{\rm l}}^{V_{\rm u}}\frac{\partial B(z,N^{*})}{\partial z}\frac{\partial \Gamma(z)}{\partial N^{*}}{\hbox{d}}z $$

(A.10)

Finally, we use the fact that, for all $N^{\ast}$, $\Gamma (V)=0$ and $\Gamma (V)=1$, so that its derivative at those points is zero. Therefore, we end with:

$$ \mathop{\int}\nolimits_{V_{\rm l}}^{V_{\rm u}}B(z,N^{*})\frac{\partial \gamma(z)} {\partial N^{*}}= -\mathop{\int}\nolimits_{V_{\rm l}}^{V_{\rm u}}\mathop{\frac{\partial B(z,N^{*})}{\partial z}}\limits_{(+)}\mathop{\frac{\partial}{\partial N^{*}}(F(z)^{N^{*}})}\limits_{(-)}{\hbox{d}}z \ge 0 $$

(A.11)

and the result follows immediately.

Proof of Proposition 4

For this case, it is convenient to rewrite the equilibrium condition (A.2) as:

$$p^{\rm e}=1 - \hbox{G}(\Pi (p^{\rm e})) $$

(A.12)

where G is the cumulative function corresponding to the density g. Next, let us begin by considering an arbitrary change in g. Since $\Pi(p^{\rm e})$ is not affected by the changes in g, it follows that changes in the initial equilibrium p ^e depend only on changes of G(Π(p)) when evaluated at that point. Formally, consider two different distribution functions, G₀ and G₁, with p ₀ and p ₁ as the corresponding equilibrium proportions of informed bidders. Clearly

$$ \Omega_{1}(\Pi(p_{0})) \ge \Omega_{0}(\Pi (p_{0})) \quad \hbox{iff}\quad \hbox{G}_{0}(\Pi(p_{0})) \ge \hbox{G}_{1}(\Pi (p_{0})) $$

(A.13)

where the subscript {0, 1} identifies the pre- and post-change functions. Therefore, p ₁ ≥ p ₀ iff G₁(Π (p ₀)) ≤ G₀(Π (p ₀)).

The intuition behind this is obvious. For any value of p, including the equilibrium p ^e, the value of information is independent of G. Thus, the original equilibrium can only be changed with changes in the proportion of bidders with information costs below or above the corresponding value of information.

To advance a further step, consider the case where changes in g induce an increase in p ^e and, consequently, a decrease in [(1−p)N−1]. The latter effect acts in the same way as an exogenous decrease of N. Hence, it causes a decrease of the integral in the expression (Equation (3)) for the expected winning bid, E(b), and a simultaneous shift of mass of the density $\gamma_{N^{\ast}}$ to the left. Of course, to have a strict change of p, the change in G must be big enough to overcome the discrete nature of p.

Proof of Proposition 5

The proof of this proposition hinges on showing the proportion of bidders choosing to become informed is the same in the two cases – the hypothetical case where the probability the auction is real is α and the information costs are distributed as g(x) and the real case with the auction definitely real and information costs distributed as $\hat g(x) \equiv \alpha g(\alpha x)$. If the proportion of bidders is the same, then bids will be unaffected by the fact that after the auction a lottery is held to determine if the auction was real.

Equation (3) defines the proportion of informed bidders in the hypothetical case, except that profits must be diluted by the probability that the auction is real, α:

$$ \Omega(\alpha \Pi(p^{e})) \ge p^{e};\quad p^{e}+\frac{1}{N}\ge \Omega(\alpha \Pi(p^{e}+\frac{1}{N})) $$

(A.14)

With the purely real auction, costs are distributed as $\hat g(x) \equiv \alpha g(\alpha x)$ which implies $\hat{\Omega}(x)\equiv \Omega (\alpha x)$. It is easy to see that this implies that the conditions defining the proportion of informed bidders in the purely real auction are the same as in Equation A.14. Thus the two auctions yield identical outcomes.