Balancing Cost Effectiveness and Incentive Properties in Conservation Auctions: Experimental Evidence from Three Multi-award Reverse Auction Mechanisms

Abstract

Government agencies are increasingly using economic incentives to encourage landowners to adopt conservation practices. Auctions enable agencies to identify land conservation practices with low opportunity costs. At the same time, landowners’ opportunity costs contain useful information for government agencies to rank conservation priorities. This paper introduces a new reverse auction mechanism that performs well both from the cost effectiveness and cost-revelation perspectives and compares three multi-award reverse auction mechanisms. The first mechanism is called the Uniform Price Reverse (UPR) auction, where each winning bidder is paid the lowest rejected bid. The second mechanism is called the First Price Reverse (FPR) auction, where winning bidders are paid their submitted bids. The third, novel, mechanism is called the Generalized Second Price Reverse (GSPR) auction, where each winning bidder is paid the bid that is immediately higher. Theoretically, I derive the equilibrium bidding strategy for each auction mechanism and show that a symmetric equilibrium strategy may not exist under the GSPR auction. Empirically, lab experiment results show that UPR and GSPR auctions lead to a higher efficiency level compared to FPR, while UPR auction yields the lowest auctioneer surplus and is the least cost effective. As a result, GSPR maintains good incentive properties similar to UPR and presents potential large cost-saving opportunities to the auctioneer.

Appendices

Appendix

A: Equilibrium Strategy in GSPR

Following Gomes and Sweeney (2014), the bidder i’s (with a private cost c) expected payment in equilibrium is denoted by E(p(c)), by the Revelation Principle, the E(p(c)) has to satisfy

$$\begin{aligned} c\in \underset{\hat{c}}{argmax} E(p(\hat{c}))-\sum _{k=1}^MG_k(\hat{c})c. \end{aligned}$$

(22)

According to the Integral-form Envelop Theorem (Milgrom and Segal 2002), equation (22) implies that,

$$\begin{aligned} E(p(c))-\sum _{k=1}^MG_k(c)c=\sum _{k=1}^M\int _c^{\bar{c}}G_k(\tilde{c})d\tilde{c}+E(p(\hat{c})), \end{aligned}$$

(23)

where $E(p(\hat{c}))=0$ as the highest cost bidder expects to receive zero payment with a zero probability of winning the auction. According to the definition of GSPR,

$$\begin{aligned} E(p(c))= & {} E(b_{(k+1)})\nonumber \\= & {} \sum _{k=1}^{M}G_k(c)E(s(c_{(k+1:N)})|c_{(k:N)}\le c\le c_{(k+1:N)})\nonumber \\= & {} \sum _{k=1}^{M}G_k(c)E(s(c_{(1:N-k)})| c\le c_{(1:N-k)})\nonumber \\= & {} \sum _{k=1}^{M}G_k(c)\int _c^{\bar{c}}s(\tilde{c})\frac{(N-k)(1-F(\tilde{c}))^{N-k-1}f(\tilde{c})}{(1-F(\tilde{c}))^{N-k}}d\tilde{c}. \end{aligned}$$

(24)

Based on equations (23) and (24)

$$\begin{aligned} \sum _{k=1}^MG_k(c)c+\sum _{k=1}^M\int _c^{\bar{c}}G_k(\tilde{c})d\tilde{c}=\sum _{k=1}^{M}G_k(c)\int _c^{\bar{c}}s(\tilde{c})\frac{(N-k)(1-F(\tilde{c}))^{N-k-1}f(\tilde{c})}{(1-F(c))^{N-k}}d\tilde{c}. \end{aligned}$$

(25)

where

$$\begin{aligned} G_k(c)= \begin{pmatrix} N-1 \\ k-1 \end{pmatrix} (1-F(c))^{N-k}(F(c))^{k-1}. \end{aligned}$$

(26)

Differentiating w.r.t. c on both sides,

$$\begin{aligned} \sum _{k=1}^M\frac{d G_k(c)}{d c}c\,=\,& {} -s(c)\frac{f(c)}{1-F(c)}\sum _{k=1}^{M}(N-k)G_k(c)\nonumber \\&+\sum _{k=1}^M\frac{\frac{dG_k(c)}{d c}(1-F(c))^{N-k}+G_k(c)(N-K)(1-F(c))^{N-k-1})f(c)}{(1-F(c))^{2N-2k}}\nonumber \\&\qquad \int _c^{\bar{c}}s(\tilde{c})(N-k)(1-F(\tilde{c}))^{N-k-1}f(\tilde{c})d\tilde{c}\nonumber \,=\,& {} -s(c)\frac{f(c)}{1-F(c)}\sum _{k=1}^{M}(N-k)G_k(c)\nonumber \\&+\sum _{k=1}^M\left( \frac{dG_k(c)}{d c}\frac{N-k}{(1-F(c))^{N-k}}+G_k(c)\frac{(N-k)^2f(c)}{(1-F(c))^{N-k+1}}\right) \nonumber \\&\qquad \int _c^{\bar{c}}s(\tilde{c})(1-F(\tilde{c}))^{N-k-1}f(\tilde{c})d\tilde{c}. \end{aligned}$$

(27)

Differentiating equation (26),

$$\begin{aligned} \frac{\partial G_k(c)}{\partial c}= & {} \begin{pmatrix} N-1 \\ k-1 \end{pmatrix}f(c)[1-F(c))^{N-k-1}F^{k-2}(c)[(k-1)(1-F(c))-(N-k)F(c)]\nonumber \\= & {} \frac{G_k(c)}{(1-F(c))F(c)}f(c)[(k-1)(1-F(c))-(N-k)F(c)]\nonumber \\= & {} (k-1)\frac{f(c)}{F(c)}G_k(c)-(N-k)\frac{f(c)}{1-F(c)}G_k(c). \end{aligned}$$

(28)

Combining equations (27) and (28),

$$\begin{aligned}&\sum _{k=1}^M\left( (k-1)\frac{f(c)}{F(c)}G_k(c)c-(N-k)\frac{f(c)}{1-F(c)}G_k(c)c\right) \nonumber \\&\quad =-s(c)\frac{f(c)}{1-F(c)}\sum _{k=1}^{M}(N-k)G_k(c)\nonumber \\&\qquad +\sum _{k=1}^M\left( (k-1)(N-k)\frac{f(c)}{F(c)(1-F(c))^{N-k}}G(c)\right) \int _c^{\bar{c}}s(\tilde{c})(1-F(\tilde{c}))^{N-k-1}f(\tilde{c})d\tilde{c}. \end{aligned}$$

(29)

Rewrite the above equation,

$$\begin{aligned}&\sum _{k=1}^M(N-k)\frac{f(c)}{1-F(c)}G_k(c)(s(c)-c)\nonumber \\&\quad =-\sum _{k=1}^{M}(k-1)\frac{f(c)}{F(c)}G_k(c)c\nonumber \\&\qquad +\sum _{k=1}^M\left( (k-1)(N-k)\frac{f(c)}{F(c)(1-F(c))^{N-k}}G(c)\right) \int _c^{\bar{c}}s(\tilde{c})(1-F(\tilde{c}))^{N-k-1}f(\tilde{c})d\tilde{c}. \end{aligned}$$

(30)

Therefore, we have

$$\begin{aligned} s(c)=c+\sum _{k=1}^M\gamma _s(c)\int _c^{\bar{c}}(c+s(\tilde{c}))(1-F(\tilde{c}))^{N-k-1}f(\tilde{c})d\tilde{c}. \end{aligned}$$

(31)

where

$$\begin{aligned} \gamma _s(c)=\frac{(k-1)(N-k)\frac{G_s(c)}{F(c)(1-F(c))^{N-k-1}}}{\sum _{t=1}^M(N-t)G_t(c)}. \end{aligned}$$

(32)

Specifically, when $N=5$, $M=2$, equation (31) implies

$$\begin{aligned} \frac{(s(c)-c)(4G_1(c)+3G_2(c))F(c)(1-F(c))^2}{3G_2(c)}=\int _c^{\bar{c}}(c+s(\tilde{c}))(1-F(\tilde{c}))^{2}f(\tilde{c})d\tilde{c}, \end{aligned}$$

(33)

or

$$\begin{aligned} (s(c)-c)\frac{(1-F(c))^2(1+2F(c))}{3}=\int _c^{\bar{c}}(c+s(\tilde{c}))(1-F(\tilde{c}))^{2}f(\tilde{c})d\tilde{c}, \end{aligned}$$

(34)

which can be further simplified to

$$\begin{aligned} \int _c^{\bar{c}}(c+s(\tilde{c}))(20-\tilde{c})^{2}d\tilde{c}=(s(c)-c)\frac{(20-c)^2(c+5)}{3}. \end{aligned}$$

(35)

Rewrite the above equation as

$$\begin{aligned} s(c)=c+\frac{3\int _c^{\bar{c}}(c+s(\tilde{c}))(20-\tilde{c})^{2}d\tilde{c}}{(20-c)^2(c+5)}. \end{aligned}$$

(36)

Non-existence of Symmetric Equilibrium Take the first order derivative with respect to c,

$$\begin{aligned} s^{\prime }(c)=1+\frac{3((20-c)-(c+s(c))(20-c)^2)-(s(c)-c)(3c^2-7c+200)}{(20-c)^2(c+5)}. \end{aligned}$$

(37)

When $s(c)>0$ and $c\in [5,20]$, according to equation (37), we can show that $s^{\prime }<0$, which violates the requirement that $s^{\prime }(c)$ must be strictly increasing for an efficient equilibrium, and a symmetric equilibrium does not exist in our case (Caragiannis et al. 2011; Gomes and Sweeney 2014).

B: Additional Regression Results and Figures

See Tables 11, 12 and Figs. 5, 6, 7, 8, 9, 10 and 11.

Table 11 Regression results, efficiency level, buyer value 15

Table 12 Regression results, efficiency level, buyer value 25

Fig. 5

Cumulative bids distribution under different reverse auction mechanisms

Fig. 6

Session specific bid median, UPR

Fig. 7

Session specific bid median, FPR

Fig. 8

Session specific bid median, GSPR

Fig. 9

Bids distribution by relative rank in a group, last 25 periods. Notes: Figures are based on the data collected from the last 25 periods. The black-lined boxes show the interquartile range, the line in the box is the median, and the vertical line segments stretch to 5% and 95% percentile

Fig. 10

Allocation of realized surplus and total realized social surplus. Notes: Figure a shows the allocation of realized surplus between the auctioneer and bidders across different reverse auction mechanisms by period. Figure b shows the realized total social surplus across different reverse auction mechanisms by period

Fig. 11

Realized efficiency under different auction mechanisms. Notes: This figure shows the realized efficiency across different reverse auction mechanisms by period