Division of Research Graduate School of Business Administration The University of Michigan August 1975 ' EMPIRICAL BAYES ANALYSIS OF LEASE AUCTIONS Working Paper No. 116 by R.W. Andrews The University of Michigan FOR DISCUSSION PURPOSES ONLY None of this material is to be quoted or reproduced without the express permission of the Division of Research.

Introduction and Problem Statement Many business and industrial transactions are decided through the use of sealed competitive bidding. This is common in contract bidding —e.g., construction jobs, defense contracts, plumbing and heating —in which the low-dollar bidder wins the contract and is entitled to perform the required services. It is also the case in acquiring property rights, for example, purchasing or leasing acreage for mineral or oil rights. In this case the high-dollar bidder wins the rights to the property, Since Friedman's article [5] in 1954, much has been written about analyzing a competitive bidding situation. Most of what has been written is concerned with proposing a strategy for a bidder (see Arps [2], Brown [3], Friedman [5], and LaValle [7]). Notable exceptions are Crawford [4] and Pelto [101, who analyze bids without proposing a strategy. Stark [11] gives a comprehensive bibliography on competitive bidding. Herein, the concern is with the analysis of a lease auction. Specifically, this report gives a procedure for estimating the bidworth of a tract of land (or off-shore continental shelf) which will be used for oil and/or gas rights. Empirical Bayes as well as Bayes methods are employed and therefore other experiences of similar bidding situations are required. In the second section, a Bayes estimator with the associated Bayes risk is derived based on a multiplicative model for the bid worth of a lease. In section three, an Empirical Bayes estimator is proposed on the assumption that other experiences of

-2 - similar bidding situations exist. The mean squared error (M.S.E.) of the Empirical Bayes estimator is compared with the M.S.E. of the maximum likelihood estimator and with the Bayes risk. Section three also investigates a case in which the prior distribution is assessed incorrectly. Since the Empirical Bayes procedure does not depend on an assessment of the prior distribution, we see that the Empirical Bayes estimator's M.S.E. is smaller than the risk that was realized using a Bayes procedure in conjunction with an incorrectly assessed prior distribution. In the concluding section data from the bid tabulations of U.S. offshore sales is analyzed and a strategy is suggested based on the sequence of Empirical Bayes estimators. The development of this report first considers the following problem: There is a group of k tracts which will be let for bids. The bid worth of a tract is denoted by ej; for j = 1,2,....k. We have the bids for all k tracts, where each bid is denoted by Xij,the th th ith bid on the jt tract. Therefore, the problem is structured as (:); j = 1, 2,.... k with: 3 x - (Xlj' Xj'.***jXnA) j j ' nj In the second section we establish the distribution of the bids and find a connection between the parameters of the distribution of X.ij and the bid worth ej The geometric mean of Xi, designated by 13 3 3

-3 - T = Exi3)l/ is sufficient for e.; therefore the problem is reduced to T j;j = 1, 2,.... k. Using Empirical Bayes methods we estimate 1, 62, 03,....3 the bid worth of all the tracts. Enroute to the Empirical Bayes solution a Bayes solution is given which enables us to evaluate the accuracy of the Empirical Bayes procedure using Monte Carlo studies. One of the benefits in solving the problem as stated above is that it leads to the next step, a bidding strategy. Throughout this report it is assumed that some of the Xj's have an element, denoted by XOj, which is the bid of the individual company or bidding combine which is attempting to assess k+l, k+2 the bd k+l' 0k+2' k+r. worth of the next r tracts. Using the Empirical Bayes solution, a sequence of bids X0 = (Xo k+l Xo k+2 ''' X k+r) is proposed. ABayes Estimator for Bid Worth Consider the situation in which the bid worth, designated by 0, of an item or lease is formed by the product of ml- different factors. If D1, D2,.., D, and F represent these factors then

-4 - 'In e F I D. - t-1 t=l f The common factor, F, discounts the actual worth. For example, using the straight forward method of appraisal of individual tracts as given in Crawford [4], the discounted gross oil reserves would be determined by 0 FD2D34D5DD6D D in which D1 number of pays; D = average pay thickness; D = average length of reservoir; D4 = average width of reservoir; D5 fraction of pore space occupied by hydrocarbons; D6 = fraction recovery; 6 D7 = average worth per unit volume of oil produced. Throughout this report we will be concerned with bid worth, and we assume that bid worth is a certain percentage of actual worth. This percentage, designated by F, is assumed the same for all bidders. The factor F is based on average deferment, interest rates, average rate of return before taxes, and dry hole risk0 Since a bidder cannot assess D1,,29 'a Dm exactly, these A A A quantities must be estimated by D!, D2,... Dm which are assumed to be unbaised estimators. The resulting bid is m I X F n D. t=l

-5 - If the estimators D1, D2,.., Dm are assumed independent, we have (2.1) E[X] = 0. Extensive studies with bids for oil and mineral rights indicate that for a specific tract the bids are lognormally distributed [1], [4], [10]. If X1, X2,... X are n bids for a specific tract, we state that this sequence of random variables is independent and identically * 2 2 distributed lognormally with parameters p and a, and with known. In practice, c can be estimated from the bids on all k tracts. Using (2.1) we can write that for a given ' 2 2 X, X2,... Xn - i.i.d. A (ln -.5cr, c ), and therefore 2 2 Tie - A (lne -.5a2, a /n). Since 8 is formed by a multiplicative process, a lognormal prior,G(e), is assigned, 2 0 A (as3). We find that the resulting Bayes estimator of 0 under squared error loss is snS2 OB exp f 2. O+2n$2 ) 9 - T exp 2 2 ) with a resulting Bayes risk given by (2.2) R(G) = exp 2a + 202) [ 1- exp( 2 } ] e2 + 2

-6 - (In the next section, this Bayes risk is compared with the M.S.E. of the proposed Empirical Bayes estimator through the use of Monte Carlo studies.) The resulting risk can sometimes be evaluated if the prior distribution is assessed incorrectly. For example, in using the log2 - normal prior, assume that 8 is assessed exactly; however choose a'+ e for the first parameter instead of the correct a. In this case the estimator we would use would be 8^ = ~B exp -- 2 2 B B ka2 + nJ with the resulting risk (2,3) Risk exp 2a+2 1 - exp 2 - ex - /2 2' 21 2 7 This risk will be used to demonstrate the worth of the Empirical Bayes procedure when similar experiences are available. An Empirical Bayes Estimator for Bid Worth In bidding for oil and gas rights, usually a group of tracts is announced for sale. A collection of companies or consortiums bid for the rights to these tracts. After bids for these k tracts have been received, the geometric mean of the bids for each tract can be found, and the problem can be represented as given in (1.1) with Tij i A(lne -.502, c2/n); i = l,2,...k. The maximum likelihood estimator of., adjusted for bias is = Ti exp{.52 - (a/21) } 0 Tiexp {.5 - ( /2n)

-7 -and (3.1) A2 ( t3, 1L) 9 |,f ~ A (ln8 -. ' 2/n). OJ1i7 A Una.- (ai /2n), a n). Since we want to estimate 8,, the bid worth of the -- tract, for i = 1,2,,..k, we propose the Empirical Bayes estimator as given in Krutchkoff [6] and Lemon [8]: k A A I j f(%eij) (3.2) _,k X f(iej) with f(e,0lj) = (-/a5i 2) exp -/2 - lnj+ 2/2 Notice that in this problem we can use all the k experiences to find the Empirical Bayes estimator of the i-h experience, since the bids for allk tracts are collected at the same time. In this situation we can find an Empirical Bayes estimator of the first experience, i.eo, 01, which would not be possible if the data were collected sequentially. The Empirical Bayes estimator weights the classical estimators, A th,i's, according to how close the data from the jt experience is to the th data from the i- experience. The weighting used is the conditional likelihood of EL given that 8. =. ' I i is1

-8 -The following two definitions will enable us to comment on the general small sample properties of 0.: M.S.E. of 0. 1 (3.3) R and 3~3) R = M.S.E. of i, and Var (ilei) Var (0,) In the sequentially-collected data case, extensive Monte Carlo studies on (3.2), with a normal and binomial conditional distribution (Lemon [9]), have shown that the M.S.E. of 0. is smaller than the M.S.E. of 0. for k > 2, i.e., if one or more other experiences exist. Furthermore, these studies have shown that for k increasing to approximately fifteen, R decreases. That is, the improvement levels off at approximately fourteen or fifteen other experiences. Figure 1 (see Figure 6 of Lemon [9]) demonstrates how R decreases as k increases. This plot was made using a normal conditional distribution. These extensive Monte Carlo studies also demonstrated that the amount of improvement of the Empirical Bayes estimator over the classical estimator is dependent on the prior distribution and the conditional distribution only as they effect Z, the ratio of the conditional variance to the prior variance.

1.0.9. R o5 4 -o3 o2-.1- Z = 2.0 IIlll -- I- '- i-, 1 2 34 5 10 15 20 25 30 40 50 Number of Experiences (k) Fig oI. Ratio of M.SoEo's using a normal conditional distribution. In further simulation studies, which were executed with a lognormal prior distribution and a lognormal conditional distribution of the form (3.1), the above comments held true with two exceptions. The plot of R versus the number of experiences was not always monotone decreasing since the value of the M.SoE. depended on the actual realized value of @0 Also, for the reduction in M.S.E. to be appreciable, the value of Z had to be greater than one. The output from a Monte Carlo simulation with 500 replications is given in Figure 2. For this run the estimated value of Z is 2.52. Z must be estimated since the

-10 -conditional variance depends on the realized value bf 0. Notice the large value of R at the thirty-eighth experience. This resulted because the realized value of 0 for this experience was 2.17 standard deviations from the prior mean. This meant that the actual value of Z was 1.87. Since the Empirical Bayes estimator weights the other experiences, this also helps to account for the procedure's relatively poor showing against values of 0 distant from the prior mean. Similar comments could be made for experiences five, twenty-nine, and forty-nine. Notice that the M.S.E. of the Empirical Bayes estimator is appreciably smaller than the M.S.E. of the classical estimator for almost all of the experiences. Line B in Figure 2 is the ratio of the Bayes risk (2.2) to the expected mean squared error (E.M.S.E.) of 0j, and line B* is the ratio of the realized risk of eB (2.3) to the E.M.S.E. of 0j. This realized risk was found by using eB, which is the Bayes estimator for the prior distribution A (5.4,.004), although the correct prior distribution should have been A (5.3,.004). Table 1 gives the ratio B* for some values of e, i.e., if A (5.3,.004) is the correct prior distribution and A (5.3+e,.004) is used as the prior distribution, the ratio of the realized risk (2.3) to the E.M.S.E. of ej is B*o Notice that in all but three experiences the Empirical Bayes M.S.E. is smaller than the realized risk if C is.10. This indicates that if an error is made in assessing the prior distribution, the Empirical Bayes procedure can be substantially better than the procedure incorrectly assumed to be the Bayes procedure.

-11-........... ' 0(1.06) \B*.8.7 o6 o5.4.3 B.2.1 Z = 2,52...I.... I - 1- \ --- —- I -- 1 2 3 4 5 10 15 20 25 30 40 5( Number of Experiences (k) Fig. 2. Ratio of M.S.E.'s using a lognormal conditional distribution. TABLE 1 B* = Realized Risk/E.M.S.E of 8,; 0 - A (5.3,.004) E —.20 -.10 -.05 0.0.05.10.20 3 2.04.76.41.28.42.83 2.62

-12 -For k = 15, another Monte Carlo study was executed using (3.2) in which each 6j (j = 1,2...15) was estimated using the simulated observations in all fifteen experiences. This is representative of the outer continental shelf data which will be analyzed in the next section. The value of Z for the data analyzed in the next section was estimated to be 1.49. Therefore, the estimated value of Z for this study was set at 1.49. For each of the fifteen experiences, Table 2 gives the R values (3.3) and the number of standard deviations that the realized 8 is from the prior mean, designated by do The largest value of R occurred when 0 was 2.74 standard deviations from its mean. The ratio of the Bayes risk to the E.M.SoE. of the classical estimator was.387 for this study. Outer Continental Shelf (OCS) Data with a Proposed Strategy The Empirical Bayes estimation procedure was employed on tabulated bids of fifteen tracts [12]. The data is given in Table 3, For the value of a2 we used 2 - k-1 j=l with in Xj n. 1 In X J,i1 ij

TABLE 2 Reduction in MoS.E.; k = 15, Estimate of Z = 1.49 Experience No. d R 1 -o41.59 2 -.31.58 3 o.00 51 4.40.55 5.51.51 6.50.49 7 2.74.88 8 -.11.57 9 -.80.76 10 1.32.57 11 -.29.59 12 -.21.56 13 -.09.54 14 -.60.72 15.17.52

-14 -For the data in Table 3, a = 1.81. Since E[i60i] = e we can estimate the prior variance using [6] ~t.^ As, Vare = Vars -E[Var( | )]. For this data the approximate value of Z is 1.49 which indicates that the Empirical Bayes estimator will show some reduction in mean squared error but not as great as would have occurred if Z were larger, Based on the sequence of Empirical Bayes estimates of bid worth, a strategy is proposed for the next set of tracts that are offered for sale. Since we have estimates of the.-'s and since it has been substantiated that the bids, given 6, are lognormally distributed, we can determine a representative quantile for any bidder. A particular bidding firm should develop a bid in its usual manner; then, based on the previous empirical data, the firm should adjust this bid so that it will estimate the quantile position of the winning bid. If u is the quantile of order q from a lognormal distribution q 2 with parameters (P,a ), and vq is the quantile of order q from the standard normal [1], then (4.1) u =exp(P + v a). q q Consider the bids in Table 3 denoted with an A following each bid. These bids are from a specific bidding firm. Since 2 2 we can solve for v in (4.1) by setting 0j = 0j and a = 1.81. This can be plhed fo a f f As n ids The mean of thes be accomplished for all of firm A's thirteen bids. The mean of these

TABLE 3 2 Tablulated Bids in Dollars Per Acre-Z 1.49; a = 1,81 m. ---. A'T 1 0 q I, S 6 7 8 I —aC -- UI - - '- __________ - _ I -,.~ ll~-~* --- —----- -------. — l- - - T- - r Bids 1003 666 (A) 354 350(B) 536.4 1058.7 1663.6 2008 1434 1369 (A) 894 583 577 350(B) 52 619.1 1368.-6 - 1629.0 22010(' 10619 8919 4416 (A) 3090 263Q(B) 2230 1073 892 88 2598.8 5 77:-.5 5772-.5 -I 13338 10695 (B) 1448 (A) 1070 724 301 266 53 952. 8 2106:-3 2627.8 -1 1225 (B) 1195 (A) 382 266 76 408.0 842.7 1265,3 I 701(B) 580(A) 274 237 403.1 795.7 1349.9 I 10619 7905 (A) 5072 (B) 4581 3172 2606 1854 1767 1133 636 2871.1 6492.-8 5915.4 -T 7541 5405 3618 (A) 1974 1963 1402 (B) 1303 1073 73 Geometric Mean, 0 0 T 1631.2 3651.9 4439,7 I e-j tF~1 - 1. _ _ =,_,._, =.. I I I T.. i- f Mn Q -in 11 12 13 14 15 lL00.~L. ~t~S~a I -' I ~v __ - II _. J — - I A. - __ _ - -—. -. I- I - I t Bids Geometric Mean, T e 36805 19290(B) 7948 5818 (A) 1360 1156 892 88 2824.6 6244.4 5751.7 -T 15932 3505(B) 2008 1339 (A) 631 370 282 1387.2 3017.5 3853.2 I 1225 (B). 681:..7: 389 76 396.3 782,2 1335.4 701 (B):244 198 418,8 826.6 1384.0 I nnn I 8157 7616 7559 (A) 5784(B) 2819 2681 2087 2065 1111 367 2855.7 6458.0 5908.7 -r 3706 2387 1928(B) 1785 (A) 654 307 177 73 I 14197 7184 (B) 2-008 1820 (A) 334. 35 1278.0 2720.6 35l1. 6 I 728.1 1609.6 1948.8

-16 -thirteen v 's is then used as an indication of the quantile for firm q q bids can also be found and designated vWA. The A subscript denotes that we only considered the winning bids on those tracts for which firm A was bidding, in this case all tracts except rnumbers 11 and 12. After all bids for a new sequence of tracts are developed by firm A, they are multiplied by exp{.i(WA - VA) which adjusts the bid to the average quantile of the winning bids. As a demonstration of this procedure we will reuse the data in Table 3 and indicate the results. This is a most favorable situation in that we are using the same bids for estimating the parameters and demonstrating the strategy. In the case of firm A, A =.42, WA = 1.17, and.exp{VWA - VA} = 2.75; so the bids firm A would have submitted using this strategy are: Tract No. 1 2 3 4 5 6 -A-Bid 1832 3765 12146 3983 3287 1595 7 8 9 10 13 21741 9951 16002 36003 20790 14 15 4909 5005. This would have resulted in firm A winning the rights to eight tracts. In addition to the number of winning bids, to indicate how well a specific firm, say firm C, has bid we define:

-17 -W Sum of winning bids for firm C Sum of the 2 place bids on the tracts that C won For A's revised bidding strategy, W = 1.94. This same procedure was applied to the fifteen bids from firm B (see Table 3) with the resulting bids being: Tract No. 1 2 3 4 5 6 B-Bid 764 764 5739 23338 2673 1530 7 8 9 10 11 "1068 3059 42093 7648 2673 12 13 14 15 1530 12622 4207 15676. These bids show that firm B would have won the rights to seven more tracts using this strategy. The value of W for B increased only slightly from 1.28 to 1.30 with the addition of these seven winning bids. The ratio of the sum of all fifteen winning bids to the sum of all the second place bids was 1.75. Concluding Remarks For a specific firm, there is usually a fixed total amount of capital that is available for a group of tracts 12], say Y dollars. Therefore, it is suggested that the initial bids be formulated under the restriction that Y 'exp{a(VA - VWA)} be the total amount exposed. Then the resulting bids will total Y dollars.

-18 -Since the success of this entire procedure depends heavily on accurate estimates of the j's, this report has emphasized that the E. o the Empral Baes e matr, is smaller than the M.S M.S~E. of the Empirical Bayes estimator, 0, is smaller than the M.S.E. of the classical estimator. Simulation studies have shown that the MNPS-E. of 6 will be smaller if Z is larger. Therefore, this procedure will yield better results for larger values of Z.

-19 - References [1] Aitchison, J. and Brown, J. A. C. The Lognormal Distribution. New York: Cambridge University Press, 1957. [2] Arps, John J. "A Strategy for Sealed Bidding." Journal of Petroleum Technology, 17 (September 1965): 1033-9. [3] Brown, K. C. "Bidding for Offshore Oil: Toward an Optimal Strategy," Journal of the Graduate Research Center, Southern Methodist University Press, 36 (October 1969): 1-71. [4] Crawford, P. Bo "Texas Offshore Bidding Patterns," Journal of Petroleum Technology, 22 (March 1970): 283-9. [5] Friedman, L. "A Competitive Bidding Strategy." Operations Research, 4 (February 1956): 104-12. [6] Krutchkoff, R. G. "Empirical Bayes Estimationo" The American Statistician, 26 (December 1972): 14-16. [7] LaValle, I. H. "A Bayesian Approach to an Individual Player's Choice of Bid in Competitive Sealed Auctions." Management Science, 13 (March 1967): 584-97. [8] Lemon, G. H. and Krutchkoff, R. G. "An Empirical Bayes Smoothing Technique." Biometrika, 56 (1969): 361-5. [9] Lemon, G. H. "Smooth Empirical Bayes Estimators: With Results for the Binomial and Normal Situations." Proceedings of the Texas Tech Symposium on Statistics (August 1969), pp. 110-40. [10] Pelto, C. R, "The Statistical Structure of Bidding for Oil and Mineral Rights." Journal of the American Statistical Association, 66 (September 1971): 456-60. [11] Stark, R. M. "Competitive Bidding: A Comprehensive Bibliography." Operations Research, 19 (1971): 484-90. [12] U.S. Department of the Interior, Bureau of Land Management, New Orleans Office, Outer Continental Shelf Statistical Summary 1973-1975.