Bidding Optimum Bonus for Federal Offshore Oil and Gas Leases
Authors: Lohrenz John 
Published: December 2007

Summary.
How a bidder should bid for federal offshore oil and gas leases offered by bonus bidding is detailed. Quantitative answers are given for bidders seeking to maximize value as well as reserves. The winner's curse is delineated. Further, it is shown how bidding as a joint venture rather than solo can diminish bidders' values.

Introduction
Consider an item of value V for which Bidder j offers a payment of value bj and each of n' other bidders offers a payment of value bi that is not necessarily equal for all n'bidders. Throughout, let all values be tax-adjusted and appropriately discounted present values. Bidder j "wins" with probability . The expected value to Bidder i. of the offered item is
(1)
Eq. 1 implies that n' is a stipulated nonnegative integer. When the number of bids competing with j's bid is a distribution of n',
(2) and (3)
The notation used hence will be E and p .. with the appropriate condition, n'or, implied by the context. Furthermore, let V= 1 such that E, bj and bi are fractions of V in Eq. 1.
The foregoing describes the application treated here. The item of value offered is an oil and gas lease. A bonus in a sealed bid is the payment of value offered by bidders. The bidder offering the highest bonus "wins" the lease.
How Should a Bidder Bid? Entirely new quantitative answers are given here. The answers depend, first of all, on what a bidder seeks most with bidding. Two distinct
objectives can be considered:maximizing oil and gas reserves ormaximizing present values. The mathematical rule invoked with the first objective is to bid such that E=O.
with the second objective, E is maximized. A reserve maximizer seeks reserves constrained only by the expectation of not losing economic present values. A value maximizer seeks the largest possible increase in economic present values with whatever reserves are coincident.
First, bidding on a single lease with no constraints on the magnitude of the bid is considered. The next section defines a model and its assumptions leading to optimum bids when the number of competing bids, n', is stipulated. The third section extends the model by considering an uncertain number of competing bids. Then the assumptions of the model are justified. Joint venture bidding, the reasons for it, and the costs compared with solo bidding are discussed in the fifth section. Finally, the last section summarizes the conclusions.

Bidding Against a Stipulated Number of Competing Bids for a Single Lease
Consider Bidder j and n' bidders-individually indistinguishable-named i, all of whom will bid on an offered lease. The true, but unknown to all bidders, value of the lease is V=1. Bidder j estimates the value of the lease as, each of the n' other bidders estimates a unique value of the lease, . Assume that both j and all i obtain their estimates, and, as samples from a lognormal distribution with the geometric mean of V= 1 and variances and, respectively. Further assume that each Bidder i submits a bid, equal to the product of the individual bidder's and a bid fraction, common to all n' Bidders i. Consider Bidder j's bidding an amount, obtained by multiplying by j's bid fraction, . The outcome to Bidder j, E, is defined by the following function: (4)
The frequency of j's winning -- i.e., submitting a bid higher than each of all n' bids from Bidders i -- can be written as follows: (5)
Eq. 5 is rigorous. PSN is the cumulative unit normal probability function, and xmax, n'is the largest of n' samples from a unit normal distribution.
JPT
P. 1102^