Competitive bidders for oil and gas production assets seek to divide shares of "pies" of unknown value that each asset represents. Unlike with real pies, however, bidders' strategies can easily imply an expected negative share and the seller can reap more than 100% of the unknown asset values.
An Introductory Example. Suppose the underlying "true" value of a certain asset is 1, but no one knows that value. Instead, we and a single competitor can each estimate a (different) value. We and our single competitor both makes estimates of the "true" underlying value as samples from a lognormal distribution with a variance (based on natural logarithms) of 1. Assume all estimates are unbiased with respect to the "true" underlying value; our estimates will be higher than the underlying value 50% of the time and so will our competitor's. Thus, we and our competitor have the same estimation acumen.
Suppose, further, that our single competitor always bids against us and bids an amount equal to 30% of his or her estimate of the "true" underlying value, i.e., uses a bid factor of 0.3. If our bid is higher than our competitor, we "win", we pay the amount we bid, and then learn the "true" value of the asset is 1. Our net value is [1 - (our winning bid)]. If we paid more than 1, we "won" the bid competition, but lost net value.
Figs. 1–4 show the outcomes for this example when we make bids using bid factors from 0 to 0.5 applied to our estimate. Fig. 1 for the frequency of winning, F(W), shows that if we use the same bid factor as our competitor, we both win half of the time; that is merely a passed checkpoint with intuition. More aggressiveness, i.e., using a higher bid factor, increases our frequency of winning; bidding more conservative, vice versa. Fig. 2 shows the expected value that accrues to the winning bidder, E(V W). The value accruing is the "true" underlying value, 1, less the winning bid. Note that when we win, albeit infrequently, using a low bid factor, we can expect to net a rather high proportion of the "true" underlying value. In agreement with intuition, Fig. 2 also shows that when we use the same bid factor, 0.3, as our competitor, we will get the same net return from our bidding when winning.
A key outcome when bidding is the product of the frequency of winning and the net value accruing to the winning bidder, the net expected value of bidding, F(W)E(V W). Fig. 3 shows the net expected values for our competitor, ourselves, and the seller of the asset. We have a maximum net expected value when our bid factor, C=0.18; above our C=0.44, we have a negative expected value.
Note that, still in agreement with intuition, the curves for we and our competitor on Fig. 3 intersect when we use the same bid factor, 0.3. One might expect that bidders bidding with a bid factor of 0.3 should net 70% of the underlying "true" value, but Fig. 3 shows that here both we and our competitor get only 12% apiece. What happened to the other 76%? Fig. 3 shows the seller got the "other" 76%. Note that, although unknown when bidding, the underlying pie is the asset which has a "true" value of 1. Bidding merely pie is the asset which has a "true" value of 1. Bidding merely divides that pie between we, our competitor, and the seller; the sum of all curves at any bid factor on Fig. 3 is unity. Unlike real pies, however, bidders can get negative pieces of the pie. Likewise, a seller can get more than the whole pie. For example, when our bid factor is 0.5, we have a negative net present value on Fig. 3, our competitor a slightly positive outcome, and the seller an outcome greater than 1. In any bidding situation, graphs such as Fig. 3 show how the net expected value pie is divided.
What is wrong with the rationale that supposes that using a bid factor, C, where C less than 1, should lead to a net expected value of (1- C) of the "true" underlying value?