INTRODUCTION

If exploration is just luck, it seems we spend a lot of unnecessary money gathering and analyzing data only to be surprised when the drill bottoms out. On the other hand, if skill holds the upper hand, how can we identify it, nurture it, and reward it?

In this paper we will look at an interpretation of the exploration decision process and see if we can separate skill from luck to help mobilize a more effective search for this declining resource. As we shall see, it will be very hard to make this separation -- in fact, if you would like a hard and fast answer that "skill represents 72.3% of the exploration decision," you'll be very disappointed. Before laying the foundation for the analysis, however, I can report that luck plays a more significant role than most of us would like to admit.

We'll take a simple profit equation, look to see which elements are responsible for most of the uncertainty, and then create two "players," one with traditional skills and the other with good sense but no specific knowledge to allow skillful estimates. We'll see how the two players compare over a long run of drilling by simulating exploration on a computer.

GAMES OF SKILL AND CHANCE

Consider for a moment the roulette wheel and the prospect for solving its motion with skill. To accurately predict the outcome of a spin, you need only to know the initial rotational forces, the wheel friction, the elasticity of the ball and the wheel surfaces, the surface angles around the wheel where the ball might bounce, the initial velocity and spin of the ball, and the friction coefficient between the ball and the wheel. With this input data and the proper equations, it should be child's play to make a killing in the casinos. I would call this an engineering solution to the problem of "where will the ball fall?"

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There may be other ways to decide a bet. A fellow claiming ignorance of physics might opt for some simple record keeping. Over time, he observes that red comes up about half the time, black about half and green only on rare occasions. He would make his bets accordingly. Often there comes a person who hasn't the time or inclination to "study" the problem; he just makes a hunch and plays it.

Who will do best? Who will expend the most labor in getting an answer? In roulette, the engineering or scientific approach holds no advantage whatsoever over either the record keeper or the hunch player. While it may be possible to put together a set of equations describing the motions of the ball and wheel, it doesn't seem likely that the measurements of the various forces and velocities would ever be accurate enough to give even a whisper of a decent prediction, especially in the time allowed. The skill-short hunch player can on the average do much better than the engineer with his giant computer because while the average winnings may be the same (negative, of course), the hunch player does not have the burden of the large upfront expense for study and equipment. If the pla er without skill can perform as well as the player with skill, then that stands as evidence that the game is 100% luck. We shall use this kind of comparison later for the exploration business.

Luck rules the game of roulette not because of a theoretical impossibility to arrive at solutions, but because of an impossibility to find these solutions within normal time and budget constraints.

THE PROCESS

Allow me to assume that our main goal is to turn a profit. Because of our training and expertise, we choose to do that in the oil and gas exploration business. Accepting the premise that to make money in exploration, we actually have to find something of value, how do we go about it? -- we drill (Figure 1). Of all the exploration tools we use, only the drill actually finds commercial oil or gas. The other tools often give useful information about the possibility of hydrocarbon accumulations, but in themselves find nothing. (Realize that even the drill sometimes falters in its ability to give specific results, but no one doubts its superiority over other tools.)

If given the task of just finding oil or gas with no economic constraints, we'd simply drill the earth on 40 acre spacings. This economic constraint makes our business exciting. Anyone can find oil; the challenge is to do it profitably.

Thus the near-sciences of geology, geophysics, and economics are inexorably intertwined--and because uncertainty arises from the imperfection of our tools, we must add to the three near-sciences those branches of mathematics known as probability theory and statistics.

The exploration decision has always had to deal with the economic trade-offs of how to high-grade areas prior to using the expensive drill, a process that involves data gathering and interpretation. This is the "soft" part of the business -- soft in the sense that besides its ambiguity, the data tend to be quite sparse, leading to several alternative explanations.

PLATEAUS OF KNOWLEDGE

We use knowledge to reduce uncertainty, and as explorationists, we want to gather information that increases knowledge and allows us to discriminate among prospects and areas. We recognize a few really significant jumps in knowledge level:

(1) we identify a sedimentary basin;
(2) source rocks exist;
(3) nature provides a reservoir quality rock;
(4) we have some kind of anomaly; and
(5) oil or gas is present in commercial quantities.

Notice that we must credit the drill with most of these jumps in our knowledge level. The principal exception to that rule is (4) where we generally use a geophysical tool. Also notice that the first two processes mostly affect assessment of discovery chance. The last three affect both chance and size. I have limited this particular study to the transition from (4) to (5)--that is, what role does luck play once we've

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identified an anomaly? (identification isn't always a "eureka" kind of event but rather a slow-moving and sometimes painful process. For our purposes, we'll assume a sharp demarcation between identification and definition which, if not strictly true, will not materially affect the results.) This stage of exploration can be very expensive with the running of additional seismic lines for prospect definition, expensive computer processing, and land purchase (especially the offshore bonuses); it seems a worthwhile place to begin.

PROFICIENCY MEASUREMENTS IN EXPLORATION

To compare luck with skill, we'll have to find some agreeable way to measure skill -- no simple task. Here is a list of possibilities: (a) money made; (b) oil and gas found; or (c) success ratio. As it turns out these obvious choices don't work out very well because the outcome of decisions involving luck may not have much relationship to the quality of the decisions. An explorationist may have a very high success ratio because he looks for sure things -- which tend to be small. The amount of oil found and money made could result from one or two lucky strikes in an otherwise uneventful career.

For these reasons, I would propose several more fundamental measures of skill,

Fig. 1. Flow chart showing the ideal in the oil and gas exploration business -- turning a profit.

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objective ones that anyone can use:

A. Precision of reserves estimates;

(1) productive area estimates,
(2) net pay thickness,
(3) recoverability (bbls/acre-ft or mcf/acre-ft).

If you're serious about measuring exploration skill, you will probably want to check not only the precision of reserves estimates but also of productive areas, pay thicknesses, and recovery factors. How well can an explorationist differentiate between the smaller, thinner, tighter sands and those of more economic interest characterized in Figure 2?

B. Precision of chance estimates (or chance components);

(1) source and timeliness,
(2) reservoir quality rock,
(3) structural or stratigraphic elements for trapping,
(4) sealing mechanism.

These components (and you may add others) act like links in a chain (Figure 3) -- one break and the whole prospect falls apart. For example, we're really not interested in the "degree" of reservoir rock but only about the likelihood that the rock will be of the minimal quality necessary to allow economic production. The degree or variation in rock quality above this minimum properly belongs in reserves determination. An as yet unpublished manuscript by Dr. Peter R. Rose proves the value of breaking chance apart in this manner.

In this paper, we will deal only with reserve estimates because, as it turns out, they seem to bring on most of the profit uncertainty in exploration. Look at the following measure of precision:

[EQUATION]

or

[EQUATION]

If you're accountable, you ought to deliver about what you say you will; the sum () of the estimates should equal the sum of the actuals. By using confidence intervals and other tools of the mathematical statistician, you can take most of the guess-work out of these measurements and at the same time build in the necessary corrections for the number of data points you have. You may be surprised at how few points it takes to see important trends in estimating.

DEFINITIONS

Skill: The ability coming from one's knowledge to do something well. Well means unbiased and with reasonable precision.

Luck: A combination of circumstances or events, operating by chance to bring good or ill.

Fig. 2. The major elements of reserves estimates.

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Notice that skill does not require the ability to explain how an estimate is arrived at -- which therefore allows hunches and intuition and whatever other senses an explorationist can bring.

PROFIT AND THE CRITICAL INGREDIENTS

A simple profit formula:

Profit = Volume Price - Costs

If you want to know which of the three items on the right side of the equal sign holds the most importance, you would look at the profit-per-unit change in each of the three. If we have 1,000 units of Volume for a Price of $10 each and a total Cost of $2,000, for every $1 change in Cost, it's easy to see that we have a corresponding $1 change in Profit. Not accidentally, the coefficient of Costs is "1." For every $1 change in Price, we will see a $1000 change in Profit because each dollar will be multiplied through the 1000 units of Volume. Again, not accidentally, the coefficient of Price is Volume, 1000 units. So whatever profit formula we use, the coefficients of the variables will play an important role.

What if we want to look at the uncertainty in Profit, knowing that Price will be exactly $10 -- say the contract has already been signed. While the Price coefficient is quite high, Price uncertainty is zero and thus Price can have no influence whatsoever on the uncertainty in Profit. This reasoning suggests the following formulation for the uncertainty influence of a variable:

( Profit)/( Price) Std. Dev. of Price

where the "" means change and Std. Dev. is the standard deviation of Price, a measure of the degree of its uncertainty. See the Appendix for details on influence calculations.

Consider the following profit formula:

[EQUATION]

where:

Profit = expected present value profit of exploratory venture;
Reserves = barrels in millions;
Price = $/bbl at the well head;
Net = explorer's portion after royalty;
Cost = cost per barrel to develop and produce;
Chance = probability of finding commercial quantities of oil;
Keep = (1 - Tax rate);

Fig. 3. The chain of chance.

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Dryhole = total cost of bonus, exploratory drilling, seismic, etc.;
(1 - Chance) = chance of failure;
Time 1 = discount factor for total revenue;
Time 2 = discount factor for development and operating costs;
Time 3 = discount factor for dryhole expense.

These three time factors allow us to enter the essence of time and present value discounting without all the trouble of multi-year cash flows and is quite similar to finding a center of mass in mechanics. We estimate the centers of mass of revenue and cost flows and look up the discount factor in a table for the corresponding number of years. The Appendix contains the details on the coefficients. Also, you may notice that we have no fixed costs for development, instead including them with the variable costs. It keeps the formula simpler and will cause no problem for the analysis, because true fixed costs tend to be small compared to the total.

I chose exploratory plays from three representative areas to gain some understanding of how the variables influence profit -- Mid-Continent, Gulf of Mexico, and Alaska's Bering Sea. Table 1 gives the assumptions, chosen to yield about a 12% expected rate of return (discounted cash flow) including about 7% inflation. The numbers are reasonable approximations of what you would see on representative discovery sizes of 250 thousand bbls, 18 million bbls, and 1.4 billion bbls, respectively. The findings of the study, however, do not depend on the exact numbers below. In fact, don't be fooled by what appears to be two- and three-digit accuracy. In this business, it would be presumptuous to put too much faith in the first digit, let alone worry about the 2nd and 3rd.

Though key to this study, no other number in the table eludes us quite like reserves uncertainty because so little data has been published. The numbers used come from personal experience, internal unpublished reports, and the USGS study revealed in Science,(FOOTNOTE 1) which showed no bias and a variance of about 1.9 for its estimates compared to later determined true(?) values. As Figure 4 shows, such a variance translates to an uncertainty factor of about 6; that is, USGS estimates were within a factor of 6 about 80% of the time.

Table 1.

FOOTNOTE 1. Science, August 1979: American Association for the Advancement of Science, p. 489-490.

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The concept of uncertainty factor may be unfamiliar, but the idea is quite simple. Traditionally, people describe their uncertainty in an estimate by attaching a ± so many units or % ("We can estimate the cost ± 10%," or, "That well should cost $1 million ± $100 thousand.") When uncertainty becomes very large, the ± representation doesn't work well so we introduce an alternative method -- divide and multiply instead of add and subtract. Multiply a base number by the factor to get the high side of the range and divide by the factor to get the low side, hence the strange combination of symbols for division and multiplication on the ordinate of the graph in Figure 4. The Appendix shows details of the calculation.

Figure 5 illustrates the magnitude of uncertainties for the 3 areas compared to their respective failure costs, and results from a Monte Carlo simulation using the profit equation given above and the data from Table 1. Generally, as this ratio grows larger, investors should avail themselves of more risk-sharing strategies to avoid being completely at the mercy of luck -- the bad kind.

RESULTS OF UNCERTAINTY INFLUENCE CALCULATIONS

To no one's surprise, reserves stand out as the most consequential variable in the profit calculation (see Figure 6). Of course that results from the particular set of uncertainties chosen in Table 1. You will find the individual uncertainty influences in Table 2.

These uncertainties remain even after experts have done most of what they can. Don't look at Price, for example, and think that some econometric study will

Fig. 4. The relationship between variance and the uncertainty factor.

Fig. 5. Ratio of present worth uncertainty to cost of exploration failure.

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reduce that part of the uncertainty, leaving the total smaller. Economists are notorious for poor estimating performance, even 3 to 6 months at a time. Expect no help 10 to 20 years ahead.

Chance belongs to the explorationist, but without direct evidence of hydrocarbons, it's a tough variable to improve on. In the Gulf of Mexico, you know the chance of success on a reasonable-looking structure is about 20-30%; it's not 50% and not 5%. On the other hand, in Alaska you probably have no success chances above 20%, at least not for commercial quantities of oil (5% to 10% may be closer). Historical success ratio in the area of interest is a base you will generally want to embrace.

Development and operating cost uncertainties result mostly from reserves uncertainties -- deep thin sands being more costly than shallow thick ones. Explorationists generally do know something about depth, but there are variations and surprises.

That leaves us with reserves estimates as the major attack point for reduction of risks in petroleum exploration, and the best place to test theories about the role of luck in our business.

SETTING UP THE LUCKY PLAYER

Remember, I said earlier that the plan was to go for luck, not stupidity. So one reasonable strategy to test Chance outcomes is to let Luck look at the average historical discovery size in an area and use it for every anomaly he sees. On the average he would be right, but you'd think that individual estimates might be horrible, leading to poor performance. Mr. Luck will always choose 18 MMBO as his answer, absurd as it may sound. Almost no one would actually employ such a strategy; we only use it here as a pure luck device and therefore a benchmark against which to test skill.

In a competitive environment, this kind of estimator might run into trouble in those areas with larger potential. The competition would be more aggressive, thus shutting him out. So any test we perform should look at the effects of such competition.

SETTING UP THE SKILLFUL PLAYER

Even the skillful explorer will have his bad days -- not all that he estimates turns out to be what he thinks, but he should at least be unbiased (that is, correct on average). Our skillful explorationist does the following. He identifies an anomaly and sets about to size it with more work -- seismic, etc. If he looks at two prospects which truly have 10 MMBO and 50 MMBO respectively, he will tend to think the second larger. If he looks at a lot of prospects that turn out to contain 5 MMBO, he will guess some at 2 MMBO and some at 8 MMBO, others at 4 MMBO and at 6

Fig. 6. Contribution to total uncertainty by the component variable.

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MMBO. He may even have a couple of 1 MMBO estimates and a 13 MMBO estimate. The average of all these, of course, is 5 MMBO:

(2 + 8 + 4 + 6 + 1 + 1 + 13)/7 = 5

The uncertainty ascribed to this explorationist appears in Table 1.

THE SIMULATION

We set up a basin with field sizes similar to those in the Gulf of Mexico--average = 18 million bbls (108 bcf)--and used a spread or variance consistent with the USGS study cited earlier. First draw one of these "true" field sizes and let it serve as the target prospect identified by both Skill and Luck. Skill chooses a value somewhere around "true" with an estimating uncertainty from Table 1. Luck, remember, always picks 18. Repeat this exercise 10,000 times.

COMPETITION

Rather than use the more severe offshore competition brought about by the rules of Federal OCS leasing, I used instead an onshore representation. If you're able to acquire a prospect for drilling, chances are high that at least one other explorationist has already passed on that prospect. By virtue of getting the prospect, you probably think it has more value than others. In the simulation described above, and assuming 100% chance of competition, we determine which of our two players gets the prospect as follows. If Skill estimates reserves of greater than 36 MMBO (twice Luck's estimate) then Skill gets it. If Skill estimates less than 9 MMBO (1/2 Luck's estimate) then Luck gets it. With a Skill estimate between 9 and 36 MMBO the prospect goes by lot with Skill's chance of getting it roportional to the size of his estimate.

For zero chance of competition, half the prospects go to Skill and half to Luck with no regard for the size of the estimates. At 50% chance of competition, half the prospects see full competition and half see none.

RESULTS

To see how each player performed, we merely kept track of his estimates and actuals on those prospects he acquired. Figure 7 shows the estimating prowess of the skillful player as a function of the degree of competition he faces. Decimal fractions show the chance that one other competitor shows up. Note the apparent degradation in estimating ability as the probability of competition increases. For zero competition we have evidence of good skill, which, afterall, served as the basis for

Fig. 7. Relationship between induced bias and competition.

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the simulation. Since, however, Luck does as well, we're hesitant to concede the match to Skill just yet.

By the time we get to a 100% chance of a competitor, skill's ratio has gone sour--to about 1.40. That means that on the average, the skillful estimator foretells about 40% more reserves than really exist, a trait guaranteed to cause financial grief over time.

Some readers may identify this induced bias problem as the "winner's curse"--a competitive phenomenon wherein the estimator making the worst error in overestimating reserves tends to win the opportunity. This most unwanted event happens even to the explorationist whose estimates are correct on the average. Many people think of the curse applying only to offshore lease sales, but actually, it pervades every competitive situation. You dismiss it at your own peril. (To keep this analysis simple, we've only looked at one other competitor when in actuality we may find several. The winner's curse affects players more severely as competition increases.)

At zero competition, Luck shows no bias either. As competition increases, Mr. Luck has a problem of greatly increasing induced bias, but to him it suggests a cure. If when he wins against competition, he tends to estimate too high, then he could just reduce his estimate in the first place to counter. By moving his estimate downward to 50 or 75% of 18 MMBO, perhaps he can beat Mr. Skill. And as Figure 8 shows, that's exactly what happens. Luck overtakes Skill by applying expertise in the theory of gambling--without the expense of even one more seismic line. As the chance of competition increases, Luck hedges with "trial and error" multipliers of 1.0, .87, .75, .60, and .47 respectively. As a practical matter, Luck would have a tough time estimating the degree of competition enabling hi to calculate these multipliers. Most any reasonable attempt, however, would prove better than none. Game theorists suggest that Skill could also play this hedging strategy to purge this unwelcome form of bias. But tied as he is to his maps, will he?

A seemingly nonsensical strategy demonstrates clear superiority over a more conventional one when amended with some good sense about playing the odds. Luck, it turns out, doesn't waste his efforts on piddling-sized fields either. At worst he gets about one half as many barrels as does Skill per anomaly identified--a side result of the simulation.

What about precision of these estimates? Using the uncertainty factor as a measure, Figure 9 shows that Luck does about as well as Skill. At zero competition, Skill can get within a factor of 5 about 80% of the time while Luck's factor is nearly 6. (The winner's curse correction doesn't seem to affect these numbers.) Taking the

Fig. 8. Luck removes bias by applying expertise in the practice of gambling.

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ratio of these two numbers, we find that Skill's uncertainty is 85% of Luck's. This number comes directly from the assumptions about uncertainty made in Table 1, but I doubt any other reasonable guess would change the outcome significantly. It took thousands of trials in the simulation to separate the two methods; in real life, with the much smaller number of opportunities people have, you would be hard-pressed to see any difference. Luck could reduce his uncertainty further with just a primitive assessment of size, because anomaly identification also yields a rough idea of magnitude.

CONCLUSIONS

Now for a stab at the "how much skill, how much luck" question. For this part of exploration--anomaly definition and size or value determination--Mr. Luck can do substantially as well as Mr. Skill, and better with recognition of the winner's curse problem. We'd have to conclude that this part of exploration is mostly luck especially since a significant portion, say 25%, of the uncertainty lies beyond the reach of the explorationist, a number we get from Table 2.

To get the amount of luck in value assessment, we choose the ratio of Skill's overall normalized uncertainty to luck's (Figure 6 and Table 2). Both start with the largely irreducible 25% from price, costs, etc. Absent a reliable direct hydrocarbon finding tool, the chance portion of about 10% should also be the same. The remaining reserves assessment piece (65%) comes from Figure 9 where the simulation showed skill with about 85% of luck's uncertainty.

[EQUATION]

Luck, therefore scores very well, even against, an unemotional, unbiased opponent. If Skill were to give in to the normal temptations of overoptimism and salesmanship, the Luck strategy portrayed here would overwhelm.

Fig. 9. Comparison of Luck versus Skill in precision of estimates, using the uncertainty factor as a measure.

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For companies with many risk-spreading opportunities and no gambler's ruin or gambler's hurt situations, bias provides much more danger than does the garden variety of uncertainty they face. First, go after bias, then uncertainty.

Conventional wisdom attacks uncertainty with more seismic lines, more studies, more manpower, more processing, and more weekend work--much like the engineering assault on the roulette wheel. All have helped, but Nature seems to get harder to read as we get smarter in reading.

A less orthodox but potentially more rewarding strategy would allow the power of diversity to work for us. Dust off Chamberlin's Multiple Working Hypotheses ideas (FOOTNOTE 2) recognizing that no one has any lock on truth. A little work spread over lots of ideas could yield a much more accurate answer through the simple process of averaging. Melding the creative talents of several explorationists in this manner can reduce the uncertainty factor dramatically. Averaging 2 independent estimates of reserves moves the factor from about 6 to less than 4--an odds player's delight. (See Figure 11 and Appendix for calculation detail.) Putting the money saved into

Fig. 10. Analysis of costs versus reserves, Alaska.

Table 2.

FOOTNOTE 2. Journal of Geology, February-March 1931: The University of Chicago Press, p. 155-165

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more drilling would, it seems, contribute more effectively to the ultimate goal of finding oil and gas.

If our business does have large uncertainties which do not readily yield to traditional hard work, then the organization will have to put less emphasis on work-related improvements and more on how to achieve high performance with amorphous information. No doubt this will be the toughest hurdle for those managers who accept on faith that toil and labor bring forth better answers. The roses may well accrue to those managers who spend more time learning how to play the odds a Luck did with the winner's curse in the example given.

Do not misread any of this as degrading the marvels of science as they apply to exploration. That phase of exploration relating to the identification of prospects puts much emphasis on seismic technology, for example. We did not attempt to quantify the degree of luck in that segment of the search because of the scarcity of data. (At the minimum, you would have to know how many wells fail because the seismic map wasn't close enough to Nature's map.) Nor would I suggest abandoning the use of and search for better tools. The study only demonstrates that the value of delineation information may be much less than generally perceived and the value of knowledge on how to play the probabilities may be much more.

APPENDIX: UNCERTAINTY FACTORS

Uncertainties of the magnitude we see in exploration don't fit the usual mode of ± so many barrels. A central guess of 5 million bbls could, for example, have an upside potential of 15 million bbls--an addition of 10 million to the base. But how could you subtract 10 from 5? It makes no sense to say 5 ± 10. We could, however, get an upside from multiplying the base by 3 and dividing by 3 to get a downside yielding a range of 5/3 to 5 3 or 1.7 to 15.

Table 1 presented reserves uncertainties based on logarithms of barrels rather than the barrels themselves. Remembering that we exponentiate log barrels to get back to barrels,

e^S.D. = a factor of uncertainty,

Fig. 11. Analysis illustrating the reduction in uncertainty factor as the number of independent estimates increases.

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that should include the true value about 68% of the time, by definition of normal distribution standard deviation. Using probability tables for the normal distribution, we find that ± 1.28 standard deviation units produces a range that should include the actual value 80% of the time. So the 80% uncertainty factor given on the graphs comes from the simple formula:

e^1.28S.D. = 80% factor.

Besides the convenience of the factor, it has the additional strength of being theoretically consistent with the use of the log-normal distribution for describing reserves uncertainties.

UNCERTAINTY INFLUENCES

The simple formula below shows the relationship between profit and the more important variables in petroleum exploration. By taking partial derivatives and multiplying by a measure of uncertainty, we can find the influence, I, of each variable on total profit uncertainty:

[EQUATION]
[EQUATION]
[EQUATION]
[EQUATION]
[EQUATION]
[EQUATION]
[EQUATION]
[EQUATION]
[EQUATION]
[EQUATION]

AVERAGING AND THE REDUCTION OF UNCERTAINTY

Statistical theory says that the variation or uncertainty of averages becomes smaller by the square root of n where there are n independent estimates making up the average. Averages of samples of 2, therefore, show a standard deviation smaller by (1/2) .7. Going back to the first section of this Appendix, we see that the reductions in standard deviation would have to be exponentiated in order to work through to the uncertainty factor. Taking the assumed standard deviation for estimates in the Gulf of Mexico of 1.22 (FOOTNOTE 3) we get:

[EQUATION]

The next figure illustrates the reduction in uncertainty factor as you increase the number of independent estimates. It says you can take two not-so-good estimates, average them, and get an estimate that may be much better than the one obtained by doing the more work. That means money saved and better reserve estimates.

Getting independent estimates of reserves for use in averaging, however, will not be an easy task.

For those intrigued by oddities, Figure 9 shows decreasing uncertainty with increasing competition--another legacy of the winner's curse. It means when you measure your own estimating uncertainties with real reserves data, you will calculate too small a number because you will not have mixed in the data from those deals you didn't make and wells you didn't drill.

FOOTNOTE 3. The 1.22 must look ridiculously accurate after those earlier comments about the quality of the numbers. Actually, I first estimated a variance of 1.5 and took the square root to get the std. dev. of 1.22.

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