Decision trees are a useful way to describe alternative scenarios and select the decision that maximizes the NPV or whatever the decision maker is trying to optimize. In decision theory, the most desired outcome is based on the goals and preferences of the decision maker. The reservoir engineer can use decision trees to describe complex scenarios with multiple decisions and multiple probabilities. This discussion can only be considered a brief introduction. In constructing a decision tree we use rectangles to represent decision nodes and circles to represent probability nodes. Two or more decisions can be associated with each decision nodes and multiple nodes can be associated with a probability node. A probability node representing betting one thousand dollars on number “30” at a roulette wheel in Las Vegas looks like this:
The single bet on number 30 can easily be evaluated as to its expected value as follows:
In other words, a single $1,000 bet on number 30 (or any other number) has a negative expected value of 52.63 dollars. Similar analyses will show negative expectations for each of the gambling games explaining the fabulous hotels and inexpensive “all you can eat” buffets in Las Vegas. But is it crazy to play roulette or make other decisions selecting lower expected values than other alternatives? No, the decider may have a different use for $35,000 than $1,000. Maybe he owes a debt that it is immediately due and has a major negative result if he is unable to generate $35,000 right away. This particular preference for risk is actually unusual; most people have less utility for expected outcomes that have large negative impacts. This analysis does not mean that every player will lose money playing roulette. It is a relatively straightforward exercise to model a roulette wheel with various strategies in which a significant fraction of the players win. It is the aggregate EMV of all players over the long run that is negative.
Suppose someone gives you the chance to play a game in which a fair coin is flipped. In the case of heads, you receive $2.00 and for tails you get nothing. You will no doubt be happy to play this game as it has an expected monetary value (EMV) of $1.00. How much would you be willing sell your ticket for? It is unlikely anyone will pay you much more than $1.00 and if you sell it for much less you are “giving away” EMV. Now consider another game. In this game you have to buy a ticket. In this game a heads pays $3.00 and a tail pays $1.00. How much would you be willing to pay for this ticket? The EMV of this game is $2.00 and if you pay any less than that you are (on an expected value basis) gaining money. Would you pay more than $1.00? If you paid $1.00, the second game becomes equivalent to the first with the net result of a head being $3.00 – $1.00 = $2.00 and the result of a tail would be $1 -$1 = 0. Is there a difference in how much you are willing to sell your ticket for in the first game and what you are willing to pay for it in the second game? Decision makers often make decisions on other than an expected value basis based on how much investment exposure is necessary.
Let’s consider another set of decisions. In the first option, you pay €1,000 by investing in a very small percentage (0.1%) of a drilling well that you anticipate has a 50% chance of success (or a coin flip for heads if you prefer). In the case of a discovery you win a series of cash flows with an NPV of €4,000 while a dry hole pays nothing. The EMV is 0.5*€4,000 – €1,000 = €1,000. Are you interested in this investment? If you believe these numbers and have 1,000 euros (€) to invest it is an obvious decision to participate in the project. Now let’s look at the 100% working interest position. In this case you need to invest €1,000,000 and have a 50% chance of €4,000,000. Assume that your net worth is just enough that you could come up with the money by mortgaging your house, cashing in your retirement and borrowing all of the money that you can; it is unlikely that you would accept such an investment opportunity. A single inve