In this video, I'll talk about decision trees, the expected value of sample information,
and the expected value of perfect information.
People make decisions all the time.
For example, you can decide what to eat for breakfast, omelets or pancakes?
This is a very simple decision.
Even if you made a bad decision, it doesn't cost you too much.
In the business world, however, the decision making process is not so straightforward.
Instead of making just one decision, usually a series of decisions must be made, one following
another.
And a bad decision could potentially bankrupt a company.
A lot of uncertainties may be associated with the business decision making process.
Under such circumstances, we can use a tool called the decision tree to help us make the
optimal decisions.
In a decision tree, a decision node is denoted by a square.
It represents a time when a decision is made.
An event node is denoted by a circle.
It represents a time when the outcomes occur.
A branch of a decision tree is a terminal branch if no decision nodes or event nodes
come out of this branch.
Let's see an example.
An oil company owns a land and wants to decide whether to drill to find oil.
They have two options.
Option one, they can directly decide to drill for oil or decide to sell the land without
drilling.
Based on the information they have right now, the land has a 25% chance of containing oil
and a 75% chance of being dry, that means, no oil.
Option two, they can do a seismic experiment and then decide whether to drill or sell the
land.
A seismic experiment is conducted by creating a shockwave and send it into the ground.
The reflected shockwave will be collected and analyzed.
The reflection from oil will show certain patterns.
Although the results are not 100% accurate, we can use the seismic experiment to improve
the preliminary estimates of the probabilities of whether the land contains oil.
The cost of the seismic experiment is $2 million.
There is a 40% chance that the result will be favorable, and 60% chance that the result
will be unfavorable.
If the result is Favorable, the chance of containing oil is 60%, which is significantly
improved from 25% without conducting the seismic experiment.
If the result is Unfavorable, the chance of containing oil is 10%.
The drilling operation will cost $5 million.
If oil is found, the company's payoff is $40 million.
If the land is sold without drilling, the company's payoff is $4 million.
The question is, to maximize the company's earnings, what strategy should they follow?
Let's try to build a decision tree for this project.
At the very beginning, they need to make a decision about whether they should do the
seismic experiment.
Because it's a decision node, we represent it with a square.
They have two options, no seismic experiment or do the seismic experiment.
If they don't do the experiment, they need to make another decision immediately: Drill
or sell?
If they sell the land, they don't need to make further decisions and they don't need
to observe the outcome anymore.
So this is a terminal branch.
If they decide to drill, they need to observe the outcome.
So this is an event node.
We represent it with a circle.
The outcome could be "the land contains oil" or "the land is dry".
They are both terminal branches.
Come back the beginning.
If they decide to do the seismic experiment, there are two potential outcomes, favorable
or unfavorable.
If the result is favorable, they need to decide whether to drill or sell.
If they drill, there are two potential outcomes, oil or dry.
If the result is unfavorable, they still need to decide whether to drill or sell.
If they drill, there are two potential outcomes, oil or dry.
This is the structure of the decision tree.
We need to put specific numbers on specific branches.
The cost of doing the seismic experiment is $2 million.
So, we put -2 here.
The cost of the drilling operation is $5 million.
So we put -5 for every branch that is marked as "drill".
The company will earn $4 million by selling the land.
So we put +4 for every branch that is marked as "sell".
If oil is found, the company will earn $40 million.
So we put +40 for every branch that is marked as "oil".
Now, let's calculate the final payoff for each scenario.
There is no seismic experiment, so no cost is related to this branch.
The drilling operation will cost $5 million.
So, we minus 5.
If oil is found, the earning is $40 million.
So, we plus 40.
So the final payoff is 35.
We can calculate the final payoffs for other scenarios by following the same rule.
For this scenario, it's just -5.
For this scenario, it's 4.
For this scenario, it's -2-5+40=33.
For this scenario, it's -2-5=-7.
For this scenario, it's -2+4=2.
For this scenario, it's -2-5+40=33.
For this scenario, it's -2-5=-7.
For this scenario, it's -2+4=2.
Now, let's put the probabilities on specific outcome branches.
If they drill without the seismic experiment, the probability of finding oil is 25%.
The probability of being dry is 75%.
If the company does the seismic experiment, there is a 40% chance that the result is favorable,
and a 60% chance that the result is unfavorable.
If it's favorable and the company decides to drill, the probability of finding oil is
60%.
The probability of being dry is 40%.
If it's unfavorable and the company decides to drill, the probability of finding oil is
10%.
The probability of being dry is 90%.
Now, let's find the expected payoff for this event node.
The payoff is 35 with a probability of 25% and -5 with a probability of 75%.
So, the expected value is 35*25%+(-5)*75%, which is 5.
We put 5 here for this node.
For this decision, if they drill, the payoff is 5.
If they sell, the payoff is 4.
5 is greater than 4.
So drill is a better decision.
We put 5 here for this node.
Let's do that for other event nodes and decision nodes.
The expected value for this node is 33*60%+(-7)*40%, which is 17.
For this decision, if they drill, the payoff is 17.
If they sell, the payoff is 2.
So drill is a better decision.
We put 17 here for this node.
The expected value for this node is 33*10%+(-7)*90%, which is -3.
For this decision, if they drill, the payoff is -3.
If they sell, the payoff is 2.
So sell is a better decision.
We put 2 here for this node.
The expected value for this node is 17*40%+2*60%, which is 8.
For this decision, if no seismic experiment, the payoff is 5.
If they do the seismic experiment, the payoff is 8.
So do the seismic experiment is a better decision.
We put 8 here for this node.
Now, this decision tree is complete.
The optimal strategy is: do the experiment; If favorable, drill; If unfavorable, sell.
The expected payoff is z=$8 million.
We can split this entire decision tree into two parts.
The top one is a decision tree without experimentation.
The bottom one is a decision tree with experimentation.
We would like to know how much value the experiment has brought to the decision making process.
It's called the value of experimentation, or the value of sample information.
To do this, we need to know two other values.
We need to determine the expected value with original information (EVWOI).
It's the largest expected value that would be obtained without experimentation.
Then we need to determine the expected value with sample information (EVWSI).
It's the largest expected value with experimentation but we assume the experimentation cost is
0.
The expected value of sample information: EVSI = EVWSI – EVWOI
If EVSI ≤ real testing or the experimentation cost, then testing is not worth it.
Otherwise, the testing brings more value than its cost.
In this example, the expected value with original information, or the largest expected value
that would be obtained without experimentation, is 5.
The expected value with sample information, or the largest expected value with experimentation
excluding the experimentation cost, is 8+2=10.
The expected value of sample information is 10-5=5, which is greater than the real testing
cost, 2.
So, the testing is totally worth it.
The expected value of sample information measures the value of some limited and incomplete information.
We can also calculate another index called EVPI, which measures the expected value of
perfect or complete information.
To do this, we need to know two other values.
First, we still need to calculate the expected value with original information (EVWOI).
Then, we need to calculate the expected value with perfect information (EVWPI).
I'll talk about how to calculate this one later.
The expected value of perfect information: EVPI = EVWPI – EVWOI
EVPI is an upper bound of EVSI.
That means the value of conducting experimentation will always be less than or equal to the value
of perfect information.
To calculate EVWPI, we assume that all uncertain events that can affect the company’s payoff
still occur with the given probabilities, that means, 25% oil and 75% dry.
But the company finds out whether the land has oil or not before making the decision
to drill or sell.
We can calculate the final payoff for each scenario.
For this scenario, it's 40-5=35.
For this scenario, it's just 4, because if the company sells the land, it cannot earn
the payoff of 40.
For this scenario, it's -5.
For this scenario, it's 4.
After that, we know that drill is the better decision for this node and sell is the better
decision for this node.
Then the expected value with perfect information is 35*25%+4*75%, which is 11.75.
The expected value with original information is still 5.
The expected value of perfect information is 11.75-5=6.75, which is greater than the
expected value of sample information, 5.
Okay, I talked about decision trees, the expected value of sample information, and the expected
value of perfect information in this video.
Thanks for watching.