DECISION TREE AND VALUE OF INFORMATION – CASE

03 May DECISION TREE AND VALUE OF INFORMATION – CASE

DECISION TREE AND VALUE OF INFORMATION

Required Reading

Introduction to Decision Trees

Consider the decision to choose between two different alternatives. Imagine that you have a decision where there are three, four or more choices. Then consider that you may have to estimate the probabilities of two or more future states for choice. How do you model this decision in such a way so that you can visualize it easily, analyze the data, get results, and then even make changes for sensitivity analysis? The answer is to use a Decision Tree (D-tree) model.

Let’s take a very simple, generic problem. You have two options to choose from, A and B. let’s say two different stocks choices. You determine that there are three basic outcomes: the market goes high, the market goes medium, and the market goes low. Given the research that you have done, here are the two possible investments and outcomes for a specific period of time, say a year:

Table 1: Possible investments and outcomes

	Option A	Option B
Market level	High	Medium	Low	High	Medium	Low
Probability	0.4	0.3	0.3	0.6	0.25	0.15
Payoff	10	6	2	8	4	3

You can do the calculations in Excel, like you did in module 2. But consider that you may have more complex problems. So using a D-tree is the best way to go. Here is the way this problem looks using a D-tree in Excel using the Simple Decision Tree tool.

The blue square denotes a decision node. A node is a place in the tree where this branches. You see two branches from the blue decision node. These represent the two choices. The choice on the top is labeled Option A and the bottom is Option B (keeping it generic). You could label these with specific names of the stocks. Each option shows the future states. In this example, we have determined that there are two completely separate futures, indicated by the fact that the probabilities are different. But each future has three possible states: high, medium, and low, which are based on the possible behavior of the market, depending on which stock you might buy. The green circle denotes an Uncertainty node. You see the three branches coming off each of these nodes. We have labeled these High, Medium, and Low for each uncertainty node corresponding to each choice of stock. We have entered the probabilities on each branch coming off the uncertainty node. For Option A you see the probabilities 0.4, 0.3, and 0.3 and the corresponding payoffs at the end of the branches, 10, 6, and 2. For Option B you see the probabilities 0.6, 0.25, and 0.15 and the corresponding payoffs at the end of the branches, 8, 4, and 3.

Once we have created the d-tree and entered the labels, the probabilities and the payoffs, the d-tree automatically calculates the best choice. Note that on the top branch you see the Expected Value of this choice is 6.4. The bottom branch shows the Expected Value of 6.25. You also see that on the top branch, there is this symbol >>> next to Option A, which means it is the preferred choice. Then next to the blue square, you see the value of 6.4 which is the value of the best choice, Option A.

You can change the numbers and do sensitivity analysis and play “what-if” games to test out different theories and scenarios. You can add branches to any node or delete branches from any node. Note that there are few “rules” when creating D-tree models, mostly which conform to probability theory. First, at the Choice node, you need to include all of the relevant choices, including the one, do nothing. We have done so here, but for a complete analysis, there is always the status quo or do nothing. Then each chance node must have branches that correspond to a set of mutually exclusive and collectively exhaustive outcomes. Mutually exclusive means that only one of the outcomes can happen. In our example, for either stock, the market can only do one thing, go high, medium or low. Collectively exhaustive means that no other possibilities exist and one of the specified outcomes has to occur. While not a hard and fast rule, usually time is represented from Left to Right. First the Decision Maker chooses an option, then action takes place over time and ultimately one of the future states occurs. Of course in our model we represent all possible future states.

The following instructions on Decision Tree apply to Windows users. If you are a Mac user and do not have any access to a Windows PC, you may try one of the options below (please note that the first two options would incur additional costs):

1. Install Excel and third party add-in in Mac such as  http://treeplan.com/trial/ .

2. Install Parallels software ( http://www.parallels.com/ ) to create a Windows virtual desktop in Mac;

3. Create the decision tree manually. You may use SmartArt Graphics in Office applications (Word, Excel and PowerPoint) to do decision trees. Refer to the following links on SmartArt Graphics:

Create a SmartArt graphic. (n.d.). Retrieved from  https://support.office.com/en-US/article/Create-a-SmartArt-graphic-FAC94C93-500B-4A0A-97AF-124040594842

Learn more about SmartArt graphics. (n.d.). Retrieved from  https://support.office.com/en-US/article/Learn-more-about-SmartArt-Graphics-6ea4fdb0-aa40-4fa9-9348-662d8af6ca2c

About SmartArt graphics in Office 2016 for Mac. (n.d.). Retrieved from  https://support.office.com/en-US/article/About-SmartArt-graphics-in-Office-2016-for-Mac-607b1b23-435a-4bac-98b6-0c9ad0f4ef85

Download the Excel Add-in called  SimpleDecisionTree_V1.4.xla  and add it into your Excel.

Watch this video showing how to include an Add-in:  http://permalink.fliqz.com/aspx/permalink.aspx?at=62469fea72a2420dae1ea19682bb9372&a=5fae3cf0f1624f39b0341263a6541ea0

Watch this video showing how to use the Simple Decision Tree tool:  http://permalink.fliqz.com/aspx/permalink.aspx?at=38aa3b81400343c9bc2b78461f884789&a=5fae3cf0f1624f39b0341263a6541ea0

Practice creating a decision tree by using this example problem:

You have a choice of which game to play: flipping a coin or rolling a single die. It costs you $10 to play the game. If you choose the coin flip, you will win $15 for a heads, but will win only $3 for a tails. If you choose the die roll, you will win 3 times the amount showing on the die. For example, a 4 pays $12 and a 6 pays $18, while a 1 pays only $3.

Once you have created your d-tree for this problem, download this Excel file to check it:  Decision Tree-Which Game to Play . Then you are ready for Case 4.

Deciding to use an Expert: the Value of Information

Wouldn’t it be nice if you could find an expert that could predict the future with 100% accuracy? (Imagine how rich this expert would be.) All you would have to do is pay this expert and he would tell you whether the future will go one way or the other, and then you would know what to do. Your decision would be easy. But that is a fantasy. Some experts are good at predicting the future and we can find out what their track record is.

For example, suppose you are in the last stages of product development of two similar product designs, a new smart phone. One smart phone has many new advanced features but will be very costly to produce and sell with a high price. The other smart phone has only a few new features and would be considered to be only average in its value and have only an average price on the market.  We need to decide which one of these two smart phone options we are going to finish and take to market, but only one. We estimate the two possible future states of the market demand: the market will generally want a High Value, High Price smart phone (Pr = 0.4), or an Average Value, Average Price smart phone (Pr = 0.6). This is the decision tree for this decision including the payoffs.

You know an Expert that you can consult who has a good track record of predicting the future in these situations. This is the track record:

	Market behavior
Expert	High value	Avg Value
Says “wants high value”	0.85	0.08
Says “wants avg value”	0.15	0.92

When the Market actually is demanding High Value, the expert is correct (predicts “High value”) 85% of the time, and is wrong (predicts “Avg. value) 15% of the time. When the Market is actually demanding Avg. value, the expert is correct (predicts “Avg. value) 92% of the time, and is wrong (predicts “High value”) 15% of the time. We can also state these using conditional probability statements:

Pr( Expert Says “wants high value” | Market demand is High value) = 85%

Pr( Expert Says “wants avg value” | Market demand is High value) = 15%

Pr( Expert Says “wants high value” | Market demand is Avg value) = 8%

Pr( Expert Says “wants avg value” | Market demand is Avg value) = 92%

We are showing these probability statements because in a moment you will be asked to read an article that explains Bayes’ Theorem, which uses this kind of probability statements. Here is the modified decision tree if you were to consider consulting an Expert. Note that you have not yet consulted the expert, but are only considering it.

Aggregating multiple probability distributions: when using several experts, combine their distribution assessments into one for use by

In this scenario, with the consideration of consulting an expert, you do not know what he/she will say, and therefore these are unknown future states. He could say the market “wants High V” or “wants Avg V”. We need to determine these probabilities. And note that the probabilities of which actual future market will occur will be different depending on the expert’s prediction.

We need to use Bayes’ theorem to help us determine these new probabilities. Go to this website page which provides an easy to understand explanation of Bayes’ theorem using a medical example, testing for cancer:  http://betterexplained.com/articles/an-intuitive-and-short-explanation-of-bayes-theorem/

Now that you have an understanding of Bayes’ Theorem and “flipping” the probabilities, here are the calculations for the smart phone example.

	0.4	0.6
	Market behavior
Expert	High value	Avg Value
Says “wants high value”	85%	8%
Says “wants avg value”	15%	92%

This is the expert’s track record. Multiply the probabilities (your estimates) of actual market behavior (0.4, and 0.6) down the column to get conditional probabilities.

	Expert
38.8%	Says “wants high value”	34.0%	4.8%
61.2%	Says “wants avg value”	6.0%	55.2%
100.0%

Then you add across to get the probabilities of what the expert might say:

Pr(Expert Says “wants high value”) = 38.8%

Pr( Expert Says “wants avg value”) = 61.2%

And added together these equal 100%, since the expert must say one or the other. Now calculate the conditional probabilities of the future states given what the expert says:

	Conditional
	Market behavior
Expert	High value	Avg Value
Says “wants high value”	87.6%	12.4%	100%
Says “wants avg value”	9.8%	90.2%	100%

For example, in the top row: 0.34 / 0.388 = 0.876, and 0.048 / 0.388 = 0.124. Now we can enter these probabilities into the decision tree and determine what the EVM payoff is for consulting the expert. If this EVM is higher than the EVM for not consulting the expert, then there is value in the information provided by the expert.

Finally, we need to decide if the value of this information is enough for us to pay for it. If this value is more than what the expert charges, then we should consult the expert.

Download the Excel file  BUS520 Module 4 SLP Examples-Practice.xlsx  that provides examples and a Practice Exercise.

Watch this video showing how to determine the value of information:  http://permalink.fliqz.com/aspx/permalink.aspx?at=f616909cae2d4d06834359502f672aff&a=5fae3cf0f1624f39b0341263a6541ea0

Practice determining the Value of Information; do the Practice Exercise in the Excel file.

Optional Reading

Charlesworth, D. (2013). Decision analysis for managers: A guide for making better personal and business decisions. New York, NY: Business Expert Press. Available in the Trident Online Library.