Decision Tree

Another way to present the consequences together with probabilities and outcomes is decision tree. A decision tree can be a more efficient representation of the present uncertainties, alternatives and consequences than a consequence table. 

When drawing a decision tree, using quantitative measures, the hardest part is how to assign chance (probability) because it involves a high degree of subjective probability estimation where it is easily affected by heuristics and biases. 

 

 

Example 3

Using Example 3 in the previous article <Uncertainty and Chance>, with only one uncertainty labelled as "1", it is even much clearer to visualise the uncertainty and consequence of the decision. Descriptive consequences are seen in Figure 1, whilst quantitative measures in Figure 2.

Figure 1. Decision tree of only one uncertainty in Example 3 using qualitative consequences.

 

Figure 2. Decision tree of only one uncertainty in Example 3 using quantitative consequences.

Difference between Decision Tree and Consequence Table

There are a couple of difference, the first one is chance of occurrence, second is linkage of subsequent decisions with the possible consequences.

1. A consequence table doesn't allow you to consider/ jot down the chance of happening. You just compare the consequences of all objectives (criteria) you listed out. It allows you to cancel out some objectives in the table using Even Swap method, but it doesn't account for chance.
   
2. As for a consequence table, it can only illustrate the consequences for one decision problem. For decision tree, it can connect the outcomes and consequences from one decision with that from other decisions. 
3. While there are more than one uncertainty, the uncertainties are connected and that will branch out different consequences at varying probabilities, so a decision tree is particularly useful when it comes to a decision where it got different decision points mingling with different uncertainties.
   

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Example 4: Decision Tree for One Decision Problem

A man, Mr K, has a business conflict with his business partner. Mr K is thinking about to sue his partner for money compensation for his loss.  Below is the information:

If takes legal action:

• The cost of undertaking the lawsuit: $450,000
• The possible outcome and [chance] of the lawsuit are: Win [75%]; Lose [25%]
• If win, the compensation being awarded is estimated from $500,000 to $1,500,000 according to the similar cases in the past and the past record of this judge ordered.
• If lose, no compensation received and lose the legal fee of $450,000

If not take legal action:

• No need to pay the cost of $450,000
• Mr K's business partner agrees to recompense him $600,000 if Mr K gives up suing him.
  

Now, the decision problem is: To sue or not to sue.

 

Based on the above information, we can draw a decision tree below (Figure 3):

Figure 3. Decision tree based on the preliminary information.

Further Analysis

To analyse the situation more clearly, we will do the following:

1) Firstly we should break the possible outcome of the money awards into components (range):

• Small award: $500,000 - $750,000
• Medium award: $750,001 - $1,000,000
• Large award: $1,000,001 - $1,250,000
• Very large award: $1,250,001 - $1,500,000
  

2) Assign chance to each of the components. Since we have no knowledge about the probability of the individual award range, so we put an equal probability for all at 25% chance. Refer to <Subjective Probability Estimation> and <Principle of Insufficient Reason>.

For better comparison, I employ the median of the range to represent the award of 'Small', 'Medium', 'Large' and 'Very large'. Table 1 depicts the possible outcomes with the divided money range.

Table 1. Illustration of the divided money award with chance in case of winning the case.

3) Account all other costs for the total consequence in money equivalent value, such as time sacrifice and emotional energy put in for the lawsuit. We convert these intangible cost to a tangible value, which is money. The conversion is done by trade-off, i.e. how much would Mr K be willing to pay (WTP) to rid himself from the lengthy court litigation. This is the money worth for the indeterminable variables like time, psyche, emotional energy, anxiety, family annoyance, etc. 

Although in fact the conversion of emotional energy sacrifice to money term should be different in different winning situation, and in different money award outcomes, for simplicity I just assume such a conversion at different situations/ money ranges are all the same. 

    Total consequence (in money equiv.) = Median money award - WTP for time sacrifice  - WTP for emotional energy - cost 

 

For small award:

    Total consequence (in money equiv.) = $625k - $20k - $25k - $450k = $130k 

Table 2 summarises the individual sum of money and calculate the total consequence in terms of money.

Table 2. Calculation of total consequence in terms of money value for undertaking the lawsuit.

4) In order to draw a simplified decision tree with all uncertainties in terms of money equivalent consequences, we also should calculate the money loss in case of losing the case (see Table 3).

Table 3. Illustration of money loss in case of losing the case.

Now, with the information of Tables 1 to 3, a clearer decision tree can be drawn as in Figure 4.

Figure 4. Decision tree using total consequences in terms of money for comparison.

From the decision tree in Figure 4, the situation is much clearer now.

• If Mr K takes legal action, he has 25% chance that he will lose the case and end up losing $450k legal fee. This is the worst consequence of all, even the chance is relatively small, it is still possible.
  
• At the same time, he has 75% of winning the case rewarding from $500k - $1.5m (nominal amount awarded).
  
• Once lawsuit is started, he expects that there will be a lengthy legal process which means he needs to put in money, time and emotional energy into it. This means at the minimum he has 75% chance of having $130k net gain (minus all costs) and maximum of $925k net gain (see Table 2). These are assumed to be the same for winning or losing situation (in fact, the emotional energy input to losing situation should be higher).
  
• If he chooses not to take legal action, he will have 100% chance of getting the compensation of $600k (net gain) from his business partner without any loss.
  

If you were Mr K, what would you do?

Expected Value Theory

Another objective reference we can take is by comparing the total expected value of taking legal action  vs that of not taking legal action. 

Table 4 summarises all individual Expected Values for each money rewards / loss including winning or losing. The total Exptected Value of taking legal action is the summation of all individual Expected Values.

• Total Expected Value of taking legal action = 75% × 25% × $130k + 75% × 25% × $380k + 75% × 25% × $630k + 75% × 25% × $925k- 25% × $495k = $263,437

                The result is shown in Table 4.

• Total Expected Value of not taking legal action = 100% × $600k = $600k

So by comparison, it is clear that, a rational choice is that Mr K should choose not to take legal action, because:

Expected Value of taking legal action ($263k) < Expected Value of not taking legal action ($600k)

Even though objectively Mr K should take the option of not taking legal action, there is something we should consider, which is risk tolerance.

Table 4. Total Expected Value of taking legal action.

Conclusion

A decision tree can only help us visualise all uncertainties and their outcomes more clearly, or even it can connect all decision problems for you. But it cannot decide for you. It is still down to you to make the decision and bear all the responsibilities for any unpleasant consequences come upon your own head in future. This is why, it is of tremendous importance for every person to learn how to make decision.

Expected Value theory is deployed here to help make a rational choice, but there are something we have missed so far, which is personal risk tolerance level. This involves some sort of emotional element for decision making. I will have another article posted for it <Assessing Risk Tolerance for Decision Making>.

 

More Topics:

Smart Choices (1) | Focus and Defining Problem

Smart Choices (2) | Eight Elements in PrOACT Approach

Smart Choices (3) | Comparing with Consequence Table

Trade-offs in Decision Making

Uncertainty and Assigning Chance

Subjective Probability Estimation 

Principle of Insufficient Reason

Assessing Risk Tolerance for Decision Making

風險決策的兩個理論: 期望值 & 期望效用

Reference

John S. Hammond, Ralph L. Keeney, Howard Raiffa, Smart Choices - A Practical Guide to Making Better Life Decisions, Broadway Books, 1999.

 

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