Welcome to Decision Analysis!
Ever found yourself paralyzed by a tough choice? Should you invest in a new business, or keep your money in the bank? Should a company launch a new product now or wait for more market research? Decision Analysis is the mathematical toolkit we use to take the guesswork out of these big life and business questions. By the end of these notes, you’ll be able to map out complex scenarios and calculate the most logical path forward.
Don't worry if this seems like a lot of diagrams at first—it’s essentially just a logical "map" of what could happen!
1. The Building Blocks: Decision Trees
A Decision Tree is a chronological diagram that shows all the possible paths a decision can take. To draw one, we use specific symbols that you need to recognize immediately.
Key Terms and Symbols
- Decision Nodes (Squares \(\square\)): These represent points where you have to make a choice. You choose which branch to follow.
- Chance Nodes (Circles \(\bigcirc\)): These represent points where luck or external factors take over. You don't choose here; instead, outcomes happen based on probabilities.
- Branches: The lines coming out of nodes. Branches from a chance node must have probabilities that add up to 1.
- Pay-offs: The final values at the very end of the branches (on the right-hand side). These are usually financial (profit or loss).
Quick Review: Remember, you control the squares (\(\square\)), but nature controls the circles (\(\bigcirc\))!
2. Expected Monetary Value (EMV)
The Expected Monetary Value (EMV) is the average amount of money you would expect to make if you repeated the decision many, many times. It helps us compare different "chance" outcomes on a level playing field.
How to calculate EMV at a Chance Node:
To find the EMV, you multiply each possible outcome by its probability and add them together.
\(\text{EMV} = \sum (\text{Probability} \times \text{Pay-off})\)
Example: A chance node has two outcomes:
1. Win £100 with probability 0.6
2. Lose £20 with probability 0.4
\(\text{EMV} = (0.6 \times 100) + (0.4 \times -20) = 60 - 8 = £52\)
Key Takeaway: The EMV isn't necessarily a value you will get; it's the long-term average. It’s the "logical" value of that specific gamble.
3. "Folding Back" the Tree
To solve a decision tree, we always work backwards (from right to left). This process is often called "rolling back" the tree.
Step-by-Step Process:
- Start at the far right with the Pay-offs.
- Move left to the first node.
- If it's a Chance Node (\(\bigcirc\)): Calculate the EMV and write it above the node.
- If it's a Decision Node (\(\square\)): Look at the values on the branches coming out of it. Choose the best one (usually the highest value for profit). Write this value above the node and "prune" (put a double slash // through) the branches you didn't choose.
- Continue moving left until you reach the very first node. The value there is your final answer!
Common Mistake: Many students try to work from left to right. Don't do it! You can't know the value of a choice until you know what the outcomes of that choice are worth.
4. Utility Theory
Sometimes, money isn't everything. Would you rather have a 100% chance of £1,000,000 or a 50% chance of £3,000,000? Mathematically, the second option has a higher EMV (£1.5m vs £1m), but most people would take the guaranteed million! This is where Utility comes in.
What is Utility?
Utility is a measure of "satisfaction" or "usefulness." We assign utility values (usually between 0 and 1) to monetary amounts.
- \(U = 0\) is usually the worst outcome.
- \(U = 1\) is usually the best outcome.
Did you know? Using utility helps model "risk-averse" people (who hate losing) or "risk-seeking" people (who love the thrill of the gamble).
Calculating with Utility:
In your exams, you might be given a "utility function" or a table. You simply replace the £ signs with utility values and perform the exact same "fold back" procedure. Instead of EMV, you are calculating Expected Utility.
Key Takeaway: If a question mentions "Utility," ignore the £ values for your calculations and only use the \(U\) values provided!
5. Summary and Tips for Success
Memory Aid: "Square is me, Circle is the sea"
(You control the Square; the Circle is like the ocean—unpredictable and based on chance!)
Quick Checklist for Exam Questions:
- Do your probabilities at every chance node add up to 1.0?
- Have you subtracted any costs? (If it costs £5,000 to enter a project, subtract that from the final pay-offs).
- Did you work Right to Left?
- Did you clearly mark discarded branches with double slashes //?
Final Encouragement: Decision Analysis is one of the most practical parts of Further Maths. Once you get used to the "rolling back" logic, these marks are some of the most consistent ones you can get in Paper 4D. Keep practicing those tree diagrams!