Where Should We Eat? The Five Rules of Decision Analysis Explained
A gentle introduction to the discipline that connects data science to actionable value
As a data scientist, I find the field of decision analysis an underrated tool in my work. When my passion for statistics and programming drags me deep into the weeds, decision analysis pulls my focus back to the big picture impact.
The core of decision analysis consists of five rules that unlock a powerful quantitative framework. These rules were initially formulated by Professor Ronald Howard and are outlined in his book, The Foundations of Decision Analysis [1].
These rules outline the “algebra” of decision-making. They provide the ability to capture subjective beliefs and preferences in a mathematically rigorous framework.
In this post, we will explore each of the rules of decision analysis to answer one of the great mysteries of the ages — where should we eat for dinner tonight?
The dinner decision
We are trying to decide which of the two following restaurants we want to eat at for dinner:
- Roulette’s Tacos: Roulette’s is a new (imaginary) taco place in Austin, TX, so you know it’s good. Their salmon tacos are stellar when they have them in stock. Unfortunately, the only other meal they serve is a salad, which is disappointing.
- Reliable Randy’s Pizzaria: Reliable Randy’s has a good pepperoni pizza. They run out of pepperoni occasionally, but they always have cheese pizza as a backup.
The rules
We will now break down this decision using the five rules of decision analysis. By the end, we will know where to eat tonight.
1. The Probability Rule
We must quantify all uncertainties in the decision with probabilities.
The first rule instructs us to assign probabilities to quantify any uncertainties in the decision. These probabilities are inherently Bayesian rather than frequentist because they capture the subjective beliefs of the decision-maker.
There are two uncertainties in the dinner decision:
- Whether Roulette’s Tacos will have tacos when we arrive.
- Whether Reliable Randy’s will have pepperoni when we arrive.
Based on past experiences (data), we believe that there is a 75% chance that Roulette’s tacos will have tacos and a 90% chance that Reliable Randy’s will have pepperoni.
The probabilistic assessment is where data science contributes to the decision analysis workflow. Perhaps we could use modeling techniques to analyze data to unlock more granular insights. For example, a good data scientist might compute a probability that accounts for the time of day we go to the restaurants. In fact, a core concept of decision analysis is the value of information (VOI), which quantifies the value of a data scientist’s work based on its ability to update the beliefs of a decision-maker. See here for a more detailed discussion of VoI.
In decision analysis, the role of data science is to inform the beliefs of the decsion-maker.
Based on our beliefs, we can represent the decision with the following decision tree:
Notice that in decision trees, squares represent decisions, and circles represent lotteries.
The red square represents the decision regarding which restaurant we will choose. Once we choose a restaurant, we will face an uncertain lottery regarding which meals they will have available.
2. The Order Rule
We must order all possible outcomes based on our preferences.
In the dinner decision, there are four possible outcomes: tacos at Roulette’s, salad at Roulette’s, pepperoni pizza at Reliable Randy’s, and cheese pizza at Reliable Randy’s.
We must order the outcomes in a non-cyclical manner. For example, we cannot prefer tacos to salad, salad to cheese pizza, and cheese pizza to tacos because that results in a cycle of preferences.
In this case, ordering our preferences is straightforward. Tacos are the best option. After that, pepperoni pizza is our second choice, followed by cheese pizza. Salad is disappointing, so we will put it at the bottom. Our ordering is then:
- Tacos at Roulette’s
- Pepperoni Pizza at Reliable Randy’s
- Cheese Pizza at Reliable Randy’s
- Salad at Roulette’s
3. The Equivalence Rule
If we prefer A to B and B to C, there exists a probability p that makes us indifferent between B and a lottery with p chance of A and 1-p chance of C.
The equivalence rule allows us to equate each possible outcome to a lottery between the best and worst outcomes — tacos and salad. Let’s start with pepperoni pizza.
We need to determine the indifference probability, p, such that:
We prefer tacos to pepperoni pizza and pepperoni pizza to salad. So naturally, there must be some amount of salad risk we’re willing to accept for a chance at delicious tacos instead of a sure-fire pepperoni pizza.
Consider two limiting cases:
- If p=1, we would be indifferent between a guaranteed pepperoni pizza and guaranteed tacos. We’re not; we prefer tacos!
- If p=0, we would be indifferent between a guaranteed pepperoni pizza and a guaranteed salad. We’re not; the pizza is much better!
So, p is somewhere between 0 and 1.
The indifference probability only describes our preferences. It is not calculated from data.
I am indifferent between a guaranteed pepperoni pizza and a lottery with an 80% chance of tacos and a 20% chance of salad. So, we will use p=80% for Reliable Randy’s pepperoni pizza.
We also need to determine our indifference probability for cheese pizza. Cheese pizza is almost as good as pepperoni pizza and still much better than salad. My indifference probability is 70%, so I agree with the following statement:
4. The Substitution Rule
If we are indifferent between B and a lottery between A and C, we must be willing to substitute B for the lottery.
The substitution rule allows us to exchange pepperoni pizza and cheese pizza in the decision tree with the lotteries we determined in the last section. This procedure gives the following decision tree:
The substitution is helpful because now our decision tree only has two possible outcomes — tacos and salad. Notice that the lotteries for Reliable Randy’s Pizzaria can be simplified using the law of total probability.
The total probability of tacos with the Reliable Randy’s branch is (0.90 x 0.80) +(0.10 x 0.70) = 79%. We find that the total probability of salad for the branch is 21% with the same procedure or by recognizing that the total probability must sum to 100%.
The simplification results in the following decision tree:
5. The Choice Rule
If we prefer A to C, and we face multiple lotteries between A and C, we must choose the lottery with the highest probability of A.
This rule is intuitive. We have reduced our decision tree to two lotteries between tacos and salad. We prefer tacos to salad, so we must choose the lottery with a higher taco probability, which is Reliable Randy’s Pizzaria.
Roulette’s Tacos is actually a lottery between tacos and salad with a 75% chance of tacos. From the rules of decision analysis, we determined that Reliable Randy’s Pizzaria is equivalent to a lottery between tacos and salad with a 79% chance of tacos.
It looks like we’re having pizza tonight!
Conclusions
In this post, we explored the five rules of decision analysis. These rules are the foundation of a powerful quantitative decision-making framework. Although the dinner example we explored is trivially simple, it helps build the intuition for more complex decisions.
The core of decision analysis is translating beliefs and preferences into rational actions.
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References
- Howard, Ronald A. “The Foundations of Decision Analysis.” IEEE transactions on systems science and cybernetics 4.3 (1968): 211–219.
