The Central Idea
Many decisions have the same basic structure:
- we choose an action;
- something uncertain happens;
- the combination produces an outcome;
- the outcome has a consequence, payoff, cost, or loss.
A decision table or decision tree records that structure so that the alternatives can be compared systematically.
Decision Problems Have Choices and States
A decision is an action that the decision maker can choose. A state of nature is a condition that affects the result but is not under the decision maker's direct control.
Suppose a business must choose between a small production run and a large production run. Demand may later be low or high.
The production level is the decision. Demand is the uncertain state.
What Is a Decision Table?
A decision table displays decisions and uncertain states in a rectangular arrangement.
One common layout places decisions in rows and states in columns:
| Decision | Low Demand | High Demand |
|---|---|---|
| Small Run | $20,000 | $35,000 |
| Large Run | -$5,000 | $60,000 |
Each interior entry is the payoff produced by one decision-state combination.
What Is a Payoff?
A payoff is a numerical measure of the consequence of an outcome.
Depending on the problem, a payoff may represent:
- profit;
- revenue;
- cost savings;
- utility;
- time saved;
- performance points;
- or another quantity we prefer to make larger.
Loss Tables Reverse the Direction
Some decision tables display losses or costs instead of payoffs.
With payoffs, larger values are usually preferred. With losses or costs, smaller values are usually preferred.
The table must therefore be interpreted according to what the numbers mean, not merely according to their size.
Decision Tables Separate Structure from Calculation
A useful habit is to construct the table before trying to decide what action is best.
This separates two questions:
- What can happen?
- How should the alternatives be compared?
That distinction becomes especially important when probabilities or utilities are introduced later.
Example: A Simple Product Decision
A company can introduce either a standard product or a premium product. Market response may be weak or strong.
| Decision | Weak Market | Strong Market |
|---|---|---|
| Standard Product | $40,000 | $70,000 |
| Premium Product | $10,000 | $110,000 |
The premium product has more upside but also performs worse when the market is weak.
Dominance
A decision dominates another decision when it performs at least as well in every relevant state and strictly better in at least one state.
For example:
| Decision | State 1 | State 2 |
|---|---|---|
| A | 12 | 18 |
| B | 9 | 15 |
Decision A dominates Decision B because \(12>9\) and \(18>15\).
Dominated Alternatives Can Often Be Removed
If one decision is worse than another in every possible state, the dominated decision normally does not need further analysis.
Removing dominated alternatives can simplify a large decision table before probabilities or more advanced criteria are applied.
What Is a Decision Tree?
A decision tree represents a decision problem as branches. The branches show how the situation unfolds over time.
A tree is especially useful when:
- events happen in stages;
- later decisions depend on earlier outcomes;
- probabilities attach naturally to branches;
- the order of events matters.
Decision Nodes
A decision node represents a point where the decision maker chooses among available actions.
Decision nodes are traditionally drawn as squares.
A simple structure might be read:
Choose: Launch now or delay.
Chance Nodes
A chance node represents an uncertain event whose outcome is not chosen by the decision maker.
Chance nodes are traditionally drawn as circles.
Branches leaving a chance node may represent outcomes such as:
- high demand or low demand;
- success or failure;
- approval or rejection;
- rise or fall;
- working component or failed component.
Terminal Nodes
A terminal node marks the end of a path through the tree.
The terminal node records the consequence associated with the entire chain of preceding choices and events.
A terminal value may be a payoff, loss, utility, cost, or another numerical measure.
A Tree Is Read from Left to Right
Decision trees are usually built from the present toward the future.
Each complete path from the starting node to a terminal node represents one possible history of the decision problem.
Probabilities Belong on Chance Branches
Suppose demand can be high with probability \(0.7\) or low with probability \(0.3\).
The chance branches may be labeled:
\[ P(\text{high})=0.7 \]and
\[ P(\text{low})=0.3. \]The probabilities leaving the same chance node must sum to \(1\).
Decision Branches Do Not Receive Probabilities
A decision branch represents a choice, not a random event.
If the decision maker can choose either A or B, we do not normally assign probabilities to those branches merely because both are possible.
Probabilities describe uncertainty. Decision branches describe control.
Expected Value at a Chance Node
When numerical payoffs and probabilities are available, the expected value at a chance node is the probability-weighted average of the branch values.
\[ \boxed{E[X]=\sum_i p_i x_i}. \]For two outcomes,
\[ E[X]=p_1x_1+p_2x_2. \]Example: Expected Value of a Small Production Run
Suppose a small production run produces:
- $20,000 if demand is low;
- $35,000 if demand is high.
Let
\[ P(\text{low})=0.4,\qquad P(\text{high})=0.6. \]Then
\[ E[\text{small}] =0.4(20{,}000)+0.6(35{,}000). \] \[ =8{,}000+21{,}000. \] \[ \boxed{E[\text{small}]=29{,}000}. \]Example: Expected Value of a Large Production Run
Suppose a large production run produces:
- -$5,000 if demand is low;
- $60,000 if demand is high.
Using the same probabilities,
\[ E[\text{large}] =0.4(-5{,}000)+0.6(60{,}000). \] \[ =-2{,}000+36{,}000. \] \[ \boxed{E[\text{large}]=34{,}000}. \]Choosing by Expected Monetary Value
If expected monetary value is the chosen decision criterion, compare the expected values of the available alternatives.
\[ E[\text{small}]=29{,}000 \]and
\[ E[\text{large}]=34{,}000. \]Under this criterion, the large production run is preferred.
Expected Value Does Not Describe a Guaranteed Outcome
An expected value is a weighted average across possible outcomes.
In the large-run example, the expected value is $34,000, but neither actual outcome is $34,000. The realized result is either -$5,000 or $60,000.
Expected value summarizes the distribution; it does not promise the average on any single trial.
From a Decision Table to a Decision Tree
A one-stage table can be converted into a tree.
Begin with a decision node:
- branch to Small Run;
- branch to Large Run.
Each decision branch then reaches a chance node:
- Low Demand;
- High Demand.
Each final branch ends at the corresponding payoff.
From a Decision Tree to a Decision Table
A simple one-stage decision tree can also be compressed into a table.
Use:
- one row for each available decision;
- one column for each state of nature;
- one cell for each terminal payoff.
Trees emphasize sequence. Tables emphasize comparison.
Why Trees Become More Valuable in Sequential Decisions
Suppose a company can first run a market test and then decide whether to launch.
The second decision depends on the test result.
A flat table becomes cumbersome because it must list complete strategies. A tree naturally shows the order:
- choose whether to test;
- observe a test result;
- choose whether to launch;
- observe market demand;
- receive the final payoff.
Strategies Are Complete Plans
In a sequential problem, a strategy is not merely the first choice.
It is a complete rule describing what the decision maker will do at every later decision node that might be reached.
For example:
Run the test; launch if the signal is favorable; do not launch if the signal is unfavorable.
Rollback Analysis
Decision trees are usually evaluated from right to left. This process is called rollback or fold-back analysis.
- Start at the terminal outcomes.
- Compute expected values at chance nodes.
- Choose the preferred branch at decision nodes.
- Continue backward until reaching the beginning of the tree.
Chance Nodes Average; Decision Nodes Choose
This distinction summarizes much of decision-tree analysis.
At a chance node:
\[ \text{value}=\sum_i p_i x_i. \]At a decision node, under a maximum-payoff criterion:
\[ \text{value}=\max\{\text{available branch values}\}. \]If the quantities are losses or costs, the decision node may instead choose the minimum.
Example: A Two-Stage Decision
A company can pay $5,000 for a market test before deciding whether to launch a product.
Suppose a favorable test occurs with probability \(0.6\) and an unfavorable test occurs with probability \(0.4\).
After a favorable test, launching has expected value $50,000. After an unfavorable test, launching has expected value -$10,000. Choosing not to launch yields $0.
Rollback the Later Decisions First
After a favorable test:
\[ \max\{50{,}000,0\}=50{,}000. \]After an unfavorable test:
\[ \max\{-10{,}000,0\}=0. \]Therefore, the rational strategy under expected monetary value is:
launch after a favorable test and do not launch after an unfavorable test.
Now Roll Back the Test
Before including the testing cost, the expected value of the information produced by the test is
\[ 0.6(50{,}000)+0.4(0)=30{,}000. \]Subtract the $5,000 testing cost:
\[ 30{,}000-5{,}000=25{,}000. \]The test branch therefore has value
\[ \boxed{25{,}000}. \]Conditional Probabilities Can Change Later Branches
In realistic sequential problems, a test result may change the probability of later events.
For example:
\[ P(\text{high demand}\mid\text{favorable test}) \]may be much larger than
\[ P(\text{high demand}\mid\text{unfavorable test}). \]The tree should use the probabilities appropriate to each node.
Branch Probabilities Must Match the Information Available
A probability written on a branch should describe uncertainty at that point in the tree, given everything already observed along the path.
Using an unconditional probability after new information has arrived can distort the decision.
Joint Path Probabilities
The probability of following an entire path through successive chance nodes is found by multiplying the conditional branch probabilities along that path.
If
\[ P(A)=0.6 \]and
\[ P(B\mid A)=0.7, \]then
\[ P(A\cap B)=0.6(0.7)=0.42. \]Expected Value Can Be Computed by Paths
If a tree has several terminal paths, one equivalent method is to multiply each terminal payoff by the probability of reaching it and then add.
\[ E[X]=\sum_{\text{terminal paths}} P(\text{path})\cdot\text{payoff(path)}. \]Rollback is usually clearer for sequential decisions because it preserves the decision structure.
Decision Tables Are Better for Compact Comparisons
A decision table is often preferable when:
- there is one decision stage;
- the states are clearly defined;
- each decision-state pair has one payoff;
- the main task is side-by-side comparison.
Decision Trees Are Better for Timing and Sequence
A decision tree is often preferable when:
- there are multiple stages;
- new information arrives over time;
- later choices depend on earlier observations;
- conditional probabilities matter;
- the order of actions and events matters.
Tables and Trees Are Representations of the Same Logic
The underlying mathematics does not depend on whether the problem is shown as a table or a tree.
Both representations organize:
- available decisions;
- uncertain states;
- probabilities;
- consequences;
- comparison rules.
Common Error: Confusing Decisions with Random Events
A decision is something the decision maker chooses. A chance event is something uncertain that occurs.
Mixing the two can lead to incorrect probabilities and an incorrectly structured tree.
Common Error: Probabilities Do Not Sum to One
The probabilities leaving a chance node should cover the possible outcomes represented at that node.
Therefore,
\[ \boxed{\sum_i p_i=1}. \]If the sum is not \(1\), either the probabilities are inconsistent or some possible outcomes are missing.
Common Error: Averaging Payoffs Without Probabilities
Suppose outcomes of $10,000 and $50,000 have probabilities \(0.9\) and \(0.1\).
The ordinary average
\[ \frac{10{,}000+50{,}000}{2}=30{,}000 \]is not the expected value.
The expected value is
\[ 0.9(10{,}000)+0.1(50{,}000)=14{,}000. \]Common Error: Ignoring Costs on Earlier Branches
If a test, survey, prototype, inspection, or research step costs money, that cost must be included in the value of paths that use it.
Otherwise the analysis overstates the value of gathering information.
Common Error: Choosing Too Early in a Sequential Tree
A later decision should be evaluated using the information that will be available when that decision is actually made.
This is why rollback begins at the right side of the tree and works backward.
Common Error: Treating Expected Value as the Only Criterion
Expected monetary value is important, but it is not automatically the correct objective in every decision problem.
A decision maker may also care about:
- risk;
- downside exposure;
- liquidity;
- reliability;
- fairness;
- strategic flexibility;
- utility rather than money alone.
Later lessons in this section will develop these distinctions.
A Reliable Procedure for Decision Tables
- List the available decisions.
- List the relevant uncertain states.
- Determine the payoff or loss for each decision-state combination.
- Place the values in the table.
- Check for dominated alternatives.
- Add state probabilities when they are known and relevant.
- Apply the chosen decision criterion.
- Interpret the result in the context of the problem.
A Reliable Procedure for Decision Trees
- Identify what happens first.
- Draw a decision node when a choice is made.
- Draw a chance node when an uncertain event occurs.
- Label every branch clearly.
- Place probabilities on chance branches.
- Place payoffs, losses, or utilities at terminal nodes.
- Begin at the right side of the tree.
- Average at chance nodes.
- Choose at decision nodes.
- Continue backward until the initial decision is resolved.
Practice 1: Identify Decisions and States
A farmer must choose whether to plant Crop A or Crop B. The season may be dry or wet.
Identify the decisions and the states of nature.
Practice 2: Read a Decision Table
| Decision | Low Demand | High Demand |
|---|---|---|
| A | 15 | 40 |
| B | 10 | 55 |
Which decision performs better under low demand? Which performs better under high demand?
Practice 3: Identify Dominance
| Decision | State 1 | State 2 | State 3 |
|---|---|---|---|
| A | 18 | 25 | 31 |
| B | 14 | 22 | 29 |
Does one decision dominate the other?
Practice 4: Expected Value
A decision produces $10,000 with probability \(0.3\) and $50,000 with probability \(0.7\).
Find the expected value.
Practice 5: Compare Two Decisions
Decision A produces $30,000 with probability \(0.4\) and $50,000 with probability \(0.6\).
Decision B produces $5,000 with probability \(0.4\) and $70,000 with probability \(0.6\).
Which has the larger expected monetary value?
Practice 6: Probability Check
A chance node has branch probabilities
\[ 0.25,\qquad0.50,\qquad0.35. \]Is the chance node valid as written?
Practice 7: Joint Path Probability
A first event occurs with probability \(0.8\). Given that event, a second event occurs with probability \(0.6\).
Find the probability of the complete path containing both events.
Practice 8: Roll Back a Chance Node
A chance node has:
- payoff \(100\) with probability \(0.2\);
- payoff \(40\) with probability \(0.8\).
Find the value of the chance node.
Practice 9: Roll Back a Decision Node
A decision node offers two already-evaluated branches worth \(52\) and \(47\).
Under a maximum-value criterion, what is the value of the decision node?
Practice 10: Table or Tree?
A firm can conduct a survey. After observing the survey result, it can choose among three launch strategies. Market demand is then revealed.
Which representation is likely to be more natural: a decision table or a decision tree? Explain why.
Practice Answers
Practice 1
The decisions are:
- plant Crop A;
- plant Crop B.
The states of nature are:
- dry season;
- wet season.
Practice 2
Under low demand, Decision A is better because
\[ 15>10. \]Under high demand, Decision B is better because
\[ 55>40. \]Practice 3
Decision A dominates Decision B because
\[ 18>14,\qquad25>22,\qquad31>29. \]Practice 4
\[ E[X]=0.3(10{,}000)+0.7(50{,}000). \] \[ =3{,}000+35{,}000. \] \[ \boxed{E[X]=38{,}000}. \]Practice 5
For Decision A:
\[ E[A]=0.4(30{,}000)+0.6(50{,}000) \] \[ =12{,}000+30{,}000 \] \[ =42{,}000. \]For Decision B:
\[ E[B]=0.4(5{,}000)+0.6(70{,}000) \] \[ =2{,}000+42{,}000 \] \[ =44{,}000. \]Therefore,
\[ \boxed{E[B]>E[A]}. \]Practice 6
No. The probabilities sum to
\[ 0.25+0.50+0.35=1.10. \]Probabilities leaving a chance node must sum to \(1\).
Practice 7
\[ 0.8(0.6)=\boxed{0.48}. \]Practice 8
\[ 0.2(100)+0.8(40) \] \[ =20+32 \] \[ \boxed{52}. \]Practice 9
\[ \max\{52,47\}=\boxed{52}. \]Practice 10
A decision tree is more natural because the problem unfolds in stages. The survey result is observed before the launch strategy is chosen, so later decisions depend on earlier information.
What to Remember
Decision tables and decision trees organize the same fundamental elements: choices, uncertainty, consequences, and comparison.
- Decision tables are compact and useful for one-stage comparisons.
- Decision trees are useful when sequence and timing matter.
- Decision nodes represent choices.
- Chance nodes represent uncertainty.
- Terminal nodes record final consequences.
- Probabilities belong on chance branches.
- Expected values are probability-weighted averages.
- Sequential trees are usually solved by rollback.
- At chance nodes, average; at decision nodes, choose.
Where We Go Next
Decision tables and decision trees give us the structural framework for comparing uncertain alternatives.
The next lesson, Expected Value and Expected Utility, develops the numerical criteria used to evaluate those alternatives and explains why the same monetary outcomes can be valued differently by different decision makers.