Decision Analysis: Expected Value and Decision Trees

Managers rarely know the future, yet they must still decide: launch or wait, expand or hold, bid or walk away. Decision analysis gives a structured way to choose under uncertainty by combining the possible outcomes, their probabilities and their payoffs into a single, defensible number. This article explains expected monetary value, the value of perfect information and decision trees through realistic business cases.

The Building Blocks

Every decision problem has the same parts:

• Alternatives – the options the decision-maker controls (order a large stock, a small stock, or nothing).
• States of nature – uncertain events outside their control (demand turns out high or low).
• Probabilities – how likely each state is, from past data, market research or expert judgement.
• Payoffs – the profit or loss of each alternative under each state, usually arranged in a payoff table.

Expected Monetary Value (EMV)

The EMV of an alternative is the probability-weighted average of its payoffs:

$$ \text{EMV}(A) = \sum_{j} p_j \, V_{Aj} $$

where \( p_j \) is the probability of state \( j \) and \( V_{Aj} \) is the payoff of alternative \( A \) in that state. A risk-neutral decision-maker picks the alternative with the highest EMV. EMV is not what will happen on any single occasion; it is the average result if the same decision were made many times.

Worked Example 1: Stocking for Eid in Karachi

A clothing retailer must decide how much stock to order for the Eid season. Demand may be high (probability 0.6) or low (probability 0.4). Payoffs, in thousands of PKR, are:

Step 1: Compute each EMV.

$$ \text{EMV(Large)} = 0.6(800) + 0.4(-200) = 480 - 80 = 400 $$

$$ \text{EMV(Small)} = 0.6(450) + 0.4(100) = 270 + 40 = 310 $$

$$ \text{EMV(None)} = 0 $$

Step 2: Decide. The large order has the highest EMV, PKR 400,000, so a risk-neutral retailer orders large.

Expected Value of Perfect Information (EVPI)

Suppose a consultant could tell the retailer in advance exactly which demand will occur. With that knowledge the retailer would order large if demand is high (800) and small if demand is low (100). The expected value with perfect information is

$$ \text{EVwPI} = 0.6(800) + 0.4(100) = 480 + 40 = 520. $$

So

$$ \text{EVPI} = \text{EVwPI} - \text{best EMV} = 520 - 400 = 120. $$

Perfect information is worth at most PKR 120,000. Any forecast or survey costing more than that is not worth buying, and since real forecasts are imperfect, a sensible budget is well below this ceiling.

Sensitivity Analysis

The probability 0.6 is only an estimate. How low could it fall before the decision changes? Let \( p \) be the probability of high demand:

$$ \text{EMV(Large)} = 800p - 200(1 - p) = 1000p - 200 $$

$$ \text{EMV(Small)} = 450p + 100(1 - p) = 350p + 100 $$

Setting them equal gives \( 650p = 300 \), so \( p \approx 0.4615 \). As long as the chance of high demand stays above about 46%, the large order remains best. Since the estimate of 60% sits comfortably above this, the decision is reasonably robust.

Decision Trees

When decisions happen in stages, a payoff table becomes awkward. A decision tree draws the problem from left to right:

• A square marks a decision node, where the manager chooses a branch.
• A circle marks a chance node, where nature chooses a branch with given probabilities.
• Payoffs are written at the ends of branches.

The tree is solved by rolling back from right to left: at every chance node compute the EMV, and at every decision node keep the branch with the highest value and prune the rest.

Worked Example 2: Should a Textile Exporter Bid?

A Faisalabad textile exporter is considering bidding for a European contract. Preparing the bid costs PKR 0.5 million. The exporter estimates a 40% chance of winning. If it wins, it has a second decision:

• Produce in-house: profit of PKR 5 million if production runs smoothly (probability 0.7) or PKR 1 million if there are delays and penalties (probability 0.3).
• Subcontract: a certain profit of PKR 3 million.

If it loses the bid, the profit is zero (apart from the bid cost). Not bidding gives zero.

Step 1: Roll back the furthest chance node (in-house production).

$$ \text{EMV(In-house)} = 0.7(5) + 0.3(1) = 3.5 + 0.3 = 3.8 \text{ million} $$

Step 2: Roll back the decision node after winning. Compare in-house (3.8) with subcontract (3.0). Choose in-house; the value of "winning" is PKR 3.8 million. Prune the subcontract branch.

Step 3: Roll back the chance node for the bid outcome.

$$ \text{EMV(Win/Lose)} = 0.4(3.8) + 0.6(0) = 1.52 \text{ million} $$

Step 4: Subtract the bid cost.

$$ \text{EMV(Bid)} = 1.52 - 0.5 = 1.02 \text{ million} $$

Step 5: Final decision. Bidding (PKR 1.02 million) beats not bidding (PKR 0), so the exporter should bid, and if it wins, produce in-house.

Sanity check: how small could the win probability be before bidding stops being worthwhile? Solve \( 3.8q - 0.5 = 0 \), giving \( q \approx 0.132 \). The estimated 40% is about three times this break-even, so the recommendation is safe even if the estimate is optimistic.

Beyond EMV: Risk Attitudes

EMV assumes the decision-maker is risk-neutral. A small family business might prefer the small order in Example 1, because a PKR 200,000 loss could be devastating even though the average outcome is worse. Decision analysts handle this in several ways:

1. Maximin (pessimistic): choose the alternative whose worst payoff is best. Here, small (worst case 100) beats large (−200).
2. Maximax (optimistic): choose the best best-case. Large wins with 800.
3. Minimax regret: compute the opportunity loss in each state and minimise the maximum regret.
4. Expected utility: replace money with a utility function that reflects risk aversion, then maximise expected utility.

In MBA case work, a good answer usually reports the EMV decision, the EVPI, a sensitivity range and a comment on risk.

Software

Decision trees can be built in Excel (with add-ins such as PrecisionTree or by hand with formulas), TreeAge, or Python and R scripts. For assignments, a neat hand-drawn tree with rolled-back values is often expected before any software output.

Did You Know?

• The phrase "decision analysis" was popularised by Ronald Howard of Stanford in the 1960s, who helped turn it into a professional discipline.
• Oil and gas companies were among the first major users of decision trees, deciding whether to drill exploratory wells.
• EVPI is always zero or positive: information can never lower the expected value for a rational decision-maker who is free to ignore it.

Key Takeaways

• EMV is the probability-weighted average payoff; choose the highest EMV if risk-neutral.
• EVPI sets the maximum you should ever pay for information about the future.
• Sensitivity analysis finds the break-even probability at which the decision would change.
• Decision trees handle sequential decisions; solve them by rolling back from right to left.
• Real decisions also weigh risk attitude, so report maximin or utility views where relevant.

Need help building a decision tree, computing EVPI or writing up a business case? Amathify's experts and PhD scholars provide step-by-step solutions and assignment help via WhatsApp or the Ask a Question form.