Decision-Making Under Certainty and Risk, Explained

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Decision-Making Under Certainty and Risk

Quantitative Analysis · Chapter 2: Decision-Making Models and RiskSection 2.2 · lesson 6 of 5510 min · free with a free account

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In this lesson

Structured Approaches When Data Is Available.

  • Separate a decision under certainty from one under risk
  • Compute EMV from a payoff table and rank alternatives
  • Bound the price of information with EVPI
  • Compare alternatives using \sigma and the coefficient of variation

Transcript

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QAN-201 - Quantitative Analysis · Section 2.2

Decision-Making Under Certainty and Risk

Some decisions arrive with every fact already on the table. Others arrive with nothing firmer than a probability. A production manager knows exact unit costs and firm customer commitments. A venture investor is guessing how a market will respond. Those two people need genuinely different tools. This section draws the line between certainty and risk. Under certainty the work is computational. You find the best known outcome. Under risk the work turns probabilistic. You weigh every outcome by how likely it is. Then you ask the sharper question. Is more information worth buying? By the end of this section, you will price that information exactly.

The Environment

Two Decision Environments, Two Toolkits

Start with the choice that governs every other choice. The decision environment determines the toolkit. Under certainty, each alternative carries one known, deterministic outcome. Pick it, and you already know exactly what happens. Under risk, several outcomes are possible for each alternative. Every one of those outcomes carries a probability. Here is the part students routinely skip. Those probabilities must be known from historical data, or reasonably estimated from available evidence. If you cannot pin the probabilities down at all, this is not risk. That is uncertainty in the strict sense, and section two point three handles it. So under risk you weigh, you measure spread, and you price information.

Under Certainty

When Every Payoff Is Already Known

Certainty sounds like the easy case, and conceptually it is. The procedure is a comparison, nothing more. Calculate or look up the payoff for every alternative. Then select the one that best satisfies the objective. Maximum profit. Minimum cost. Fastest delivery time. Here is the counter-intuitive part. The challenge is computational, not conceptual. A real problem can carry thousands of variables and constraints. The logic never changes. Find the feasible alternative with the best deterministic payoff. That is exactly why linear programming dominates this environment. Production scheduling, transportation routing, blending, and workforce assignment all qualify. But only while the underlying parameters are treated as fixed. Chapters nine through eleven build that machinery in full.

The Core Criterion

Expected Monetary Value

Now the environment that actually needs analysis. Risk. The workhorse criterion here is expected monetary value. It is the probability-weighted average of an alternative's possible payoffs. Multiply each payoff by its probability, then add the products. But what is that number really telling you? It is a long-run average outcome. It assumes the same decision, repeated many times, under the same distribution. That assumption deserves more attention than it usually gets. A once-in-a-decade capital decision still gets scored by a long-run average. The rule could not be simpler. Highest EMV for profit, lowest expected cost for a cost problem. Everything that follows is a refinement of that one number.

Valuing Information

Expected Value of Perfect Information

Suppose someone offers you a perfectly reliable forecast. It tells you, before you choose, which state of nature will occur. You would then always pick the alternative best suited to that state. The question is what such a forecast is worth. Expected value of perfect information answers precisely that. Take the expected value with perfect information, and subtract the best EMV without it. For each state, take the payoff of the best alternative for that state. Then weight those best-state payoffs by the state probabilities. The gap between the two is the value of resolving the uncertainty. And that gap is a hard ceiling on the price of information. Charge more than it, and the analysis can never recoup the cost. That holds however accurate the consultant turns out to be.

Measuring Risk

Variance and Standard Deviation of Payoffs

EMV has a blind spot, and it is a serious one. It locates the centre of a payoff distribution. It says nothing whatsoever about the spread. Two alternatives can share an identical expected value, and still expose you to completely different ranges of results. So we measure dispersion directly. Variance is the probability-weighted average of squared deviations from the EMV. Square each gap between a payoff and the EMV, weight it, and sum. Standard deviation is simply the square root of that variance. Why take the root at all? Because it comes back in the same units as the payoffs themselves. Dollars, not squared dollars. That is what makes it interpretable. Now watch what dispersion cannot settle on its own. A risk-neutral manager is indifferent when the expected values are equal. A risk-averse one prefers the stable option. A risk-seeker chases the upside.

Normalizing Risk

The Coefficient of Variation

Standard deviation only means something when the scales match. Picture a fifty thousand dollar standard deviation on a project. If that project's expected value is one hundred thousand dollars, you should worry. The same fifty thousand against an expectation of five million is a rounding error. Same risk number. Completely different situation. The coefficient of variation fixes this. Divide the standard deviation by the EMV. You get risk per unit of expected return, with the scale divided out. It is often quoted as a percentage. Now a small product launch and a major capital investment are comparable. Holding return constant, a lower ratio is the more attractive risk profile.

Worked Example

Capacity Expansion: The Payoff Table

Let us put all of it to work on one decision. A manufacturer of industrial components is planning capacity for next year. Three alternatives sit on the table. A small expansion, low capital outlay and limited upside. A medium expansion in between. And a large expansion, high outlay, large upside. But a serious downside if demand disappoints. Four demand states are forecast, from low through to high. Their probabilities come from sales-pipeline analysis. Twenty per cent, thirty, thirty, and twenty. Look hard at the large expansion under low demand. That cell is a loss.

Step 1

EMV for Each Alternative

Step one. Compute the expected value of each alternative. Multiply every payoff by its probability, then sum across the row. The small expansion comes to ninety-three thousand dollars. The medium expansion, one hundred and five thousand. The large expansion, one hundred and fourteen thousand. So on this criterion alone, the large expansion wins. It is worth pausing on how thin that lead is. Nine thousand dollars separates large from medium. And that lead is bought with a possible loss of forty thousand. The criterion ranks the alternatives. It hides what sitting on top actually costs.

Step 2

Pricing Perfect Information

Step two. Price perfect information. Ask what you would do if you knew the demand state in advance. Under low demand you would choose small, for fifty thousand. Under medium-low demand, small again, for eighty thousand. Under medium-high, you would switch to large, for one hundred and eighty. Under high demand, large again, for two hundred and eighty. Weight those four best payoffs by their probabilities. The total is one hundred and forty-four thousand. Subtract the best expected value without information, one hundred and fourteen. The answer is thirty thousand dollars. So no market study or expert forecast here is worth more than that, however accurate it claims to be.

Step 3

Comparing Risk Profiles

Step three. Look at the risk, not just the return. Compute the standard deviation of payoffs for each alternative. The small expansion sits near twenty-eight thousand dollars. The medium expansion, about sixty thousand. The large expansion, nearly one hundred and fifteen thousand. Now divide each by its own expected value. Small comes in at zero point three zero. Medium at zero point five seven. Large at just over one point zero. The large expansion's spread is roughly equal to its expected return. A risk-averse manufacturer might rationally take the medium expansion instead. Lower return, but far lower risk per dollar of that return. The right answer depends on the firm's financial cushion, not on expected value alone.

Formula Drill

Three Alternatives, Four Demand States

One short drill to make the computation automatic. A retailer is setting a seasonal order size. Three order sizes, four demand outcomes. The probabilities come from prior-year sales data. Fifteen per cent, twenty-five, thirty-five, and twenty-five. The small order returns ten thousand four hundred and twenty-five dollars. The medium order, fifteen thousand and fifty. The large order, seventeen thousand one hundred. So the large order maximizes expected value. But you already know that is only half the analysis. Work the same table for perfect information, and the ceiling comes to two thousand eight hundred dollars. That caps what any demand-forecasting tool is worth to this retailer. And the large order's risk ratio is zero point seven two, comfortably under one.

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FAQ

Questions students ask us

About Decision-Making Under Certainty and Risk

What is the difference between decision-making under certainty and under risk?
Under certainty, each alternative has one known outcome, so you simply pick the best payoff; the challenge is computational, which is why linear programming dominates there. Under risk, each alternative has several possible outcomes whose probabilities are known from data or reasonably estimated. You then weigh outcomes by probability, measure their spread and price information. If probabilities cannot be estimated at all, the situation is uncertainty, not risk.
How do you calculate expected monetary value (EMV)?
Expected monetary value is the probability-weighted average of an alternative's payoffs: multiply each payoff by its probability and add the products. You then choose the highest EMV for profits or the lowest expected cost for costs. EMV represents a long-run average, assuming the same decision is repeated many times under the same distribution, so it can mislead for a one-time decision and says nothing about the spread of outcomes.
How do you calculate the expected value of perfect information (EVPI)?
EVPI equals the expected value with perfect information minus the best EMV without it. To get the first term, take the best payoff in each state of nature and weight those by the state probabilities. In a capacity example, the best-by-state payoffs average $144,000 and the best EMV is $114,000, so EVPI is $30,000. No forecast or market study is worth more than that ceiling.
Why use the coefficient of variation instead of standard deviation to compare risk?
The coefficient of variation, standard deviation divided by expected value, measures risk per unit of expected return, so projects of different sizes can be compared. A $50,000 standard deviation is alarming on a $100,000 project but trivial on a $5 million one. For example, a small expansion with \sigma \approx $28,000 and EMV $93,000 has a CV of about 0.30, far safer per dollar than a large one near 1.0.

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