The Central Idea
A decision problem can be difficult for two different reasons:
- the outcome may vary even when probabilities are known;
- the probabilities or possible outcomes may themselves be uncertain.
The first situation is often called risk. The second is often called uncertainty or ambiguity.
The distinction is useful because different mathematical tools are appropriate in each case.
Risk
In one common decision-theory usage, risk refers to uncertainty that can be represented by a probability distribution.
For example, suppose a machine has:
- a \(0.98\) probability of operating normally;
- a \(0.02\) probability of failing during the next period.
The outcome is uncertain, but the model assigns probabilities to the possibilities.
Uncertainty
Uncertainty is broader. It may include situations in which probabilities are unknown, poorly estimated, changing, or not meaningfully defined.
Examples include:
- a new market with little historical data;
- a technology with no established failure record;
- a policy change whose consequences depend on human responses;
- an event whose possible outcomes have not all been identified.
Risk and Uncertainty Are Not Always Used Identically
Different fields use the words risk and uncertainty in somewhat different ways.
In this lesson, we use a practical distinction:
\[ \boxed{\text{Risk: probabilities are modeled}} \] \[ \boxed{\text{Uncertainty: the model itself is incomplete or unclear}}. \]The distinction is conceptual rather than absolute.
Variability Is Not the Same as Risk
Variability describes how spread out possible outcomes are.
Risk is a broader decision concept. It usually depends on both:
- how outcomes vary;
- which outcomes are harmful or desirable.
A variable outcome is not necessarily undesirable if the variability is mostly on the upside.
Exposure
Exposure describes how much a person, organization, or system stands to gain or lose when an uncertain event occurs.
A very unlikely event can still be important when exposure is large.
This is why probability alone is not enough to describe practical risk.
A Simple Expected-Loss View of Risk
If losses \(L_i\) occur with probabilities \(p_i\), expected loss is
\[ \boxed{E[L]=\sum_i p_iL_i}. \]This combines likelihood and consequence in one quantity.
Expected loss is useful, but it does not capture every dimension of risk.
Example: Low Probability, Large Consequence
Suppose a failure occurs with probability
\[ 0.001 \]and produces a loss of
\[ 2{,}000{,}000. \]The expected loss is
\[ 0.001(2{,}000{,}000) \] \[ =\boxed{2{,}000}. \]The event is rare, but its consequence is large enough to matter.
Expected Value Does Not Describe the Whole Distribution
Consider two alternatives:
- A: receive $50 for certain;
- B: receive $100 with probability \(0.5\) and $0 with probability \(0.5\).
Both have expected value
\[ 50. \]Yet B has greater variability and greater downside exposure.
Variance as a Measure of Dispersion
Variance measures squared distance from the mean:
\[ \boxed{\operatorname{Var}(X)=E[(X-\mu)^2]}, \]where
\[ \mu=E[X]. \]Larger variance means outcomes are more spread out around the mean.
Standard Deviation
Standard deviation is the square root of variance:
\[ \boxed{\sigma=\sqrt{\operatorname{Var}(X)}}. \]Unlike variance, standard deviation has the same units as the original variable.
Example: Comparing Dispersion
Suppose:
- A pays $50 for certain;
- B pays $0 or $100, each with probability \(0.5\).
For A,
\[ \operatorname{Var}(A)=0. \]For B, the mean is \(50\), so
\[ \operatorname{Var}(B) =0.5(0-50)^2+0.5(100-50)^2 \] \[ =0.5(2500)+0.5(2500) \] \[ =\boxed{2500}. \]Therefore,
\[ \sigma_B=50. \]Variance Treats Upside and Downside Symmetrically
Variance counts deviations above and below the mean in the same way because deviations are squared.
For many practical decisions, however, an unexpectedly high result may be welcome while an unexpectedly low result is harmful.
This motivates downside-focused measures.
Downside Risk
Downside risk focuses on outcomes below a relevant target, benchmark, or threshold.
If \(T\) is a target, a simple downside loss is
\[ \boxed{L(X)=\max(0,T-X)}. \]Outcomes above the target create no downside loss, while shortfalls are penalized.
Example: Shortfall Loss
Suppose the target is
\[ T=100. \]If the realized outcome is
\[ X=85, \]then
\[ L(X)=\max(0,100-85) \] \[ =\boxed{15}. \]If \(X=120\), the downside loss is \(0\).
Probability of Shortfall
One simple risk measure is the probability that a target will not be met:
\[ \boxed{P(X<T)}. \]This answers a direct operational question: How likely are we to fall below the threshold?
Severity of Shortfall
Probability of shortfall does not tell us how far below the target the outcome may fall.
Two decisions can have the same probability of failure but very different failure magnitudes.
Good risk analysis therefore often examines both frequency and severity.
Risk as Probability Times Consequence
A common simplified heuristic is
\[ \text{Risk}\approx \text{Probability}\times\text{Consequence}. \]This resembles expected loss when a single adverse event is being modeled.
It is useful as a first approximation but should not be treated as a complete definition of risk in every setting.
Known Probabilities
Probabilities may be reasonably well supported when they come from:
- large historical datasets;
- stable repeated processes;
- physical reliability models;
- well-tested statistical estimates;
- controlled experiments.
Even then, the probabilities are usually estimates rather than perfect facts.
Estimated Probabilities
Suppose an event occurred \(18\) times in \(600\) comparable observations.
A simple empirical estimate is
\[ \hat p=\frac{18}{600}=0.03. \]This does not mean the true probability is known exactly to be \(0.03\).
Probability Estimates Have Uncertainty
A probability estimated from data is itself uncertain.
This creates two layers:
- uncertainty about which outcome will occur;
- uncertainty about the probability model used to describe those outcomes.
The second layer is often called parameter uncertainty.
Model Uncertainty
Model uncertainty arises when we are unsure which probability model or structural relationship is appropriate.
Examples include uncertainty about:
- which variables matter;
- which distribution applies;
- whether historical relationships will continue;
- whether dependencies have been modeled correctly.
Scenario Uncertainty
Sometimes it is more honest to specify several plausible scenarios than to force a single precise probability distribution.
For example:
- slow-growth scenario;
- base scenario;
- high-growth scenario;
- disruption scenario.
Scenario analysis explores how decisions perform across these different possibilities.
Scenario Analysis
A simple scenario table may look like this:
| Decision | Weak Scenario | Base Scenario | Strong Scenario |
|---|---|---|---|
| A | 15 | 35 | 55 |
| B | -5 | 45 | 80 |
Even without agreed probabilities, the table reveals tradeoffs and vulnerabilities.
Decision Making Without Probabilities
When reliable probabilities are unavailable, several criteria may still be used.
These include:
- maximax;
- maximin;
- minimax regret;
- equal-weight or Laplace reasoning;
- robustness across scenarios.
Each criterion reflects a different attitude toward uncertainty.
Maximax
The maximax criterion chooses the decision with the largest possible payoff.
For each decision, identify its best outcome. Then choose the largest of those best outcomes.
This is an optimistic criterion.
Example: Maximax
| Decision | State 1 | State 2 | State 3 |
|---|---|---|---|
| A | 20 | 40 | 55 |
| B | 5 | 30 | 80 |
The best payoff for A is \(55\). The best payoff for B is \(80\).
Maximax therefore chooses
\[ \boxed{B}. \]Maximin
The maximin criterion chooses the decision with the best worst-case payoff.
For each decision, identify its minimum payoff. Then choose the largest of those minimum values.
This is a conservative criterion.
Example: Maximin
Using the previous table:
A has worst payoff
\[ 20. \]B has worst payoff
\[ 5. \]Maximin therefore chooses
\[ \boxed{A}. \]Regret
Regret measures how much worse a chosen action performs compared with the best action that could have been selected after the state is known.
For a payoff table,
\[ \boxed{ \text{Regret} = \text{Best payoff in state} - \text{Chosen payoff} }. \]Example: Regret Table
Start with:
| Decision | State 1 | State 2 |
|---|---|---|
| A | 30 | 50 |
| B | 20 | 80 |
In State 1, the best payoff is \(30\). In State 2, the best payoff is \(80\).
Therefore the regret table is:
| Decision | State 1 | State 2 |
|---|---|---|
| A | 0 | 30 |
| B | 10 | 0 |
Minimax Regret
The minimax regret criterion chooses the action whose largest possible regret is smallest.
In the preceding example:
\[ \max(\text{regret for A})=30 \]and
\[ \max(\text{regret for B})=10. \]Therefore minimax regret chooses
\[ \boxed{B}. \]The Laplace Criterion
When no state is judged more likely than another, one possible rule is to assign equal weights to the states.
With \(n\) states,
\[ p_i=\frac{1}{n}. \]The resulting equal-weight average is sometimes called the Laplace criterion.
Equal weighting is an assumption, not a discovery of the true probabilities.
Robust Decisions
A robust decision performs acceptably across a wide range of plausible assumptions or scenarios.
Robustness is especially valuable when:
- probabilities are uncertain;
- model assumptions may fail;
- conditions can shift;
- the cost of being badly wrong is large.
Robust Does Not Necessarily Mean Optimal in One Scenario
A robust alternative may not be the best under the most favorable scenario.
Instead, it sacrifices some peak performance in exchange for acceptable performance across many possible futures.
Risk Tolerance
Risk tolerance describes how much variability or downside exposure a decision maker is willing and able to accept.
Risk tolerance can depend on:
- available capital;
- liquidity;
- time horizon;
- ability to recover from loss;
- diversification;
- legal or contractual constraints;
- strategic importance.
Capacity and Preference Are Different
A decision maker may be psychologically willing to accept risk but lack the financial capacity to survive a severe loss.
Conversely, an organization may have high capacity for risk while still choosing conservative policies.
Good decision analysis separates willingness from ability.
Diversification
Diversification spreads exposure across several sources rather than concentrating everything in one outcome.
Diversification can reduce risk when the outcomes are not perfectly aligned.
The degree of dependence between risks therefore matters.
Correlation and Risk
If two uncertain quantities move together strongly, combining them may provide little diversification benefit.
If their movements are weakly related or negatively related, combining them can reduce overall variability.
This connects decision analysis to covariance and correlation.
Independent Risks Can Still Accumulate
Independence does not make risk disappear.
It means the occurrence of one event does not alter the probability of the other within the model.
Several independent small risks can still produce meaningful aggregate exposure.
Systemic Risk
Some risks are shared across many components of a system.
If all components depend on the same power source, supplier, market, or network, apparent diversification may fail when that common dependency is disrupted.
Shared dependencies create correlated or systemic exposure.
Tail Risk
Tail risk refers to concern about unusually extreme outcomes in the tails of a probability distribution.
Tail events may have low probability but very large consequences.
When losses are highly asymmetric, average behavior alone may hide operationally important danger.
Stress Testing
Stress testing asks how a decision or system performs under severe but plausible conditions.
Rather than asking only what is most likely, stress testing asks:
What happens if several important assumptions move against us at the same time?
Risk Registers and Structured Identification
Before measuring risk, it must first be identified.
A simple risk register may record:
- the uncertain event;
- estimated likelihood;
- potential consequence;
- exposure;
- mitigation;
- residual risk after mitigation.
Mitigation Changes the Distribution
A mitigation measure may reduce:
- the probability of an adverse event;
- the severity of the consequence;
- the duration of the loss;
- or several of these at once.
Risk analysis should compare the distribution before and after mitigation.
Residual Risk
Residual risk is the risk that remains after controls, safeguards, or mitigation measures are applied.
Few practical systems reduce risk literally to zero.
The relevant question is often whether the remaining exposure is acceptable.
Common Error: Treating Probability as Certainty
A probability of \(0.9\) does not mean the event must occur.
It means that under the model, the event receives probability \(0.9\).
High probability and certainty are different concepts.
Common Error: Treating a Small Probability as Zero
Small probabilities can matter when consequences are sufficiently large.
A useful first check is to examine the product of probability and consequence, while remembering that expected loss is not the only relevant risk measure.
Common Error: Looking Only at Expected Value
Expected value can hide:
- large dispersion;
- downside concentration;
- tail events;
- liquidity constraints;
- catastrophic failure modes.
Similar means do not imply similar risk profiles.
Common Error: Assigning Precise Probabilities Without Evidence
Numerical precision can create an illusion of knowledge.
If evidence only supports a broad range or several plausible scenarios, the model should preserve that uncertainty rather than invent a falsely precise probability.
Common Error: Ignoring Dependence
Risks that appear separate may share common causes.
Assuming independence when events are positively dependent can substantially understate aggregate exposure.
Common Error: Confusing Worst Case with Most Likely Case
A worst-case scenario is not automatically probable.
Its purpose is to explore vulnerability, not to claim that the extreme outcome is the expected outcome.
A Reliable Procedure for Analyzing Risk
- Define the decision and the relevant objective.
- List the important uncertain outcomes.
- Estimate probabilities when evidence supports doing so.
- Measure consequences or losses.
- Compute expected values or expected losses where appropriate.
- Examine variability and downside exposure.
- Check dependence among major risks.
- Stress-test severe but plausible scenarios.
- Evaluate mitigation and residual risk.
- Compare the remaining exposure with the decision maker's constraints and objectives.
A Reliable Procedure Under Deep Uncertainty
- Identify what is known and what is not known.
- Avoid assigning unjustified precision.
- Construct plausible scenarios.
- Compare decisions across those scenarios.
- Identify worst cases and major regrets.
- Look for robust alternatives.
- Identify which assumptions most affect the decision.
- Seek information that can reduce the most important uncertainty.
Practice 1: Risk or Uncertainty?
A factory component has been tested millions of times, and its one-year failure probability is estimated at \(0.004\).
Under the distinction used in this lesson, is this primarily modeled as risk or deep uncertainty?
Practice 2: Expected Loss
A failure has probability \(0.02\) and causes a loss of $80,000.
Find the expected loss from this event.
Practice 3: Variance
A random payoff is \(0\) or \(20\), each with probability \(0.5\).
Find the mean and variance.
Practice 4: Shortfall Probability
A project payoff is:
- $40 with probability \(0.2\);
- $70 with probability \(0.5\);
- $100 with probability \(0.3\).
If the target is $60, find \(P(X<60)\).
Practice 5: Maximax
| Decision | State 1 | State 2 |
|---|---|---|
| A | 15 | 50 |
| B | 5 | 80 |
Which decision is selected by maximax?
Practice 6: Maximin
Using the same table, which decision is selected by maximin?
Practice 7: Regret
| Decision | State 1 | State 2 |
|---|---|---|
| A | 25 | 45 |
| B | 10 | 70 |
Construct the regret values for both decisions.
Practice 8: Minimax Regret
Using the regret table from Practice 7, which decision is selected by minimax regret?
Practice 9: Equal-Weight Criterion
A decision has payoffs \(10\), \(30\), and \(50\) across three scenarios.
Under equal weighting, find its average payoff.
Practice 10: Identify the Modeling Problem
An analyst assigns the probability \(0.6374\) to a completely new event for which almost no data or comparable cases exist.
What is the main modeling concern?
Practice Answers
Practice 1
This is primarily modeled as
\[ \boxed{\text{risk}}, \]because the uncertainty is being represented by an estimated probability supported by extensive repeated observations.
Practice 2
\[ E[L]=0.02(80{,}000) \] \[ =\boxed{1{,}600}. \]Practice 3
The mean is
\[ \mu=0.5(0)+0.5(20)=10. \]The variance is
\[ \operatorname{Var}(X) =0.5(0-10)^2+0.5(20-10)^2 \] \[ =0.5(100)+0.5(100) \] \[ =\boxed{100}. \]Practice 4
Only the $40 outcome falls below the $60 target, so
\[ \boxed{P(X<60)=0.2}. \]Practice 5
The best payoff for A is \(50\). The best payoff for B is \(80\).
\[ \boxed{\text{Maximax chooses B}}. \]Practice 6
The worst payoff for A is \(15\). The worst payoff for B is \(5\).
\[ \boxed{\text{Maximin chooses A}}. \]Practice 7
In State 1, the best payoff is \(25\). In State 2, the best payoff is \(70\).
| Decision | State 1 Regret | State 2 Regret |
|---|---|---|
| A | 0 | 25 |
| B | 15 | 0 |
Practice 8
Maximum regret for A:
\[ 25. \]Maximum regret for B:
\[ 15. \]Therefore,
\[ \boxed{\text{Minimax regret chooses B}}. \]Practice 9
\[ \frac{10+30+50}{3} =\frac{90}{3} =\boxed{30}. \]Practice 10
The main concern is false precision. The evidence does not appear to support such a precise probability estimate.
What to Remember
Risk and uncertainty describe different limits on what we know about future outcomes.
- Risk can often be modeled with probabilities.
- Uncertainty may include unknown probabilities, incomplete models, or unidentified outcomes.
- Expected loss combines likelihood and consequence.
- Variance and standard deviation measure dispersion, not all dimensions of risk.
- Downside risk focuses on harmful shortfalls.
- Rare events can matter when consequences are large.
- Probability estimates themselves can be uncertain.
- Scenario analysis is useful when precise probabilities are unavailable.
- Maximax, maximin, and minimax regret represent different decision attitudes.
- Robust decisions seek acceptable performance across many plausible futures.
The central modeling discipline is: separate uncertainty in outcomes from uncertainty in the model used to describe those outcomes.
Where We Go Next
Risk analysis identifies the range of possible consequences and the uncertainty surrounding them. The next question is how strongly a decision depends on uncertain assumptions.
The next lesson, Sensitivity Analysis, studies how changes in inputs, probabilities, parameters, and assumptions change a model's conclusions and helps identify which uncertainties matter most.