The Central Idea

When several outcomes are possible, we often want one number that summarizes the distribution.

Expected value weights each outcome by its probability:

\[ \boxed{E[X]=\sum_i p_i x_i}. \]

Expected utility applies the same idea after converting each outcome into a utility value:

\[ \boxed{E[U(X)]=\sum_i p_i U(x_i)}. \]

The first formula averages outcomes. The second averages how those outcomes are valued.

Expected Value Is a Weighted Average

Suppose a random outcome \(X\) can take values

\[ x_1,x_2,\ldots,x_n \]

with corresponding probabilities

\[ p_1,p_2,\ldots,p_n. \]

Then

\[ E[X]=p_1x_1+p_2x_2+\cdots+p_nx_n, \]

where

\[ p_1+p_2+\cdots+p_n=1. \]

Why the Probabilities Are Weights

More likely outcomes contribute more heavily to the expected value.

If an outcome has probability \(0.8\), it receives four times as much weight as an outcome with probability \(0.2\).

Expected value therefore reflects both magnitude and likelihood.

Example: Two Possible Profits

Suppose a project produces:

Then

\[ E[X]=0.7(20{,}000)+0.3(50{,}000). \] \[ =14{,}000+15{,}000. \] \[ \boxed{E[X]=29{,}000}. \]

Expected Value Need Not Be a Possible Outcome

In the previous example, the only possible realized profits are $20,000 and $50,000.

The expected value is $29,000, even though $29,000 is not itself one of the listed outcomes.

Expected value summarizes the distribution. It does not predict the exact result of one trial.

Expected Value and Long-Run Average

When the same random process is repeated many times under stable conditions, the average observed outcome tends to move toward the expected value.

This connects expected value to the law of large numbers.

For a one-time decision, however, the decision maker experiences one realized outcome, not the long-run average.

Expected Monetary Value

When outcomes are measured in money, expected value is often called expected monetary value, or EMV.

\[ \boxed{\text{EMV}=\sum_i p_i(\text{monetary payoff}_i)}. \]

If maximizing expected monetary value is the chosen criterion, select the available decision with the largest EMV.

Example: Comparing Two Projects

Project A has outcomes:

Therefore,

\[ E[A]=0.6(40{,}000)+0.4(10{,}000) \] \[ =24{,}000+4{,}000 \] \[ =28{,}000. \]

Project B has outcomes:

\[ E[B]=0.4(60{,}000)+0.6(5{,}000) \] \[ =24{,}000+3{,}000 \] \[ =27{,}000. \]

By expected monetary value,

\[ \boxed{A\text{ is preferred}}. \]

Negative Outcomes Are Included Normally

Losses can be represented by negative payoffs.

Suppose a venture returns:

Then

\[ E[X]=0.25(80{,}000)+0.75(-20{,}000) \] \[ =20{,}000-15{,}000 \] \[ \boxed{E[X]=5{,}000}. \]

Expected Cost and Expected Loss

If the numerical outcomes represent costs or losses, the same weighted average is used.

\[ E[L]=\sum_i p_iL_i. \]

The interpretation changes: with losses or costs, smaller expected values are usually preferred.

Expected Value in a Decision Table

Consider the following payoff table:

Decision Low Demand \(0.4\) High Demand \(0.6\)
Small Run $20,000 $35,000
Large Run -$5,000 $60,000

The expected values are

\[ E[\text{small}]=0.4(20{,}000)+0.6(35{,}000)=29{,}000 \]

and

\[ E[\text{large}]=0.4(-5{,}000)+0.6(60{,}000)=34{,}000. \]

Under expected monetary value, the large run is preferred.

Expected Value in a Decision Tree

At a chance node, multiply each branch payoff by its branch probability and add.

\[ \boxed{\text{Chance-node value}=\sum_i p_i x_i}. \]

In rollback analysis, this expected value replaces the chance node when the tree is evaluated from right to left.

Linearity of Expected Value

Expected value has an important algebraic property:

\[ \boxed{E[aX+b]=aE[X]+b}. \]

More generally,

\[ \boxed{E[X+Y]=E[X]+E[Y]}. \]

This is true whether or not \(X\) and \(Y\) are independent.

Example: Adding a Fixed Cost

Suppose a project's uncertain gross payoff is \(X\), with

\[ E[X]=50{,}000. \]

If the project requires a fixed cost of $12,000, net payoff is

\[ X-12{,}000. \]

Therefore,

\[ E[X-12{,}000] =E[X]-12{,}000 \] \[ =50{,}000-12{,}000 \] \[ \boxed{38{,}000}. \]

Expected Value Does Not Measure Risk by Itself

Two choices can have the same expected value and very different distributions.

Consider:

Both have expected value

\[ \boxed{50}. \]

Yet Decision B is more variable.

Why Expected Utility Is Needed

Expected monetary value treats each additional dollar as contributing the same amount to the objective.

Real decisions often behave differently. An additional $10,000 may matter greatly to a small organization but only modestly to a very large one. A possible loss may also matter more than an equally sized gain.

Expected utility provides a mathematical way to represent those preferences.

What Is Utility?

Utility is a numerical representation of preference.

A utility function

\[ U(x) \]

assigns a utility value to each outcome \(x\).

Larger utility values represent more preferred outcomes.

Utility Is Not the Same as Money

Money may be one input to a utility function, but utility itself is not necessarily measured in dollars.

For example, a simple utility function might be

\[ U(x)=\sqrt{x}, \]

for positive wealth \(x\).

Under this function, utility rises as wealth rises, but at a decreasing rate.

Expected Utility

If outcome \(x_i\) occurs with probability \(p_i\), expected utility is

\[ \boxed{E[U(X)]=\sum_i p_iU(x_i)}. \]

A decision maker using expected utility chooses the available decision with the greatest expected utility.

The Order of Operations Matters

Expected utility is generally not the utility of expected value.

\[ \boxed{E[U(X)]\neq U(E[X])} \]

unless the utility function is linear or the particular distribution makes the two expressions coincide.

First evaluate utility at each possible outcome. Then take the probability-weighted average.

Example: Expected Utility with a Square-Root Function

Suppose

\[ U(x)=\sqrt{x} \]

and a lottery pays:

Utilities are

\[ U(100)=10 \]

and

\[ U(400)=20. \]

Therefore,

\[ E[U(X)]=0.5(10)+0.5(20) \] \[ =5+10 \] \[ \boxed{E[U(X)]=15}. \]

Compare with Utility of Expected Wealth

The expected monetary outcome is

\[ E[X]=0.5(100)+0.5(400)=250. \]

Its utility is

\[ U(E[X])=\sqrt{250}\approx15.81. \]

Therefore,

\[ E[U(X)]=15<15.81\approx U(E[X]). \]

This gap reflects the curvature of the square-root utility function.

Risk-Neutral Preferences

A risk-neutral decision maker evaluates monetary lotteries by expected monetary value alone.

A linear utility function represents this structure:

\[ U(x)=ax+b,\qquad a>0. \]

With linear utility,

\[ E[U(X)]=U(E[X]). \]

Risk-Averse Preferences

A risk-averse decision maker may prefer a certain outcome to a risky alternative with the same expected monetary value.

Risk aversion is represented by a concave utility function.

Common examples include

\[ U(x)=\sqrt{x} \]

and

\[ U(x)=\ln x, \]

for appropriate positive values of \(x\).

Diminishing Marginal Utility

Concave utility functions capture diminishing marginal utility.

This means that equal increases in wealth can produce smaller and smaller increases in utility as wealth becomes larger.

For example, moving from $10,000 to $20,000 may matter more to a decision maker than moving from $1,010,000 to $1,020,000.

Risk-Seeking Preferences

A risk-seeking decision maker may prefer a risky alternative to a certain outcome with the same expected monetary value.

This can be represented by a convex utility function.

Under convex utility,

\[ E[U(X)]>U(E[X]) \]

for many non-degenerate risky distributions.

Risk Attitude Depends on the Relevant Range

A person or organization need not display the same risk attitude at every scale or in every context.

Preferences can depend on:

Certainty Equivalent

The certainty equivalent of a risky prospect is the certain amount that gives the same utility as the prospect's expected utility.

If \(CE\) is the certainty equivalent, then

\[ \boxed{U(CE)=E[U(X)]}. \]

If \(U\) can be inverted,

\[ \boxed{CE=U^{-1}(E[U(X)])}. \]

Example: Certainty Equivalent

Return to

\[ U(x)=\sqrt{x} \]

and the equal-probability outcomes $100 and $400.

We found

\[ E[U(X)]=15. \]

Set

\[ \sqrt{CE}=15. \]

Squaring gives

\[ \boxed{CE=225}. \]

The risky prospect has expected monetary value $250 but a certainty equivalent of $225 under this utility function.

Risk Premium

For a risk-averse decision maker, the risk premium is commonly defined as the difference between expected value and certainty equivalent:

\[ \boxed{\text{Risk Premium}=E[X]-CE}. \]

In the previous example,

\[ 250-225=25. \]

Therefore,

\[ \boxed{\text{Risk Premium}=25}. \]

Interpreting the Risk Premium

The risk premium measures how much expected monetary value the decision maker is willing to give up to eliminate the risk, within the assumptions of the utility model.

A larger risk premium indicates a larger gap between the risky prospect and its certainty-equivalent value.

Jensen's Inequality and Risk Aversion

For a concave utility function,

\[ \boxed{E[U(X)]\le U(E[X])}. \]

This relationship is an application of Jensen's inequality.

It gives a compact mathematical explanation for why a risk-averse decision maker may prefer certainty to a fair gamble.

Example: A Fair Gamble

Suppose current wealth is $100 and a gamble changes wealth to:

The expected wealth is

\[ E[X]=0.5(64)+0.5(144)=104. \]

Under

\[ U(x)=\sqrt{x}, \]

expected utility is

\[ E[U(X)]=0.5(8)+0.5(12)=10. \]

The utility of expected wealth is

\[ U(104)=\sqrt{104}\approx10.20. \]

Again,

\[ E[U(X)]<U(E[X]). \]

Utility Functions Preserve Preference Order

Utility numbers are useful because they preserve ranking.

If outcome A is preferred to outcome B, a valid utility representation has

\[ U(A)>U(B). \]

The numerical utility itself is not a direct physical measurement like length or mass.

Positive Affine Transformations Preserve Expected-Utility Choices

If

\[ V(x)=aU(x)+b,\qquad a>0, \]

then \(V\) represents the same expected-utility preference ordering as \(U\).

This means that changing the utility scale by multiplying by a positive constant and adding a constant does not change which alternative is preferred.

Expected Utility in Decision Trees

A decision tree can be evaluated using utility instead of monetary payoff.

  1. Place monetary outcomes at the terminal nodes.
  2. Convert each terminal outcome to utility.
  3. Average utilities at chance nodes.
  4. Choose the branch with the larger expected utility at decision nodes.

This can produce a different decision from expected monetary value.

Example: Expected Value and Expected Utility Can Disagree

Suppose a decision maker has utility

\[ U(x)=\sqrt{x} \]

and must choose between:

Expected monetary values are

\[ E[A]=225 \]

and

\[ E[B]=250. \]

So expected monetary value prefers B.

But expected utilities are

\[ E[U(A)]=\sqrt{225}=15 \]

and

\[ E[U(B)]=0.5(10)+0.5(20)=15. \]

Under this utility function, the decision maker is indifferent between the two.

A Slightly Larger Certain Amount Changes the Choice

Suppose Decision A instead pays $256 for certain.

Then

\[ E[A]=256>250=E[B], \]

and

\[ U(256)=16>15=E[U(B)]. \]

Both criteria now prefer the certain alternative.

A Slightly Smaller Certain Amount

Suppose Decision A pays $196 for certain.

Then

\[ E[A]=196<250=E[B], \]

while

\[ U(196)=14<15=E[U(B)]. \]

Both criteria now prefer the risky alternative.

Expected Utility Requires a Preference Model

Expected value can be calculated from outcomes and probabilities alone. Expected utility requires an additional ingredient:

\[ U(x). \]

The utility function must represent the decision maker's preferences over the relevant outcomes.

Utility Should Not Be Invented Casually

Choosing a utility function is not merely an algebra exercise.

In serious decision analysis, utility may be elicited from preference comparisons, certainty equivalents, institutional objectives, or other structured evidence.

The mathematical model should reflect the decision problem rather than force the decision problem to fit a convenient formula.

Common Error: Using Percentages as Whole Numbers

If an outcome has probability \(30\%\), use

\[ 0.30, \]

not \(30\), in the expected-value calculation.

For example,

\[ 30\%(100)=0.30(100)=30. \]

Common Error: Probabilities Do Not Sum to One

For mutually exclusive outcomes that exhaust the possibilities,

\[ \boxed{\sum_i p_i=1}. \]

If the probabilities sum to more or less than \(1\), check whether an outcome is missing, duplicated, or incorrectly measured.

Common Error: Taking the Ordinary Average

Suppose outcomes \(10\) and \(50\) have probabilities \(0.9\) and \(0.1\).

The ordinary average is

\[ \frac{10+50}{2}=30. \]

But expected value is

\[ 0.9(10)+0.1(50)=14. \]

The outcomes receive probability weights, not equal weights unless their probabilities are equal.

Common Error: Applying Utility After Averaging

Expected utility is

\[ E[U(X)], \]

not generally

\[ U(E[X]). \]

For nonlinear utility functions, these are different quantities.

Common Error: Comparing Utility Numbers from Different Scales

Utility values are meaningful within a consistent preference model.

A utility value of \(80\) from one person's scale cannot automatically be compared with a utility value of \(60\) from another person's independently constructed scale.

Common Error: Ignoring the Baseline

A payoff may represent final wealth, change in wealth, profit, or another quantity.

These are not always interchangeable when utility is nonlinear.

Be explicit about what \(x\) represents before applying \(U(x)\).

A Reliable Procedure for Expected Value

  1. List every possible outcome.
  2. Assign the correct probability to each outcome.
  3. Check that the probabilities form a valid distribution.
  4. Multiply each outcome by its probability.
  5. Add the weighted outcomes.
  6. Interpret the result as an average, not a guarantee.
  7. Compare alternatives only under the decision criterion actually being used.

A Reliable Procedure for Expected Utility

  1. Identify the relevant outcome variable.
  2. Specify or obtain the utility function.
  3. Evaluate utility at each possible outcome.
  4. Multiply each utility value by its probability.
  5. Add to obtain expected utility.
  6. Repeat for each available decision.
  7. Choose the alternative with the greatest expected utility.
  8. Check whether the utility model is appropriate for the decision context.

Practice 1: Basic Expected Value

A random payoff is $20 with probability \(0.4\) and $50 with probability \(0.6\).

Find the expected value.

Practice 2: Three Outcomes

A project returns:

Find the expected value.

Practice 3: Expected Cost

A system incurs a repair cost of $1,000 with probability \(0.1\) and no repair cost with probability \(0.9\).

Find the expected repair cost.

Practice 4: Compare Expected Values

Decision A pays $30 for certain.

Decision B pays $60 with probability \(0.5\) and $0 with probability \(0.5\).

Compare their expected values.

Practice 5: Apply a Utility Function

Let

\[ U(x)=\sqrt{x}. \]

Find \(U(64)\), \(U(100)\), and \(U(144)\).

Practice 6: Expected Utility

Under

\[ U(x)=\sqrt{x}, \]

an outcome is $64 with probability \(0.25\) and $144 with probability \(0.75\).

Find expected utility.

Practice 7: Certainty Equivalent

Under

\[ U(x)=\sqrt{x}, \]

a prospect has expected utility \(12\).

Find its certainty equivalent.

Practice 8: Risk Premium

A risky prospect has expected monetary value $200 and certainty equivalent $180.

Find the risk premium.

Practice 9: Risk Attitude

A decision maker prefers $50 for certain to a gamble paying $100 with probability \(0.5\) and $0 with probability \(0.5\).

Is this behavior consistent with risk-neutral, risk-averse, or risk-seeking preferences?

Practice 10: Expected Utility versus Utility of Expected Value

Let

\[ U(x)=\sqrt{x}. \]

A lottery pays $25 or $225, each with probability \(0.5\).

Find \(E[U(X)]\) and \(U(E[X])\).

Practice Answers

Practice 1

\[ E[X]=0.4(20)+0.6(50) \] \[ =8+30 \] \[ \boxed{38}. \]

Practice 2

\[ E[X]=0.2(100)+0.5(40)+0.3(-10) \] \[ =20+20-3 \] \[ \boxed{37}. \]

Practice 3

\[ E[C]=0.1(1000)+0.9(0) \] \[ \boxed{E[C]=100}. \]

Practice 4

Decision A:

\[ E[A]=30. \]

Decision B:

\[ E[B]=0.5(60)+0.5(0)=30. \]

Therefore,

\[ \boxed{E[A]=E[B]=30}. \]

Practice 5

\[ U(64)=8, \] \[ U(100)=10, \] \[ U(144)=12. \]

Practice 6

\[ E[U(X)] =0.25\sqrt{64}+0.75\sqrt{144} \] \[ =0.25(8)+0.75(12) \] \[ =2+9 \] \[ \boxed{11}. \]

Practice 7

The certainty equivalent satisfies

\[ \sqrt{CE}=12. \]

Therefore,

\[ \boxed{CE=144}. \]

Practice 8

\[ \text{Risk Premium}=200-180 \] \[ \boxed{20}. \]

Practice 9

The gamble has expected monetary value

\[ 0.5(100)+0.5(0)=50. \]

Preferring the certain $50 to the equal-expected-value gamble is consistent with

\[ \boxed{\text{risk-averse preferences}}. \]

Practice 10

First compute expected utility:

\[ E[U(X)] =0.5\sqrt{25}+0.5\sqrt{225} \] \[ =0.5(5)+0.5(15) \] \[ =2.5+7.5 \] \[ \boxed{E[U(X)]=10}. \]

Expected monetary value is

\[ E[X]=0.5(25)+0.5(225)=125. \]

Therefore,

\[ U(E[X])=\sqrt{125}\approx11.18. \]

Hence,

\[ \boxed{E[U(X)]<U(E[X])}. \]

What to Remember

Expected value and expected utility answer related but different questions.

The key distinction is: expected value measures the average outcome, while expected utility measures the average value of the outcome to the decision maker.

Where We Go Next

Expected value gives a general way to average uncertain consequences, and expected utility allows those consequences to be evaluated through a preference model.

The next lesson, Loss Functions, develops a complementary framework in which prediction errors, poor decisions, and undesirable outcomes are assigned explicit numerical losses that can be minimized.