3. Expected Utility Theory II

Recap

Expected utility theory gives a way to compare lotteries, or probability distributions over outcomes.

The main idea is that a decision should be evaluated by the expected utility of the outcome it induces:

\[\mathbb{E}_{\omega}[u(f(d,\omega))]\]

The decision maker then chooses the decision that maximizes expected utility.

This lecture makes the idea concrete through an investment example, then states the theorem that guarantees the existence of a utility function under suitable preference axioms.

Investment Example

Suppose we have capital of one dollar and two investment alternatives: $A$ and $B$.

Alternative A

Investment $A$ gives a guaranteed return:

\[1 \text{ dollar invested in A} \mapsto 1.5 \text{ dollars}\]

So $A$ gives 1.5 dollars with certainty.

Alternative B

Investment $B$ is uncertain:

\[B = \begin{cases} 3 \text{ dollars}, & \text{with probability } 1/2,\\ 1 \text{ dollar}, & \text{with probability } 1/2. \end{cases}\]

So $B$ either triples the investment or simply returns the original investment.

Decision Variable

Let:

\[d \in [0,1]\]

be the fraction of the one dollar invested in $A$.

Then:

$d$ is invested in $A$
$1-d$ is invested in $B$

The decision set is:

\[D = [0,1]\]

States and Outcomes

There are two states of nature:

\[\Omega = \{\omega_1,\omega_2\}\]

where:

$\omega_1$: investment $B$ gives 3 dollars per dollar invested
$\omega_2$: investment $B$ gives 1 dollar per dollar invested

Both states occur with probability $1/2$.

The outcome depends on both the decision $d$ and the state $\omega$:

\[f(d,\omega) = \begin{cases} 1.5d + 3(1-d), & \omega = \omega_1,\\ 1.5d + (1-d), & \omega = \omega_2. \end{cases}\]

The outcome space is:

\[O = [1,3]\]

Average Outcome Logic

Investment $A$ gives 1.5 dollars for sure.

Investment $B$ gives:

\[\frac{1}{2}\cdot 3 + \frac{1}{2}\cdot 1 = 2\]

So if we look only at average outcome, $B$ looks better than $A$.

This logic says:

Invest everything in $B$.

In terms of the decision variable:

\[d^* = 0\]

But this ignores the risk of the lower outcome.

Worst-Case Logic

If we look only at the worst case, then for a fixed $d$ we evaluate:

\[\min_{\omega \in \Omega} f(d,\omega)\]

Here:

\[\min\{1.5d + 3(1-d),\; 1.5d + (1-d)\} = 1.5d + (1-d)\]

So the worst-case decision problem is:

\[\max_{d \in [0,1]} \left(1.5d + (1-d)\right)\]

This is maximized at:

\[d^* = 1\]

So worst-case logic says:

Invest everything in $A$.

This is conservative, but it may miss the upside from $B$.

Expected Utility Logic

Expected utility theory says we should not maximize the average outcome or the worst-case outcome directly.

Instead, choose $d$ to maximize:

\[\mathbb{E}_{\omega}[u(f(d,\omega))]\]

Since the two states have probability $1/2$, this becomes:

\[\max_{d \in [0,1]} \left[ \frac{1}{2}u(1.5d + 3(1-d)) + \frac{1}{2}u(1.5d + (1-d)) \right]\]

This decision rule depends on the utility function $u$.

The central contribution of expected utility theory is that if preferences over lotteries satisfy certain axioms, then such a utility function exists and represents those preferences.

Example Utility Function

Consider the utility function:

\[u(o) = \alpha o - o^2\]

where $\alpha$ is a scalar.

To ensure that utility is increasing on the outcome interval $[1,3]$, assume:

\[\alpha > 6\]

This means that more money gives more utility throughout the relevant outcome range.

Optimal Allocation

Plugging this utility into the expected utility expression and maximizing over $d$ gives:

\[d^* = \begin{cases} 0, & \alpha \geq 8,\\ \dfrac{8-\alpha}{5}, & 6 < \alpha < 8. \end{cases}\]

Interpretation:

If $\alpha \geq 8$, invest everything in $B$.
If $6 < \alpha < 8$, split the investment between $A$ and $B$.

For $6 < \alpha < 8$, the optimal decision is neither $d=0$ nor $d=1$. This matches the intuition that one may want some guaranteed return from $A$ and some possible upside from $B$.

Risk Attitude and the Shape of Utility

The shape of the utility function encodes the decision maker’s attitude toward risk.

For:

\[u(o) = \alpha o - o^2\]

larger $\alpha$ makes the utility function behave more like a linear function on $[1,3]$.

When utility is close to linear, the decision maker behaves more like someone who only cares about the expected outcome. That is why for $\alpha \geq 8$, the optimal decision becomes $d^*=0$, meaning everything is invested in $B$.

For smaller $\alpha$ in the range $(6,8)$, curvature matters more, and the decision maker chooses a mix of $A$ and $B$.

Moments of the Outcome

Looking only at the average outcome corresponds to caring only about the first moment.

This happens when utility is linear:

\[u(o) = ao + b\]

For nonlinear utility functions, other moments also matter.

For the quadratic utility:

\[u(o) = \alpha o - o^2\]

both the first and second moments of the outcome affect expected utility.

For a more general concave differentiable utility function, all moments of the outcome may matter.

For example:

\[u(o) = 1 - e^{-\lambda o}\]

Expanding the exponential gives a power series involving powers of $o$:

\[e^{-\lambda o} = 1 - \lambda o + \frac{\lambda^2 o^2}{2!} - \frac{\lambda^3 o^3}{3!} + \cdots\]

So expected utility may depend on:

the mean
the second moment
the third moment
higher-order moments

This explains why expected utility theory is richer than simply comparing averages.

Lotteries and Mixtures

Let $\mathcal{P}$ denote the set of all lotteries, equivalently the set of all probability distributions on outcomes.

Suppose the finite outcome set is:

\[O = \{o_1,o_2,\ldots,o_n\}\]

For two lotteries $p_1,p_2 \in \mathcal{P}$ and $\alpha \in [0,1]$, the mixture:

\[\alpha p_1 + (1-\alpha)p_2\]

is also a lottery.

For each outcome $o_j$, this mixed lottery assigns probability:

\[(\alpha p_1 + (1-\alpha)p_2)(o_j) = \alpha p_1(o_j) + (1-\alpha)p_2(o_j)\]

This can be interpreted in two equivalent ways:

first choose lottery $p_1$ with probability $\alpha$ and $p_2$ with probability $1-\alpha$
or directly form a new probability distribution by mixing the probabilities outcome-wise

Preference Axioms

Expected utility theory relies on a preference relation over lotteries.

Let $\preceq$ represent weak preference, where:

\[p_1 \preceq p_2\]

means lottery $p_2$ is at least as preferred as lottery $p_1$.

Also define:

$p_1 \sim p_2$: $p_1$ and $p_2$ are equivalent
$p_1 \prec p_2$: $p_2$ is strictly preferred to $p_1$

The theorem assumes four axioms.

Axiom 1: Completeness and Transitivity

There exists a complete and transitive preference relation on $\mathcal{P}$.

Completeness means that for any $p_1,p_2 \in \mathcal{P}$:

\[p_1 \preceq p_2 \quad \text{or} \quad p_2 \preceq p_1\]

Transitivity means the preference ordering is logically consistent.

Axiom 2: Mixture Preserves Equivalence

If:

\[p_1 \sim p_2\]

then for all $\alpha \in [0,1]$ and all $p \in \mathcal{P}$:

\[\alpha p_1 + (1-\alpha)p \sim \alpha p_2 + (1-\alpha)p\]

Equivalent lotteries remain equivalent when mixed in the same proportion with a third lottery.

Axiom 3: Mixture Preserves Strict Preference

If:

\[p_1 \prec p_2\]

then for all $\alpha \in [0,1]$ and all $p \in \mathcal{P}$:

\[\alpha p_1 + (1-\alpha)p \prec \alpha p_2 + (1-\alpha)p\]

Strict preference is preserved under common mixing.

Axiom 4: Continuity

If:

\[p_1 \prec p_2 \prec p_3\]

then there exists $\alpha \in [0,1]$ such that:

\[\alpha p_1 + (1-\alpha)p_3 \sim p_2\]

So an intermediate lottery can be matched by a suitable mixture of a worse and a better lottery.

Expected Utility Theorem

Under Axioms 1 to 4, there exists a real-valued utility function:

\[u: O \to \mathbb{R}\]

such that for all lotteries $p_1,p_2 \in \mathcal{P}$:

\[p_1 \preceq p_2 \quad \Longleftrightarrow \quad \mathbb{E}_{p_1}[u(O)] \leq \mathbb{E}_{p_2}[u(O)]\]

In words:

Comparing lotteries is equivalent to comparing their expected utilities.

This is powerful because it turns a complicated preference comparison over lotteries into an optimization problem over real numbers.

Why the Theorem Matters

The theorem says that if a person’s preferences over lotteries satisfy the axioms, then they are implicitly maximizing expected utility.

The utility function captures:

how the decision maker values outcomes
how they trade off risk and reward
why two people may make different choices under the same probabilities and outcomes

This makes expected utility theory a cornerstone for decision making under uncertainty.

Takeaways

Expected utility theory evaluates decisions by the expected utility of outcomes.
In the investment example, average outcome says invest fully in $B$, while worst-case logic says invest fully in $A$.
Expected utility can recommend an interior allocation between $A$ and $B$.
The shape of the utility function encodes risk attitude.
Linear utility makes only the mean matter.
Nonlinear utility can make higher moments matter.
Lotteries can be mixed using convex combinations.
Under reasonable preference axioms, a utility function exists.
Comparing lotteries then becomes equivalent to comparing expected utilities.