Expected Utility Theory

We see that using Expected Value is not enough to compare simple lotteries in Decision Trees

So instead of calculating Expected Values based on (numerical) consequences, we

  • replace the values of the consequences onto their utilities
  • the utilities are provided by individuals - and therefore may vary from one individual to another
  • the utility captures how an individual things about certain risk


  • suppose we have a set of alternatives $A = \{a, b, c, \ ... \ \}$
  • utility function $U_i$ of individual $i$ is a mapping $U_i: A \mapsto X$
    • where $X$ is based on numerical scale (i.e. $X \equiv \mathbb{R}$)
  • we use the Weighted Sum Model to establish the final aggregated value

The preference and indifference relations are defined as follows:

  • $\forall a,b \in A: a \ P \ b \iff U_i(a) > U_i(b)$ - the preference relation
  • $\forall a,b \in A: a \ I \ b \iff U_i(a) = U_i(b)$ - the indifference relation

Utilities for Lotteries

  • for lotteries (as defined in Decision Trees) we see the lotteries as alternatives
  • probabilities are the weights
  • $X$ is a set of consequences for which the lotteries are defined

The total utility:

  • $U(l) = \sum_{x \in X} u(x) \cdot p_l(x)$
  • where $u$ is the utility function $u: X \mapsto \mathbb{R}$ - it maps a consequence to some real number

Preference Relations

(see also Voting Theory Relations for the same ideas but in Voting Theory)

So we define the relations as:

  • $l_1 \ P \ l_2 \iff U(l_1) > U(l_2)$ - the preference
  • $l_1 \ I \ l_2 \iff U(l_1) = U(l_2)$ - the indifference

We also define the "as good as" relation as $S \equiv P \lor I$

  • $l_1 \ S \ l_2 \iff l_1 \ P \ l_2 \lor l_1 \ I \ l_2$

Note that $S$ is transitive and consistent

  • $\forall l_1, l_2, l_3 \in L(x): l_1 \ S \ l_2 \land l_2 \ S \ l_3 \Rightarrow l_1 \ S \ l_3$

$S$ alone is enough:

  • $l_1 \ P \ l_2 \iff \big[ l_1 \ S \ l_2 \big] \land \big[\overline{l_2 \ S \ l_1} \big]$
  • $l_1 \ I \ l_2 \iff \big[ l_1 \ S \ l_2 \big] \land \big[l_2 \ S \ l_1 \big]$


  • it's simple
  • takes individual preferences into account
  • we are not restricted to numerical consequences
  • there is a clear rationale why it works - the Axioms (see below)


This is a link to Arrow's Impossibility Theorem:

  • there are 5 axioms that need to be respected

Axiom 1: Ranking

When a decision maker compares two lotteries $l_1$ and $l_2$

  • he always can say if he prefers one another or he's indifferent between them


  • $\forall l_1, l_2 \in L(X): l_1 \ S \ l_2 \lor l_2 \ S \ l_1$

Axiom 2: Reduction

Suppose we have a high-order lottery $l$ over $\{l_1, ..., l_k\}$

  • we can always simplify $l$ and make a simple lottery from it
  • lotteries-simplification.png


  • lotteries-simplification-ex.png

Axiom 3: Monotonicity

for lotteries $l_1, l_2 \in L(X)$ over the same outcomes $\{x, y\} \subseteq X$

  • if (1) outcome $x$ is better than $y$ and (2) $p > q$
  • then $l_1 \ P \ l_2$
  • lotteries-monotonicity.png

Axiom 4: Independence

for high-order lotteries $l^{(1)}, l^{(2)} \in L(X)$

  • $l^{(1)}$ is over set of lotteries $\{l_1, \ ... \ , l_k\} \subset L(X)$
  • $l^{(2)}$ is over set of lotteries $\{l'_1, \ ... \ , l_k\} \subset L(X)$
  • (the sets are almost the same - they only differ in $l_1$ and $l'_1$)
  • both $l^{(1)}, l^{(2)}$ have the same probability distributions over their sets


  • if $l_1 \ I \ l'_1$ then $l^{(1)} \ I \ l^{(2)}$


Axiom 5: Continuity

$\forall x, y, z \in X$

  • if $x \ P \ y \ P \ z$ then
  • there $\exists p \in [0, 1]$ s.t.
  • lotteries-continuity.png


  • when something is given with certainty
  • we can transform it to some lottery

Axiom 3 implies that $p$ is unique



Let $S$ be a preference relation on $X$

  • $S$ satisfies the axioms
  • $\iff$
  • $\exists u \ : \ X \mapsto \mathbb{R}$ for which $l_1 \ S \ l_2 \iff U(l_1) \geqslant U(l_2)$


This principle is also used in Multi-Criteria Decision Aiding: