# ML Wiki

## 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

### Utility

• 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)

## Axioms

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

I.e.

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

### 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

Example:

### 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 $l_1 \ P \ l_2$

### 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

Independence:

• 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.

I.e.

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

Axiom 3 implies that $p$ is unique

## Theorems

### Representation

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

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

## MCDA

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