9.5.1.2 Axioms of rationality

To meet the goal of designing a utility function, it turns out that the preferences must follow rules called the axioms of rationality. They are sensible statements of consistency among the preferences. As long as these are followed, then a utility function is guaranteed to exist (detailed arguments appear in [268,831]). The decision maker is considered rational if the following axioms are followed when defining $\prec$ and $\approx$:^9.6

1. If $z_1, z_2 \in Z$, then either $z_1 \preceq z_2$ or $z_2 \preceq z_1$.
   ``You must be able to make up your mind.''

2. If $z_1 \preceq z_2$ and $z_2 \preceq z_3$, then $z_1 \preceq z_3$.
   ``Preferences must be transitive.''

3. If $z_1 \prec z_2$, then

   $$\alpha z_1 + (1 - \alpha)z_3 \prec \alpha z_2 + (1 - \alpha)z_3,$$ (9.84)

   
   
   for any $z_3 \in Z$ and $\alpha \in (0,1)$.
   ``Evenly blending in a new distribution does not alter preference.''

4. If $z_1 \prec z_2 \prec z_3$, then there exists some $\alpha \in (0,1)$ and $\beta \in (0,1)$ such that

   $$\alpha z_1 + (1 - \alpha) z_3 \prec z_2$$ (9.85)

   
   
   and

   $$z_2 \prec \beta z_1 + (1 - \beta) z_3 .$$ (9.86)

   
   
   
   ``There is no heaven or hell.''

Each axiom has an intuitive interpretation that makes practical sense. The first one simply indicates that the preference direction can always be inferred for a pair of distributions. The second axiom indicates that preferences must be transitive.^9.7 The last two axioms are somewhat more complicated. In the third axiom, $z_2$ is strictly preferred to $z_1$. An attempt is made to cause confusion by blending in a third distribution, $z_3$. If the same ``amount'' of $z_3$ is blended into both $z_1$ and $z_2$, then the preference should not be affected. The final axiom involves $z_1$, $z_2$, and $z_3$, each of which is strictly better than its predecessor. The first equation, (9.85), indicates that if $z_2$ is strictly better than $z_1$, then a tiny amount of $z_3$ can be blended into $z_1$, with $z_2$ remaining preferable. If $z_3$ had been like ``heaven'' (i.e., infinite reward), then this would not be possible. Similarly, (9.86) indicates that a tiny amount of $z_1$ can be blended into $z_3$, and the result remains better than $z_2$. This means that $z_1$ cannot be ``hell,'' which would have infinite negative reward.^9.8