# 13 Preferences over Lotteries

Suppose that \(X\) is a finite set and that \(\mathcal{L}(X)\) is the set of all lotteries over \(X\). In this section, we are interested in decision makers that have preferences over the set \(\mathcal{L}(X)\). That is, the decision makers are comparing lotteries over a set \(X\). For example, suppose that \(X=\{a, b\}\) and consider three decision makers with different preferences over \(\mathcal{L}(X)\):

- Ann prefers lotteries that give a higher probability to outcome \(a\). So, for instance, Ann has the following preferences: \[[a:1, b:0]\mathrel{P} [a:0.75, b:0.25]\mathrel{P}[a:0.5, b:0.5]\mathrel{P}[a:0.25, b:0.75]\mathrel{P}[a:0, b:1].\]
- Bob prefers lotteries that give a higher probability to outcome \(b\). So, for instance, Bob has the following preferences: \[[a:0, b:1]\mathrel{P} [a:0.25, b:0.75]\mathrel{P}[a:0.5, b:0.5]\mathrel{P}[a:0.75, b:0.25]\mathrel{P}[a:1, b:0].\]
- Carol prefers lotteries that are closer to being a fair lottery. So, for instance, Carol has the following preferences: \[[a:0.5, b:0.5]\mathrel{P} [a:0.75, b:0.25]\mathrel{I}[a:0.25, b:0.75]\mathrel{P}[a:1, b:0]\mathrel{I}[a:0, b:1].\]

It is not hard to see that Ann, Bob and Carol each have a rational preference over \(\mathcal{L}(\{a,b\})\). As explained in Chapter 10, this means that for each decision maker there is a utility function assigning real numbers to lotteries that represents their rational preference. The following utility functions on the set of lotteries over \(\{a, b\}\) represent Ann, Bob, and Carol’s preferences:

- Ann’s utility function is \(U_A([a:r, b:1-r]) = r\), for all \(r\in [0, 1]\).
- Bob’s utility function is \(U_B([a:r, b:1-r]) = 1-r\), for all \(r\in [0, 1]\).
- Carol’s utility function is \(U_C([a:r, b:1-r]) = -|r - 0.5|\), for all \(r\in [0, 1]\).

In the chapter on utility functions, we use a lowercase “\(u\)” to represent a utility function on a set \(X\). In this section, we use a capital “\(U\)” (possibly with subscripts) to represent utility functions on lotteries. This because we need to distinguish between utility functions on the set \(X\) and utility functions on the set \(\mathcal{L}(X)\).

The above utility functions are displayed in the following graph:

There is an important difference between Carol’s preferences and Ann and Bob’s preferences over the set of lotteries.

Both Ann and Bob’s preference satisfy the following property:

**Definition 13.1 **Suppose that \(X\) is a finite set, \(\mathcal{L}(X)\) is the set of all lotteries over \(X\). A rational preference \((P, I)\) over \(\mathcal{L}(X)\) is **expected utility representable** provided that there is a utility function \(u:X\rightarrow\mathbb{R}\) such that for all lotteries \(L, L'\in\mathcal{L}(X)\),

- if \(L\mathrel{P} L'\), then \(EU(L, u) > EU(L', u)\); and
- if \(L\mathrel{I} L'\), then \(EU(L, u) = EU(L', u)\).

It is not hard to see that both Ann and Bob’s preferences are expected utility representable (for Ann, consider the utility that assigns 1 to \(a\) and \(0\) to \(b\), and for Bob, consider the utility function that assigns 0 to \(a\) and 1 to \(b\)). On the other hand, Carol’s preference is **not** expected utility representable.

Towards a contradiction, suppose that Carol’s preferences are expected utility representable. Then, there is a utility function \(u:\{a, b\}\rightarrow\mathbb{R}\) such that

- Since \([a:0.5, b:0.5] \mathrel{P} [a:0.75, b:0.25]\), we have that \(EU([a:0.5, b:0.5], u) > EU([a:0.75, b:0.25], u)\). This implies that \[\begin{align*} 0.5 * u(a) + 0.5 * u(b) &= EU([a:0.5, b:0.5], u) \\ &> EU([a:0.75, b:0.25], u) \\ &= 0.75 * u(a) + 0.25 * u(b) \end{align*}\] Thus, \(0.5u(a) + 0.5u(b) > 0.75u(a) + 0.25u(b)\), and so, we have that \(u(b) > u(a)\).
- Since \([a:0.75, b:0.25] \mathrel{I} [a:0.25, b:0.75]\), we have that \(EU([a:0.75, b:0.25], u) > EU([a:0.25, b:0.75], u)\). This implies that \[\begin{align*} 0.75*u(a) + 0.25*u(b) &= EU([a:0.75, b:0.25], u) \\ &> EU([a:0.25, b:0.75], u) \\ &= 0.25 * u(a) + 0.75 * u(b) \end{align*}\] Thus, \(0.75 * u(a) + 0.25 * u(b)= 0.25 * u(a) + 0.75 * u(b)\), and so, we have that \(u(a) = u(b)\).

Putting 1 and 2 together, we have that \(u(b)> u(a) = u(b)\), which is impossible. Thus, Carol’s preferences are not expected utility representable.

To summarize, we note the following three facts about Carol’s preferences over the set of lotteries \(\mathcal{L}(\{a, b\})\):

- Carol has a rational preference on the set of lotteries \(\mathcal{L}(\{a, b\})\).
- Carol’s rational preference is representable by the utility function \(U_C:\mathcal{L}(\{a, b\})\rightarrow\mathbb{R}\) where for all \(r\in [0, 1]\), \(U_C([a:r, b:1-r]) = -|r-0.5|\).

- Carol’s rational preference is
*not*expected utility representable.

The fact that Carol’s rational preference is representable by \(U_C\) yet her rational preference is not expected utility representable means that \(U_C\) fails to satisfy the following important property of utility functions over lotteries.

**Definition 13.2 (Linear Utility Function) **A utility function \(U:\mathcal{L}(X)\rightarrow\mathbb{R}\) on a set of lotteries over a set \(X\) is **linear** provided that for all lotteries \(L_1, \ldots, L_n\in\mathcal{L}(X)\), \[U([L_1:p_1, \ldots, L_n:p_n]) = \sum_{i=1}^n p_i * U(L_i).\]

For instance, the utility function \(U_C\) is not linear since \(U_C([a:0.5, b:0.5]) \ne 0.5 * U_C([a:1]) + 0.5 * U_C([b:1])\):

- \(U_C([a:0.5, b:0.5]) = -|0.5 - 0.5| = 0\)
- \(0.5 * U_C([a:1]) + 0.5 * U_C([b:1]) = 0.5 * -|1-0.5| + 0.5 * -|0-0.5|\) \(= 0.5 * -0.5 + 0.5 * -0.5 = -0.5\)

The remainder of this chapter presents the additional axioms that are required in order to show that a decision maker’s rational preference is represented by a *linear* utility function on the lotteries.