10  Preferences over Lotteries

Suppose that \(X\) is a finite set and that \(\mathcal{L}(X)\) represents the set of all lotteries over \(X\). In this section, we focus on decision makers who have preferences over the set \(\mathcal{L}(X)\). In other words, these decision makers are comparing lotteries over the set \(X\). A key assumption is that decision makers compare lotteries based on their preferences for the outcomes.

This raises an important question: Can a decision maker’s preference over lotteries be inferred from her preference over the outcomes? For example, suppose the outcomes are \(X = \{a, b, c\}\) and the decision maker has the strict preference \(a \mathrel{P} b \mathrel{P} c\). Clearly, this decision maker would prefer the lottery \(0.1 \cdot a + 0.9 \cdot b\) over the sure-thing lottery \(c\) (since \(a\) and \(b\) are both strictly preferred to \(c\), she would rather have a chance to get either \(a\) or \(b\) than to receive \(c\) for sure).

However, given only the information about the decision maker’s strict preference over \(X\), we cannot infer how she would rank a lottery \(0.5 \cdot a + 0.5 \cdot c\) versus the sure-thing \(b\). To determine her preference between \(0.5 \cdot a + 0.5 \cdot c\) and \(b\), we need to know whether she ranks \(b\) closer to \(a\) than to \(c\), or closer to \(c\) than to \(a\). In other words, we need to understand how she compares the difference between \(a\) and \(b\) with the difference between \(b\) and \(c\). If \(u\) is a utility function representing her preferences, we are interested in how she compares differences in the utilities assigned to \(a\), \(b\), and \(c\): \[u(a) - u(b)\text{ vs. }u(b) - u(c).\]

Thus, we will use cardinal utilities when decision makers compare lotteries, as they allow us to capture the necessary information about differences in preferences between outcomes. Beyond using cardinal utilities, two key questions arise when considering how decision makers compare lotteries:

  1. How should we combine the utilities of the outcomes, when there is uncertainty about which outcome will occur, to arrive at a single number that can be used to compare lotteries?

  2. Are there additional properties beyond transitivity and completeness that constitute a rational preference over the set of lotteries?