18 Rational Decisions

Given a decision problem \((S, O, A)\), the fundamental question is: Which acts (elements of \(A\)) are choice-worthy? In other words, which acts are rational choices, and which are irrational? To answer this question, we need to represent the decision maker’s preferences over the set of outcomes \(O\). These preferences are typically modeled using a utility function \(u: O \rightarrow \mathbb{R}\). For instance, the utility associated with choosing act \(a\) in state \(s\) is denoted by \(u(a(s))\). Often, decision problems are presented by directly specifying the utilities for each outcome.

Consider a decision problem where the decision maker must choose between two acts, \(a\) and \(b\), given three possible states: \(s_1\), \(s_2\), and \(s_3\). The following table illustrates the utilities of each act in each state:

\[\begin{array}{|c|c|c|c|} \hline & s_1 & s_2 & s_3 \\ \hline a & \cellcolor{white}{\textcolor{black}{3}} & \cellcolor{white}{\textcolor{black}{1}} & \cellcolor{white}{\textcolor{black}{0}} \\ \hline b & \cellcolor{white}{\textcolor{black}{2}} & \cellcolor{white}{\textcolor{black}{3}} & \cellcolor{white}{\textcolor{black}{1}} \\ \hline \end{array}\]

The decision maker’s preference between acts \(a\) and \(b\) depends on which state is realized:

If the true state is \(s_1\), then \(u(a(s_1)) = 3\) and \(u(b(s_1)) = 2\), so the decision maker prefers \(a\) over \(b\).
If the true state is \(s_2\), then \(u(a(s_2)) = 1\) and \(u(b(s_2)) = 3\), so the decision maker prefers \(b\) over \(a\).
If the true state is \(s_3\), then \(u(a(s_3)) = 0\) and \(u(b(s_3)) = 1\), so the decision maker prefers \(b\) over \(a\).

So, whether the decision maker should choose act \(a\) or act \(b\) depends on what they believe about which state is most likely to occur.

18.1 Strict Dominance

In some cases, we can compare acts without needing to know the decision maker’s beliefs about which state will occur. Specifically, one act \(a\) can be considered definitely better than another act \(b\) if \(a\) leads to a more preferred outcome regardless of the state that is realized.

Strict Dominance: Given a decision problem \((S, O, A)\) with a utility function \(u\), we say that act \(a \in A\) strictly dominates act \(b \in A\) according to \(u\) if, for all \(s \in S\), we have \(u(a(s)) > u(b(s))\).

This means that no matter which state occurs, choosing act \(a\) always results in a higher utility than choosing act \(b\). Clearly, if \(a\) strictly dominates \(b\), then it is irrational for the decision maker to choose \(b\).

Here is an example that illustrates strict dominance.

\[\begin{array}{|c|c|c|c|} \hline & s_1 & s_2 & s_3 \\ \hline a & \cellcolor{white}{\textcolor{black}{3}} & \cellcolor{white}{\textcolor{black}{2}} & \cellcolor{white}{\textcolor{black}{1}} \\ \hline b & \cellcolor{white}{\textcolor{black}{2}} & \cellcolor{white}{\textcolor{black}{3}} & \cellcolor{white}{\textcolor{black}{4}} \\ \hline c & \cellcolor{white}{\textcolor{black}{2}} & \cellcolor{white}{\textcolor{black}{2}} & \cellcolor{white}{\textcolor{black}{0}} \\ \hline d & \cellcolor{white}{\textcolor{black}{2}} & \cellcolor{white}{\textcolor{black}{1}} & \cellcolor{white}{\textcolor{black}{0}} \\ \hline \end{array}\]

Based on this table, we can draw the following conclusions (where \(u\) is the utility described in the table):

Act \(a\) does not strictly dominate act \(b\) because, although \(u(a(s_1)) > u(b(s_1))\), we find that \(u(a(s_2)) < u(b(s_2))\) and \(u(a(s_3)) < u(b(s_3))\).
Act \(a\) does not strictly dominate act \(c\) because, even though \(u(a(s_1)) > u(c(s_1))\) and \(u(a(s_3)) > u(c(s_3))\), there is a state \(s_2\) where \(u(a(s_2)) = u(c(s_2))\).
Act \(a\) does strictly dominate act \(d\) because in all states—\(s_1\), \(s_2\), and \(s_3\)—we have \(u(a(s_1)) > u(d(s_1))\), \(u(a(s_2)) > u(d(s_2))\), and \(u(a(s_3)) > u(d(s_3))\).

Thus, act \(d\) is an irrational choice if act \(a\) is available. Consequently, a rational decision maker would not choose act \(d\). However, based on the information provided, we cannot determine which of the remaining acts the decision maker would choose. Each of the other acts could still be considered rational, depending on the decision maker’s beliefs about which state is most likely to occur.

18.2 Weak Dominance

There is a weaker form of dominance that is sometimes used to compare acts. An act \(a\) is considered weakly better than an act \(b\) if \(a\) is at least as preferred as \(b\) in every state and strictly better in at least one state.

Weak Dominance: Given a decision problem \((S, O, A)\) with a utility function \(u\), we say that act \(a \in A\) weakly dominates act \(b \in A\) according to \(u\) if, for all \(s \in S\), we have \(u(a(s)) \geq u(b(s))\) and there is some \(s\in S\) such that \(u(a(s))>u(b(s))\).

The example from the previous section illustrates weak dominance (repeated here for convenience):

Based on this table, we can draw the following conclusions (where \(u\) is the utility described in the table):

Act \(a\) does not weakly dominate act \(b\) because, although \(u(a(s_1)) > u(b(s_1))\), we find that \(u(a(s_2)) < u(b(s_2))\) and \(u(a(s_3)) < u(b(s_3))\).
Act \(a\) does weakly dominate act \(c\) because \(u(a(s_1)) \geq u(c(s_1))\), \(u(a(s_2)) \geq u(c(s_2))\), and \(u(a(s_3)) \geq u(c(s_3))\), and there is at least one state (\(s_1\)) where \(u(a(s_1)) > u(c(s_1))\).
Act \(a\) does weakly dominate act \(d\) because in all states—\(s_1\), \(s_2\), and \(s_3\)—we have \(u(a(s_1)) > u(d(s_1))\), \(u(a(s_2)) > u(d(s_2))\), and \(u(a(s_3)) > u(d(s_3))\).

The last point demonstrates the following general fact:

If an act \(a\) strictly dominates act \(b\), then \(a\) also weakly dominates \(b\).

Since act \(d\) is strictly dominated by act \(a\), we can conclude that choosing \(d\) is irrational if \(a\) is available. However, it is more controversial to argue that act \(c\) is irrational simply because it is weakly dominated by \(a\). Some argue that weak dominance does not provide a strong enough basis for ruling out an option as irrational, as there could be valid reasons for choosing \(c\) depending on the decision maker’s beliefs or risk tolerance. This debate raises more complex questions about rationality and decision-making, which are beyond the scope of this course.

18.3 Subjective Expected Utility

In most decision problems, no act strictly or weakly dominates another. In such cases, the decision maker’s beliefs about the set of states is needed to determine which acts are rational or irrational. These beliefs are typically represented by a probability function over the set of states \(S\): a function \(P: S \rightarrow [0,1]\) where \(\sum_{s \in S} P(s) = 1\).

Subjective Expected Utility: Given a decision problem \((S, O, A)\) with a utility function \(u\) and a probability function \(P: S \rightarrow [0, 1]\), the subjective expected utility of an act \(a \in A\) is defined as: \[EU(a, u, P) = \sum_{s\in S} P(s)\times u(a(s)).\] An act \(a\) is said to maximize subjective expected utility if, for all acts \(b \in A\), the following holds: \[EU(a, u, P) \geq EU(b, u, P).\]

We use the running example to illustrate the definition of subjective expected utility (repeated here for convenience):

Based on this table, where \(u\) is the utility function described in the table, and \(P\) is the probability on the set of states where \(P(s_1)=0.9\), \(P(s_2)= 0.09\), \(P(s_3)=0.01\), we can calculate the expected utilities of each action:

\[\begin{align*} EU(a, u, P) &= P(s_1)\times u(a(s_1)) + P(s_2)\times u(a(s_2)) + P(s_3)\times u(a(s_3))\\ &= 0.9\times 3 + 0.09 \times 2 + 0.01 \times 1 \\ &= 2.7 + 0.18 + 0.01\\ &= 2.89 \end{align*}\]

\[\begin{align*} EU(b, u, P) &= P(s_1)\times u(b(s_1)) + P(s_2)\times u(b(s_2)) + P(s_3)\times u(b(s_3))\\ &= 0.9\times 2 + 0.09 \times 3 + 0.01 \times 4 \\ &= 1.8 + 0.27 + 0.04\\ &= 2.11 \end{align*}\]

\[\begin{align*} EU(c, u, P) &= P(s_1)\times u(c(s_1)) + P(s_2)\times u(c(s_2)) + P(s_3)\times u(c(s_3))\\ &= 0.9\times 2 + 0.09 \times 2 + 0.01 \times 0 \\ &= 1.8 + 0.18 + 0.0\\ &= 1.98 \end{align*}\]

\[\begin{align*} EU(d, u, P) &= P(s_1)\times u(d(s_1)) + P(s_2)\times u(d(s_2)) + P(s_3)\times u(d(s_3))\\ &= 0.9\times 2 + 0.09 \times 1 + 0.01 \times 0 \\ &= 1.8 + 0.09 + 0.0\\ &= 1.89 \end{align*}\]

In this scenario, act \(a\) maximizes subjective expected utility given the specified beliefs, making \(a\) the unique rational choice. However, different beliefs can lead to different acts being considered rational. For example, if the beliefs are represented by \(P(s_1) = 0.25\), \(P(s_2) = 0.25\), and \(P(s_3) = 0.5\), then act \(b\) would maximize subjective expected utility and be considred the rational choice.

Obviously, different utility functions for the same decision problem can lead to different rankings of the acts in terms of their subjective expected utility, even if the probability distribution over states remains fixed. However, it is important to remember that these utility functions are assumed to be cardinal utility measures on an interval scale (cf. Section 10.2), which has implications for how we perform expected utility calculations.

Consider the following decision problem with acts \(a\) and \(b\), outcomes \(o_1, o_2, o_3\), and \(o_4\) and states \(s_1\), \(s_2\), and \(s_3\).

\[\begin{array}{|c|c|c|c|} \hline & s_1 & s_2 & s_3 \\ \hline a & o_1 & o_2 & o_3 \\ \hline b & o_2 & o_1 & o_4 \\ \hline \end{array}\]

Now, fix the probabilities as \(P(s_1)=0.3\), \(P(s_2)=0.4\), and \(P(s_3)=0.3\). Consider the utility function \(u_1\), as shown in the following table:

\[\begin{array}{|c|c|c|c|} \hline & s_1 & s_2 & s_3 \\ \hline a & 3 & 2 & 1 \\ \hline b & 2 & 3 & 0 \\ \hline \end{array}\]

Using \(u_1\) and the given probability distribution \(P\), we can calculate the subjective expected utility for each act:

\(EU(a, u_1, P) = 0.3\times 3 + 0.4 \times 2 + 0.3 \times 1= 2.0\)
\(EU(b, u_1, P) = 0.3\times 2 + 0.4 \times 3 + 0.3 \times 0= 1.8\)

Thus, according to \(u_1\) and \(P\), \(EU(a, u_1, P) > EU(b, u_1, P)\), indicating that act \(a\) is preferred to act \(b\).

Now, consider two different utility functions, \(u_2\) (displayed in the left table below) and \(u_3\) (displayed in the right table below):

\[\begin{array}{|c|c|c|c|} \hline & s_1 & s_2 & s_3 \\ \hline a & 7 & 5 & 3 \\ \hline b & 5 & 7 & 1 \\ \hline \end{array} \hspace{.4in} \begin{array}{|c|c|c|c|} \hline & s_1 & s_2 & s_3 \\ \hline a & 9 & 4 & 1 \\ \hline b & 4 & 9 & 0 \\ \hline \end{array}\]

All three utility functions—\(u_1\), \(u_2\), and \(u_3\)—agree on the ranking of the outcomes. However, only \(u_2\) agrees with \(u_1\) when ranking acts \(a\) and \(b\) according to subjective expected utility:

\(EU(a, u_2, P) = 0.3 \times 7 + 0.4 \times 5 + 0.3 \times 3 = 5.0\)
\(EU(b, u_2, P) = 0.3 \times 5 + 0.4 \times 7 + 0.3 \times 1 = 4.6\)

Therefore, using \(u_2\) and \(P\), \(EU(a, u_2, P) > EU(b, u_2, P)\), indicating that act \(a\) is preferred over act \(b\).

In contrast, using \(u_3\):

\(EU(a, u_3, P) = 0.3 \times 9 + 0.4 \times 4 + 0.3 \times 1 = 4.6\)
\(EU(b, u_3, P) = 0.3 \times 4 + 0.4 \times 9 + 0.3 \times 0 = 4.8\)

According to \(u_3\) and \(P\), \(EU(b, u_3, P) > EU(a, u_3, P)\), showing that act \(b\) is preferred to act \(a\).

The key observation here is that \(u_2\) is a linear transformation (Section 10.2.1) of \(u_1\), while \(u_3\) is not a linear transformation of \(u_1\). This distinction leads to the following important lemma:

Lemma 18.1 Given a decision problem \((S, O, A)\) and a probability function \(P: S \rightarrow [0, 1]\), for any acts \(a, b \in A\) and utility function \(u: O \rightarrow \mathbb{R}\):

If \(EU(a, u, P) > EU(b, u, P)\) and \(u'\) is a linear transformation of \(u\) (i.e., \(u'(\cdot) = \alpha \times u(\cdot) + \beta\) for some \(\alpha \in \mathbb{R}\) with \(\alpha > 0\) and \(\beta \in \mathbb{R}\)), then \(EU(a, u', P) > EU(b, u', P)\).
If \(EU(a, u, P) = EU(b, u, P)\) and \(u'\) is a linear transformation of \(u\) (i.e., \(u'(\cdot) = \alpha \times u(\cdot) + \beta\) for some \(\alpha \in \mathbb{R}\) with \(\alpha > 0\) and \(\beta \in \mathbb{R}\)), then \(EU(a, u', P) = EU(b, u', P)\).