16 Rational Decisions

Given a decision problem \((S, O, A)\), the fundamental question is: Which acts (elements of \(A\)) 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 represented using a utility function \(u: O \rightarrow \mathbb{R}\). Given such a utility function on outcomes, 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 depicts 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.

16.1 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.

16.2 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 6.2), 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 6.2.1) of \(u_1\), while \(u_3\) is not a linear transformation of \(u_1\). This distinction leads to the following important lemma:

Lemma 16.1 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 \(EU(a, u', P) = EU(b, u', P)\).

16.3 Exercises

Consider the following decision problem:
\[\begin{array}{|c|c|c|c|} \hline & s_1 & s_2 & s_3 \\ \hline a & 5 & 3 & 2 \\ \hline b & 4 & 4 & 1 \\ \hline c & 3 & 3 & 3 \\ \hline \end{array}\]
1. True or False: Act \(a\) strictly dominates act \(b\).
2. True or False: Act \(b\) strictly dominates act \(c\).
3. True or False: Act \(c\) strictly dominates act \(b\).
True or False: If an act is not strictly dominated by any other act, it must be a rational choice.
Consider the following decision problem with \(P(s_1)=0.5\), \(P(s_2)=0.3\), and \(P(s_3)=0.2\):
\[\begin{array}{|c|c|c|c|} \hline & s_1 & s_2 & s_3 \\ \hline a & 10 & 5 & 0 \\ \hline b & 8 & 6 & 2 \\ \hline \end{array}\]
Calculate \(EU(a,u,P)\) and \(EU(b, u P)\) Which act maximizes expected utility?

Show Answer

\(EU(a, u, P) = 0.5 \times 10 + 0.3 \times 5 + 0.2 \times 0 = 5 + 1.5 + 0 = 6.5\)

\(EU(b, u, P) = 0.5 \times 8 + 0.3 \times 6 + 0.2 \times 2 = 4 + 1.8 + 0.4 = 6.2\)

Act \(a\) maximizes expected utility since \(6.5>6.2\).
True or False: If act \(a\) strictly dominates act \(b\), then \(EU(a,u,P)>EU(b,u,P)\) for any probability distribution \(P\) over the states.
Consider the following decision matrix:
\[\begin{array}{|c|c|c|c|} \hline & s_1 & s_2 & s_3 \\ \hline A & 5 & 2 & 0 \\ \hline B & 3 & 6 & 1 \\ \hline C & 4 & 2 & 5 \\ \hline \end{array}\]
Suppose that \(P(s_1) = 0.2\), \(P(s_2) = 0.4\), and \(P(s_3) = 0.4\). Calculate the expected utility for each act and determine their ranking.

Show Answer

\(EU(A) = 0.2 \times 5 + 0.4 \times 2 + 0.4 \times 0 = 1.8\)

\(EU(B) = 0.2 \times 3 + 0.4 \times 6 + 0.4 \times 1 = 3.4\)

\(EU(C) = 0.2 \times 4 + 0.4 \times 2 + 0.4 \times 5 = 3.6\)

Ranking: \(C \succ B \succ A\) (where \(\succ\) denotes “has higher expected utility than”)
For the same decision matrix as in Question 10:
\[\begin{array}{|c|c|c|c|} \hline & s_1 & s_2 & s_3 \\ \hline A & 5 & 2 & 0 \\ \hline B & 3 & 6 & 1 \\ \hline C & 4 & 2 & 5 \\ \hline \end{array}\]
1. True or False: There exists a probability distribution over states such that act \(A\) has greater expected utility than both \(B\) and \(C\).
2. True or False: There exists a probability distribution over states such that act \(B\) has greater expected utility than both \(A\) and \(C\).
3. True or False: There exists a probability distribution over states such that act \(C\) has greater expected utility than both \(A\) and \(B\).
Consider the following decision matrix:
\[\begin{array}{|c|c|c|c|} \hline & s_1 & s_2 & s_3 \\ \hline A & 1 & 2 & 0 \\ \hline B & 3 & 6 & 1 \\ \hline C & 1 & 2 & 5 \\ \hline \end{array}\]
1. True or False: Act \(B\) strictly dominates act \(A\).
2. True or False: There exists a probability distribution over states such that act \(A\) has greater expected utility than both \(B\) and \(C\).
3. Find a probability distribution where act \(C\) has greater expected utility than act \(B\).
  
  Show Answer
  
  We need to put enough weight on state \(s_3\) where \(C\) performs much better than \(B\).
  
  For example, with \(P(s_1) = 0.1\), \(P(s_2) = 0.1\), \(P(s_3) = 0.8\):
  
  \(EU(B) = 0.1 \times 3 + 0.1 \times 6 + 0.8 \times 1 = 0.3 + 0.6 + 0.8 = 1.7\)
  
  \(EU(C) = 0.1 \times 1 + 0.1 \times 2 + 0.8 \times 5 = 0.1 + 0.2 + 4.0 = 4.3\)
  
  So \(EU(C) > EU(B)\) when most of the probability is on \(s_3\).