15 Evaluating Rational Choice Axioms

We have argued that rational decision-makers form preferences over lotteries that satisfy not only the core axioms of Transitivity and Completeness but also the Independence Axiom. However, empirical research reveals that many people deviate from these principles, particularly in experiments like the Allais and Ellsberg paradoxes. These deviations challenge the descriptive accuracy of expected utility theory and raise important questions about the nature of rationality and decision-making. How should we interpret these findings? What insights do they provide into human decision-making?

Observing deviations from expected utility theory in experiments like the Allais and Ellsberg paradoxes raises the question of whether these deviations are relevant to our understanding of rational choice. This issue brings into focus the classic distinction between normative and descriptive perspectives in decision theory.

On one hand, the fact that many people reason incorrectly about probabilities or deviate from expected utility theory does not necessarily mean that the theory itself is flawed (a point consistent with Hume’s Law: is does not imply ought). Such deviations might simply indicate that individuals are prone to errors or biases. In fact, this perspective underscores the value of studying rational choice theory: it provides a benchmark for rational reasoning that can be taught, with the potential to improve decision-making skills.

On the other hand, normative theories must also adhere to the principle that ought implies can. If we claim that people should follow expected utility theory, it must be practically possible for them to do so. If cognitive limitations prevent people from consistently applying expected utility theory, this raises concerns about the theory’s normative adequacy.

The key question, then, is whether people can consistently follow expected utility theory. If they cannot, under what circumstances do they fail, and what are the underlying reasons? Understanding the limitations of human reasoning and pinpointing the situations where deviations are most common can provide valuable insights, potentially leading to refinements in both normative and descriptive decision theories.

Whenever we observe behavior that seems to violate an axiom of expected utility theory, like the Independence Axiom, we can interpret this deviation in one of three ways:

Genuine Violations of the Axioms: One interpretation is that the decision-maker’s preferences genuinely conflict with an axiom of expected utility theory. If this is the case, it suggests that expected utility theory may not fully capture how individuals make decisions under uncertainty, pointing to the need for alternative models that better reflect actual behavior or offer principles that are more applicable or easier for people to satisfy.
Changing Preferences: Another possibility is that the decision-maker’s preferences are not stable and may shift during the course of an experiment. Factors such as fatigue, changes in emotional state, or other external influences can cause these fluctuations. In this case, what seems to be a violation of an axiom like the Independence Axiom might not indicate a genuine conflict with expected utility theory but rather a simple change in preference, reflecting temporal inconsistency rather than a fundamental deviation from rational decision-making principles.
Contextual Factors: Finally, a deviation might occur because the experimenter has overlooked relevant features of the decision context that influence preferences. Factors such as framing effects, the way information is presented, or social influences can lead individuals to make choices that seem inconsistent with expected utility theory. If these contextual influences were taken into account, the observed deviations might actually be consistent with rational decision-making principles.

The observed deviations from expected utility theory in the Allais and Ellsberg paradoxes raise questions about whether preferences are stable and how the decision-making context influences the perception of outcomes. To better understand these deviations, we will explore two key aspects of rational choice theory: the assumptions of Stability and Invariance, and the impact of how outcomes are described.

15.1 Stability and Invariance

Two critical assumptions are frequently made, often implicitly, when using models of rational choice.

First, it is typically assumed that a person’s preferences remain stable throughout the course of an experiment or decision-making process.

Stability: Individuals’ preferences are stable over the period of the investigation.

Second, it is assumed that individuals’ preferences are unaffected by irrelevant factors, such as how a decision is framed or the environment in which it is made.

Invariance: Individuals’ preferences are invariant to irrelevant changes in the context of making the decision.

These assumptions are fundamental to applying rational choice models to decision-making scenarios. Unlike formal axioms like Transitivity, Completeness, and the Independence Axiom, which ensure internal coherence of preferences, Stability and Invariance are substantive axioms. They go beyond internal consistency, imposing constraints on what decision makers should or should not care about.

This brings us to a critical dilemma for rational choice theorists:

Only Assume the Formal Axioms of Rational Choice: One option is to adhere strictly to the formal axioms of Completeness, Transitivity, Independence, and other principles without assuming Stability and Invariance. This approach preserves the internal coherence of rational choice theory but limits its practical applicability. It would consider preferences rational even if they constantly change or are influenced by irrelevant contextual factors, thereby reducing the theory’s effectiveness in explaining and predicting real-world behavior.
Assume Stability and Invariance: The alternative is to assume that preferences are Stable and Invariant, which enhances the theory’s practical relevance for prediction and explanation. However, this assumption imposes substantive expectations on decision-makers’ behavior, aligning the theory more with the perspectives of theorists rather than accurately reflecting how individuals actually behave. This approach risks making rational choice theory prescriptive—dictating how people should behave—rather than being purely descriptive of how they actually (should) make decisions.

Navigating this dilemma is crucial for developing a more accurate and applicable theory of rational choice, one that acknowledges the complexities of human decision-making while still providing a meaningful framework for understanding rational behavior.

15.2 Redescribing the Outcomes

When developing a model of rational choice, how we describe the outcomes of a decision can significantly influence preferences, even if the material consequences remain the same. Consider the following scenario from (Dreier 2004):

You have a kitten that you plan to give away, and both Ann and Bob want the kitten very much. Both are equally deserving, and both would provide excellent care for the kitten. You are sure that giving the kitten to Ann is at least as good as giving it to Bob. However, you feel that it would be unfair to Bob if you made the decision unilaterally. To introduce fairness, you decide to flip a fair coin: if it lands heads, you will give the kitten to Bob; if it lands tails, you will give the kitten to Ann.

Suppose that \(a\) is the outcome that Ann is given the kitten and \(b\) is the outcome that Bob is given the kitten. The above scenario can be represented by two lotteries:

Lottery \(L_1\): A coin is flipped, and the kitten is given to Ann, regardless of the outcome (heads or tails). This can be written as \(L_1 = 0.5\cdot a + 0.5\cdot a = 1\cdot a\).
Lottery \(L_2\): A fair coin is flipped: if it lands heads, Bob receives the kitten, and if it lands tails, Ann receives the kitten. This can be written as \(L_2 = 0.5\cdot b + 0.5\cdot a\).

Given that you believe giving the kitten to Ann is at least as good as giving it to Bob, we have either \(a \mathrel{P} b\) (given the kitten to Ann is strictly preferred) or \(a \mathrel{I} b\) (you are indifferent about who receives the kitten). According to the Independence Axiom, this means:

If \(a\mathrel{P} b\), then \(L_1\mathrel{P} L_2\); and
If \(a\mathrel{I} b\), then \(L_1\mathrel{I} L_2\).

In either case, \(L_1\) should be at least as good as \(L_2\) (i.e., \(L_1\mathrel{P} L_2\) or \(L_1\mathrel{I} L_2\)). However, this conflicts with the desire to be fair, as you strictly prefer the fair lottery \(L_2\) over \(L_1\) (i.e., \(L_2\mathrel{P} L_1\)).

This example appears to challenge the Independence Axiom as a normative principle of decision-making. One way to avoid this conclusion is to recognize that the outcome \(a\) in \(L_1\) should be described differently than in the lottery \(L_2\).

Let us distinguish between two different ways Ann can receive the kitten:

Let \(a\) represent Ann receiving the kitten in an unfair way.
Let \(a'\) represent Ann receiving the kitten in a fair way (through the coin flip).

Under this refined description, we can rewrite the lotteries as follows:

\(L_1 = 1 \cdot a\) (Ann receives the kitten unfairly, regardless of the coin flip).
\(L_2 = 0.5 \cdot a' + 0.5 \cdot b\) (who receives the kitten depends on the fair coin flip).

By redefining the outcomes to consider fairness, we remove any apparent conflict with the Independence Axiom. This is because the outcome \(a'\) where Ann receives the kitten in \(L_2\) is not the same as the outcome \(a\) in \(L_1\) in which Ann receives the kitten. Preferring \(L_2\) over \(L_1\) simply indicates a preference for giving the kitten to Ann in a fair manner rather than in an unfair manner. In other words, there is no inconsistency in thinking that giving the kitten to Ann (\(a\)) is at least as good as giving it to Bob (\(b\)), while also strictly preferring to give the kitten to Ann fairly (\(a'\)) over unfairly (\(a\)).

This example shows how critical it is to properly define outcomes in decision-making. It underscores the subtle yet significant role that concepts like fairness play in shaping rational preferences. By carefully specifying outcomes, we ensure that rational choice models align with the underlying values and principles that guide decision-makers.