3 The Rationality of Homo economicus
Principal Rationality Axioms*
Completeness Axiom
Suppose Homo economicus faces two lotteries, which we will denote as lotteries and , where both lotteries are taken from what is known as the space of available lotteries . Then, it must be the case that either , , or . What the previous sentence says is that Homo economicus either likes lottery at least as much as lottery (i.e., ), likes lottery at least as much as lottery (i.e., ), or is indifferent between the two lotteries (i.e., ). This is known as the Completeness Axiom. For future reference, we will use the equivalent terminology “weakly preferred to” rather than “likes at least as much” when referring to the preference relation .
Transitivity Axiom
Given any third lottery taken from the space of available lotteries , if and , then . In other words, Homo economicus would never fall victim to a “preference reversal,” whereby she makes choices that contradict her stated preference ranking. This is known as the Transitivity Axiom.
So that we are clear on what a lottery is, here is an example of three possible lotteries , , and .
As a final note, it is worth pointing out that the Principal Rationality Axioms imply the existence of what’s known as a utility function representing an individual’s preferences over lottery space , specifically, such that . Let’s unpack this mathematical statement. The first part of the statement (i.e., ) says that utility function magically translates an individual’s preferences for the different lotteries that make up lottery space into real numbers. The real numbers, by the way, are measured in what’s known as “utils,” or units of happiness. For example, if = 100.3, then the individual facing lottery gets 100.3 units of happiness just from the opportunity of being able to play the lottery.
The second part of the statement (i.e., ) says that the statement “lottery is weakly preferred to lottery ” (i.e., ) is equivalent to the statement “the utility level obtained from lottery is no less than the utility level obtained from lottery ” (i.e., ).
Additional Rationality Axioms*
Dominance Axiom
If in all respects (i.e., ’s expected win outcomes are larger than ’s, and ’s expected loss outcomes are lower), then . To see what we mean by “expected win” and “expected loss” outcomes, refer to the example lotteries above in the discussion of the Transitivity Axiom. Lottery ’s expected win and expected loss outcomes, respectively, are $120 (0.6 x $200) and $40 (0.4 x $100), while lottery ’s are $50 (0.5 x $100) and $35 (0.5 x $70). The final part of this axiom’s statement, , states that lottery is “strictly” preferred to lottery . An equivalent way to say “strictly preferred to” is to say “likes more than.” This is known as the Dominance Axiom.
Again referring to the example above, is by the Dominance Axiom? The answer is “no.” Although lottery ’s expected win of $120 is larger than lottery ’s expected win of $50, ’s expected loss of $40 is a