14  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.

Dreier, James. 2004. “Decision Theory and Morality.” In The Oxford Handbook of Rationality, edited by Alfred R. Mele and Piers Rawling, 156–81. New York: Oxford University Press.