# Can the Machina Paradox be solved by expanding the choice set?

In another question, the Machina paradox is mentioned as a possible counterexample to the expected utility model:

Adding to the list of paradoxes, consider Machina's paradox. It is described in Mas-Colell, Whinston and Green's Microeconomic Theory.

A person prefers a trip to Paris to watching a television program about Paris to nothing.

Gamble 1: Win a trip to Paris 99% of the time, the television program 1% of the time.

Gamble 2: Win a trip to Paris 99% of the time, nothing 1% of the time.

It's reasonable to suppose that given the preferences over items, the second gamble might be preferred to the first. Someone who lost the trip to Paris might be so disappointed that they wouldn't be able to stand watching a program about how great it is.

However, it seems to me that this can be resolved by expanding the decision space to account for possibly state-dependent utility. For example, consider a model with two time periods, $t=0$ and $t=1$. The first represents before the resolution of the uncertainty surrounding winning the trip to Paris. The second time period is after the resolution of the gamble. Now, model this potential outcomes as follows: \begin{align} A &= \{P, \emptyset\} \\ B &= \{P^C, T\} \\ C &= \{P^C, N\}, \end{align} where $A$ corresponds to the outcome where you win the trip to Paris (and then it doesn't matter what you do after that), $B$ is the outcome where you don't win the trip and you watch TV afterwards, and $C$ is the case where you don't win and you do nothing afterwards. Then, although you might like Paris over TV over nothing in one time period (...?), when considered together over time (because of some sort of complementarities) you prefer $A$ over $B$ over $C$.

My question is this. Is this a reasonable way to resolve this paradox? What are ways in which people have tried to resolve this?

• Seems reasonable, though I think it really is a question of what assumptions are being used. "Someone who lost the trip to Paris might be so disappointed that they wouldn't be able to stand watching a program about how great it is." This is an assumption that there is a hidden variable which is regret. Assuming the consumer has high regret of losing the trip he/she would not want to be reminded of the trip by the movie. Now, it would make sense to try incorporate the regret variable as a weight or something. But how we measure it? In my view it depends on consumer preferences. – Koba Nov 20 '14 at 0:12
• At the end of the last line of the penultimate paragraph, do you mean "prefer $A$ over $C$ over $B$" instead of "prefer " $A$ over $B$ over $C$" or am I missing something? – Martin Van der Linden Nov 24 '14 at 3:46

No, I would not say that this resolves the Machina paradox, because it is exactly the same as the Machina paradox: the paradox does indeed require from you to look at the three possible outcomes. The M-C/W/G book discuss only the $B$ and $C$ outcomes because it is there where the paradox focuses on whether a violation of the axiom of independence may happen.

But most importantly, Machina did not argue that all people will have preference ordering $A>C>B$. He argued that it is reasonable, for evident psychological reasons, to expect that some people may... So some others will have the ordering $A>B>C$, which does conform to expected utility framework.

The first will say "I cannot watch a movie about Paris after losing the trip - I will smash the TV!" The second will say "Well, tough luck. At least I will see it on screen and keep dreaming about it". Both seem like behaviors that could be anticipated by "usual" human beings.

The point of the paradox is not to show that Expected Utility (EU) is invalid for all people -only that it may be violated in reasonable situations, i.e. situations that may characterize a lot of people and may happen often.

What paradoxes like this examine and contemplate, is the degree to which EU represents adequately the "majority" of people in some sense, and so whether it is valid/useful/not-misleading as a core theoretical assumption in economic models, or not. And this is a matter of degree, a quantitative matter. This is true for almost all assumptions in theoretical models in social sciences.

• The point of most paradoxes in social sciences is not that the situation cannot be explained, but that the explaintion can be large and unwieldy in an empirical setting. How many states do we need for a person in reality? Under what conditions do they change states in reality? Are preference orders observable in practice, or do the majority of states sit unrealized until a critical moment, blasting our work into the dust? The paradox is simple but the treatment is not. – RegressForward May 4 '15 at 19:19

I think you are correct that this solves the Machina Paradox but I am not sure I would associate your reformulation of the model with the idea of state-dependent utility.

State-dependent utility is more than a mere extension or modification of the set of outcomes of the expected utility model. To make sense of state dependent utility, you need to have a clear distinction between states and outcomes. In state-dependent utility models, agent have preferences over lists like $( x_1, x_2, \dots , x_S)$ where each $x_i$ is a pair $(outcome, state)$.

In your example, I do not see clearly what the different states are. My understanding is that there is only one, and that the paradox is resolved by altering the set of outcomes from $\{P,T\}$ to $\{ A,B,C \}$ rather than by relying on different states. If this is correct, then there is no need to refer to state-dependency. Reformulating the outcome set seems enough.

For more on the distinction between state-dependent utility and the VnM model, I once wrote an answer about it on math.SE. See also the relevant section in Mas-Colell, Whinston and Green.