Good question. Let’s talk about analogous choices for a market, since that’s a more realistic system, and then we can bring it back to pizza.
In a market, partial preferences result from hidden state. There is never a missing preference between externally-visible states (i.e. the market’s aggregate portfolio). However, the market could have a choice between two hidden states: given one aggregate trade, the market could implement it two different ways, resulting in different wealth distributions. For instance, if I offer the market $5000 for 5 shares of AAPL, then those 5 shares can come from any combination of the internal agents holding AAPL, and the $5000 can be distributed among them in many different ways. This means the market’s behavior is underspecified: there are multiple possible solutions for its behavior. Economists call the set of possible solutions the “contract curve”. Usually, additional mechanics are added to narrow down the possible behavior—most notably the Law of One Price, the strongest form of which gives locally-unique solutions for the market’s behavior. For real markets, Law of One Price is an approximation, and the exact outcome will depend on market microstructure: market making, day trading, and so forth.
Now let’s translate this back to the original question about pizza.
Short answer: the preferences don’t specify which choice the system takes when offered mushroom or pepperoni. It depends on internal structure of the system, which the preferences abstract away. And that’s fine—as the market example shows, there are real-world examples where that abstraction is still useful. Additionally, for real-world cases of interest, the underspecified choices will usually be between hidden states, so the underspecified behavior will itself be “hidden”—it will only be externally-visible via path-dependence of later preferences.
Good question. Let’s talk about analogous choices for a market, since that’s a more realistic system, and then we can bring it back to pizza.
In a market, partial preferences result from hidden state. There is never a missing preference between externally-visible states (i.e. the market’s aggregate portfolio). However, the market could have a choice between two hidden states: given one aggregate trade, the market could implement it two different ways, resulting in different wealth distributions. For instance, if I offer the market $5000 for 5 shares of AAPL, then those 5 shares can come from any combination of the internal agents holding AAPL, and the $5000 can be distributed among them in many different ways. This means the market’s behavior is underspecified: there are multiple possible solutions for its behavior. Economists call the set of possible solutions the “contract curve”. Usually, additional mechanics are added to narrow down the possible behavior—most notably the Law of One Price, the strongest form of which gives locally-unique solutions for the market’s behavior. For real markets, Law of One Price is an approximation, and the exact outcome will depend on market microstructure: market making, day trading, and so forth.
Now let’s translate this back to the original question about pizza.
Short answer: the preferences don’t specify which choice the system takes when offered mushroom or pepperoni. It depends on internal structure of the system, which the preferences abstract away. And that’s fine—as the market example shows, there are real-world examples where that abstraction is still useful. Additionally, for real-world cases of interest, the underspecified choices will usually be between hidden states, so the underspecified behavior will itself be “hidden”—it will only be externally-visible via path-dependence of later preferences.