Suppose you offer to pay a penny to swap mushroom for pepperoni, and then another penny to swap back. This agent will refuse, failing to money pump you.
Suppose you offer the agent a choice between pepperoni or mushroom, when it currently has neither. Which does it choose? If it chooses pepperoni, but refuses to swap mushroom for pepperoni then its decisions depend on how the situation is framed. How close does it have to get to the mushroom before they “have” mushroom and refuse to swap? Partial preferences only make sense when you don’t have to choose between unordered options.
We could consider the agent to have a utility function with a term for time consistency, they want the pizza in front of them at times 0 and 1 to be the same.
If it chooses pepperoni, but refuses to swap mushroom for pepperoni then its decisions depend on how the situation is framed. How close does it have to get to the mushroom before they “have” mushroom and refuse to swap?
I previously suggested that revealing the two options as equivalent will bring the two subagents into a standstill, requiring some third factor to help decide. Which seems close to what happens if I introspect on what happens if I’m offered a choice between two foods that I think are equally good—I just decide at random or go by e.g. some force of habit that provides a slight starting point bias.
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.
Suppose you offer to pay a penny to swap mushroom for pepperoni, and then another penny to swap back. This agent will refuse, failing to money pump you.
Suppose you offer the agent a choice between pepperoni or mushroom, when it currently has neither. Which does it choose? If it chooses pepperoni, but refuses to swap mushroom for pepperoni then its decisions depend on how the situation is framed. How close does it have to get to the mushroom before they “have” mushroom and refuse to swap? Partial preferences only make sense when you don’t have to choose between unordered options.
We could consider the agent to have a utility function with a term for time consistency, they want the pizza in front of them at times 0 and 1 to be the same.
Relevant example.
I previously suggested that revealing the two options as equivalent will bring the two subagents into a standstill, requiring some third factor to help decide. Which seems close to what happens if I introspect on what happens if I’m offered a choice between two foods that I think are equally good—I just decide at random or go by e.g. some force of habit that provides a slight starting point bias.
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.