Ah, I see. You’re thinking of both theories in a math-intuition-based setting (“negatively correlates with your other copy” etc). I prefer to use a crisp proof-based setting, so we can discern what we know about the theories from what we hope they would do in a more fuzzy setting.
UDT1 receives an observation X and then looks for provable facts of the form “if all my instances receiving observation X choose to take a certain action, I’ll get a certain utility”.
UDT1.1 also receives an observation X, but handles it differently. It looks for provable facts of the form “if all my instances receiving various observations choose to use a certain mapping from observations to actions, I’ll get a certain utility”. Then it looks up the action corresponding to X in the mapping.
In problem 2, a UDT1 player who’s told to press button 1 will look for facts like “if everyone who’s told to press button 1 complies, then utility is 500”. But there’s no easy way to prove such a fact. The utility value can only be inferred from the actions of both players, who might receive different observations. That’s why UDT1.1 is needed—to fix UDT1′s bug with handling observations.
The crisp setting makes it clear that UDT1.1 is about making more equilibria reachable, not about equilibrium selection. A game can have several equilibria, all of them reachable without UDT1.1, like my problem 1. Or it can have one equilibrium but require UDT1.1 to reach it, like my problem 2.
Of course, when we move to a math-intuition-based setting, the difference might become more fuzzy. Maybe UDT1 will solve some problems it couldn’t solve before, or maybe not. The only way to know is by formalizing math intuition.
Ah, I see. You’re thinking of both theories in a math-intuition-based setting (“negatively correlates with your other copy” etc). I prefer to use a crisp proof-based setting, so we can discern what we know about the theories from what we hope they would do in a more fuzzy setting.
UDT1 receives an observation X and then looks for provable facts of the form “if all my instances receiving observation X choose to take a certain action, I’ll get a certain utility”.
UDT1.1 also receives an observation X, but handles it differently. It looks for provable facts of the form “if all my instances receiving various observations choose to use a certain mapping from observations to actions, I’ll get a certain utility”. Then it looks up the action corresponding to X in the mapping.
In problem 2, a UDT1 player who’s told to press button 1 will look for facts like “if everyone who’s told to press button 1 complies, then utility is 500”. But there’s no easy way to prove such a fact. The utility value can only be inferred from the actions of both players, who might receive different observations. That’s why UDT1.1 is needed—to fix UDT1′s bug with handling observations.
The crisp setting makes it clear that UDT1.1 is about making more equilibria reachable, not about equilibrium selection. A game can have several equilibria, all of them reachable without UDT1.1, like my problem 1. Or it can have one equilibrium but require UDT1.1 to reach it, like my problem 2.
Of course, when we move to a math-intuition-based setting, the difference might become more fuzzy. Maybe UDT1 will solve some problems it couldn’t solve before, or maybe not. The only way to know is by formalizing math intuition.