Jaynes argues that when you have symmetry in a discrete problem such that switching all the labels leaves the problem the same, you must assign equal probabilities to the available choices. (See page 34 of this.) This covers all of your scenarios except the one where Bob has the option of choosing Green, a Ball color that he does not recall, and the fill-in-the-blank scenario.
The argument only speaks to probabilities, not actions. To choose what to pick, you need utilities. For example, if being right about the color has the same utility regardless of color but it’s worse to guess wrong if the ball is red, then you’d want to pick red even if your probabilities are equal between the two alternatives.
The fill-in-the-blank problem is above my pay-grade. ;-)
Okay, “utilities” makes sense. That may have been the term I was missing.
The basic goal in all of this is preventing a system crash when there are two equal ways to move forward. Acting randomly isn’t bad and is what I would have expected people to answer. What I was looking for is how to refine “acting randomly” after the system is modified. “Utilities” sounds right to me.
And as a major disclaimer, I understand this is probably very basic to most of you (plural, as in the community). I just don’t want to start with the wrong building blocks.
There’s a well-known example in philosophy called Buridan’s Ass—a donkey is placed at the exact midpoint between two bales of hay, and being unable to choose between them (because they are identical), it starves to death. Somewhat amusingly, but also unfortunately, digital electronics can run into a similar problem known as metastability; a circuit can get stuck at a voltage roughly at the midpoint between those assigned to logic level 0 and logic level 1.
Oddly, adding a “if it’s hard to decide, choose randomly” circuit doesn’t help; it just creates another ambiguous situation at the borders of the voltage range you designate as “hard to decide”.
Jaynes argues that when you have symmetry in a discrete problem such that switching all the labels leaves the problem the same, you must assign equal probabilities to the available choices. (See page 34 of this.) This covers all of your scenarios except the one where Bob has the option of choosing Green, a Ball color that he does not recall, and the fill-in-the-blank scenario.
So then you just randomly pick between Red and Blue? What should you do if the question is fill-in-the-blank instead of multiple choice?
The argument only speaks to probabilities, not actions. To choose what to pick, you need utilities. For example, if being right about the color has the same utility regardless of color but it’s worse to guess wrong if the ball is red, then you’d want to pick red even if your probabilities are equal between the two alternatives.
The fill-in-the-blank problem is above my pay-grade. ;-)
Okay, “utilities” makes sense. That may have been the term I was missing.
The basic goal in all of this is preventing a system crash when there are two equal ways to move forward. Acting randomly isn’t bad and is what I would have expected people to answer. What I was looking for is how to refine “acting randomly” after the system is modified. “Utilities” sounds right to me.
And as a major disclaimer, I understand this is probably very basic to most of you (plural, as in the community). I just don’t want to start with the wrong building blocks.
There’s a well-known example in philosophy called Buridan’s Ass—a donkey is placed at the exact midpoint between two bales of hay, and being unable to choose between them (because they are identical), it starves to death. Somewhat amusingly, but also unfortunately, digital electronics can run into a similar problem known as metastability; a circuit can get stuck at a voltage roughly at the midpoint between those assigned to logic level 0 and logic level 1.
Oddly, adding a “if it’s hard to decide, choose randomly” circuit doesn’t help; it just creates another ambiguous situation at the borders of the voltage range you designate as “hard to decide”.