A fun subtlety about “bet or update” that just came to my mind. If you refuse to bet that X is true, you’re supposed to update enough that the bet becomes unprofitable. But that doesn’t always mean updating away from X—sometimes you update toward X.
Imagine there’s a million doors, with a pot of gold behind one of them. The host indicates one door and asks “would you like to bet a dollar at even odds that the gold is behind this door?” You know the host’s algorithm was as follows: he selected the true door with the gold and two other random doors, then selected randomly between the three. Then you would refuse to bet (because you lose a dollar with probability 2⁄3) and also refuse to update away (in fact you’d update strongly toward the door being the one with the gold, as now it has probability 1⁄3 instead of 1/million).
Reading it a bit more carefully, I guess for one-sided bets there’s a chance that you are already in the position that the bet is not profitable so you already don’t need to update. I guess the title threw me off a bit—with two sided bets you have to do one or the other (or both), with one sided you don’t.
Isn’t the point of “bet or update” is that you should be either updating on your counterparty’s credences or taking a bet that your counterparty thinks is +EV? Here, the player is updating upon observing the host point to the door, not on the bet itself.
After the player has updated on the host pointing to the door, you can require the player to take the offered bet or update as normal. Assuming the host is offering the bet as a function of his credences*, the player should update from P(gold) = 1⁄3 to P(gold) ~= 0, because the player knows that the host knows where the gold is.
*as opposed to e.g. offering the bet to provide entertainment to the audience. If the host doesn’t vary the terms of the bet based on his private knowledge of where the gold is, then the player should bet rather than update, because the bet offer doesn’t transmit any info about the host’s credences or the actual state of the world.
A fun subtlety about “bet or update” that just came to my mind. If you refuse to bet that X is true, you’re supposed to update enough that the bet becomes unprofitable. But that doesn’t always mean updating away from X—sometimes you update toward X.
Imagine there’s a million doors, with a pot of gold behind one of them. The host indicates one door and asks “would you like to bet a dollar at even odds that the gold is behind this door?” You know the host’s algorithm was as follows: he selected the true door with the gold and two other random doors, then selected randomly between the three. Then you would refuse to bet (because you lose a dollar with probability 2⁄3) and also refuse to update away (in fact you’d update strongly toward the door being the one with the gold, as now it has probability 1⁄3 instead of 1/million).
“Bet or update” assumes the possibility of taking either side of the bet. In this case I would happily take the other side of the offered bet.
It doesn’t. I wrote the “bet or update” post, so I’d know =)
Ha, so you did!
Reading it a bit more carefully, I guess for one-sided bets there’s a chance that you are already in the position that the bet is not profitable so you already don’t need to update. I guess the title threw me off a bit—with two sided bets you have to do one or the other (or both), with one sided you don’t.
Isn’t the point of “bet or update” is that you should be either updating on your counterparty’s credences or taking a bet that your counterparty thinks is +EV? Here, the player is updating upon observing the host point to the door, not on the bet itself.
After the player has updated on the host pointing to the door, you can require the player to take the offered bet or update as normal. Assuming the host is offering the bet as a function of his credences*, the player should update from P(gold) = 1⁄3 to P(gold) ~= 0, because the player knows that the host knows where the gold is.
*as opposed to e.g. offering the bet to provide entertainment to the audience. If the host doesn’t vary the terms of the bet based on his private knowledge of where the gold is, then the player should bet rather than update, because the bet offer doesn’t transmit any info about the host’s credences or the actual state of the world.