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.
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.