Here’s a toy example which should make it clearer that the probability assigned to the true state is not the only relevant update.
Let’s say that a seeker is searching for something, and doesn’t know whether it is in the north, east, south, or west. If the object is in the north, then it is best for the seeker to go towards it (north), worst for the seeker to go directly away from it (south), and intermediate for them to go perpendicular to it (east or west). The seeker meets a witness who knows where the thing is. The majority (2/3) of witnesses want to help the seeker find it and the rest (1/3) want to hinder the seeker’s search. And they have common knowledge of all of this.
In this case, the witness can essentially just direct the seeker’s search—if the witness says “it’s north” then the seeker goes north, since 2⁄3 of witnesses are honest. So if it’s north and the witness wants to hinder the seeker, they can just say “it’s south”. This seems clearly deceptive—it’s hindering the seeker’s search as much as possible by messing up their beliefs. But pointing them south does actually lead to a right-direction update on the true state of affairs, with p(north) increasing from 1⁄4 (the base rate) to 1⁄3 (the proportion of witnesses who aim to hinder). It’s still a successful deception because it increases p(south) from 1⁄4 to 2⁄3, and that dominates the seeker’s choice.
Here’s a toy example which should make it clearer that the probability assigned to the true state is not the only relevant update.
Let’s say that a seeker is searching for something, and doesn’t know whether it is in the north, east, south, or west. If the object is in the north, then it is best for the seeker to go towards it (north), worst for the seeker to go directly away from it (south), and intermediate for them to go perpendicular to it (east or west). The seeker meets a witness who knows where the thing is. The majority (2/3) of witnesses want to help the seeker find it and the rest (1/3) want to hinder the seeker’s search. And they have common knowledge of all of this.
In this case, the witness can essentially just direct the seeker’s search—if the witness says “it’s north” then the seeker goes north, since 2⁄3 of witnesses are honest. So if it’s north and the witness wants to hinder the seeker, they can just say “it’s south”. This seems clearly deceptive—it’s hindering the seeker’s search as much as possible by messing up their beliefs. But pointing them south does actually lead to a right-direction update on the true state of affairs, with p(north) increasing from 1⁄4 (the base rate) to 1⁄3 (the proportion of witnesses who aim to hinder). It’s still a successful deception because it increases p(south) from 1⁄4 to 2⁄3, and that dominates the seeker’s choice.