There are simpler examples where identifying deception seems more straightforward. e.g., If a non-venomous snake takes on the same coloration as a venomous snake, this is intended to increase others’ estimates of p(venomous) and reduce their estimates of p(not venomous), which is a straightforward update in the wrong direction.
In the fist attempt at a definition of deceptive signalling, it seems like a mistake to only look at the probability assigned to the true state (“causing the receiver to update its probability distribution to be less accurate (operationalized as the logarithm of the probability it assigns to the true state)”). Actions are based on their full probability distribution, not just the probability assigned to the true state. In the firefly example, P. rey is updating in the right direction on p(predator) (and on p(nothing)), but in the wrong direction on p(mate). And their upward update on p(mate) seems to be what’s driving the predator’s choice of signal. Some signs of this:
The predator mimicked the signal that the mates were using, when it could have caused a larger correct update to p(predator) and reversed the incorrect update to p(mate) by choosing any other signal. Also, P. redator chose the option that maximized the prey’s chances of approaching it, and the prey avoids locations when p(predator) is sufficiently high. If we model the prey as acting according to a utility function, the signal caused the prey to update its expected utility estimate in the wrong direction by causing it to update one of its probabilities in the wrong direction (equivalently: the prey updated the weighted average of its probabilities in the wrong direction, where the weights are based on the relevant utilities). We could also imagine hypothetical scenarios, like if the predator was magically capable of directly altering the prey’s probability estimates rather than being limited to changing its own behavior and allowing the preyto update.
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.
There are simpler examples where identifying deception seems more straightforward. e.g., If a non-venomous snake takes on the same coloration as a venomous snake, this is intended to increase others’ estimates of p(venomous) and reduce their estimates of p(not venomous), which is a straightforward update in the wrong direction.
In the fist attempt at a definition of deceptive signalling, it seems like a mistake to only look at the probability assigned to the true state (“causing the receiver to update its probability distribution to be less accurate (operationalized as the logarithm of the probability it assigns to the true state)”). Actions are based on their full probability distribution, not just the probability assigned to the true state. In the firefly example, P. rey is updating in the right direction on p(predator) (and on p(nothing)), but in the wrong direction on p(mate). And their upward update on p(mate) seems to be what’s driving the predator’s choice of signal. Some signs of this:
The predator mimicked the signal that the mates were using, when it could have caused a larger correct update to p(predator) and reversed the incorrect update to p(mate) by choosing any other signal. Also, P. redator chose the option that maximized the prey’s chances of approaching it, and the prey avoids locations when p(predator) is sufficiently high. If we model the prey as acting according to a utility function, the signal caused the prey to update its expected utility estimate in the wrong direction by causing it to update one of its probabilities in the wrong direction (equivalently: the prey updated the weighted average of its probabilities in the wrong direction, where the weights are based on the relevant utilities). We could also imagine hypothetical scenarios, like if the predator was magically capable of directly altering the prey’s probability estimates rather than being limited to changing its own behavior and allowing the prey to update.
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.