It seems to me that there are two distinct things which the English word ‘deceptive’ describes:
Information which leads someone to believe something false.
An action performed with the intent to present someone with deceptive information (in the first sense).
Your formalism is of the first sense, which is why it’s unintuitive that it does not take Bob’s beliefs into account.
Following is the outline of a simple formalism for the second sense:
Suppose we have two agents, Alice and Bob. Alice’s payoff is determined by the information she knows. For example, maybe she’s playing a game of poker.
Bob transmits some set of information T to Alice. Alice recieves all but some subset L(T) of T. Bob knows the value of L(T), but cannot transmit any further information to Alice.
Then, Bob predicts Alice’s total utility over the rest of her life; we call the value of this prediction Ua. Finally, he predicts what Alice’s total utility would be had all of T been transmitted (i.e. if L(T)=∅); this prediction is called Ut.
The information which Bob attempted and failed to transmit to Alice, L(T), is deceptive (2) with respect to Alice if and only if Ua<Ut.
I don’t think any one formalism can cover both senses. For one thing, it would need to represent the four possible states of “deceptiveness” as a binary “deceptive” property.
I think that it might be best to consider different terms to describe the two different senses of “deceptive.”
It seems to me that there are two distinct things which the English word ‘deceptive’ describes:
Information which leads someone to believe something false.
An action performed with the intent to present someone with deceptive information (in the first sense).
Your formalism is of the first sense, which is why it’s unintuitive that it does not take Bob’s beliefs into account.
Following is the outline of a simple formalism for the second sense:
Suppose we have two agents, Alice and Bob. Alice’s payoff is determined by the information she knows. For example, maybe she’s playing a game of poker.
Bob transmits some set of information T to Alice. Alice recieves all but some subset L(T) of T. Bob knows the value of L(T), but cannot transmit any further information to Alice.
Then, Bob predicts Alice’s total utility over the rest of her life; we call the value of this prediction Ua. Finally, he predicts what Alice’s total utility would be had all of T been transmitted (i.e. if L(T)=∅); this prediction is called Ut.
The information which Bob attempted and failed to transmit to Alice, L(T), is deceptive (2) with respect to Alice if and only if Ua<Ut.
I don’t think any one formalism can cover both senses. For one thing, it would need to represent the four possible states of “deceptiveness” as a binary “deceptive” property.
I think that it might be best to consider different terms to describe the two different senses of “deceptive.”