I do not have a formal proof, but here is an outline:
Consider a message one player can transmit to the group .
If it is beneficial to transmit , it must be detrimental to a subset , and at least neutral for .
The subsets and are in zero-sum conflict, so as an entity does not benefit from transmitting .
Thus, for to be beneficial to a player within , it must be at the expense of the other players, contradicting the definition.
Also, if there are several coalition-forming messages, and one of them leads to the highest payoff whether or not it is true, it is always beneficial to transmit that message, so it is cheap talk.
Therefore, the highest-payoff coalition-forming message is either true or silence.
The third step is a little tricky. What if the coalition forming itself benefits everyone in ? Well then they will form a coalition with the true message, “we should build a coalition”. The fifth step is also tricky. What if there are many possibilities for your secret, and the coalition you build with, “I am type 1″ is good when you are types 1, 2, or 3, but bad when you are type 4? Then everyone should expect you to transmit this when you are types 1, 2, or 3, so if it really does build a coalition it must be equivalent to transmitting, “I am type 1, 2, or 3, but definitely not 4”.
In that sentence, you are not arguing that the lie is no better than silence, you are arguing that it is no better than some truthful message. (This is technically still a falsification of my previously-stated interpretation of your claim.)
This argument is based on the assumption that the other players already know all circumstances under which you would transmit this message, so there’s no harm in admitting them.
I now realize that if all players have perfect knowledge of the exact conditions under which you would transmit some message, then the actual informational payload of every message is that those conditions are true. (Even with a randomized strategy, you can just interpret the RNG output as part of the conditions.) You might as well literally say “message #27”. Classifying the message itself as truth or lie becomes academic, because no one is expected or intended to pay attention to its face-value claim, and in fact there’s no reason for it to make a face-value claim at all. (Under this very strong assumption.)
So if we’re going to assume players have perfect knowledge of each others’ strategies (including what messages they send under what circumstances), I no longer think it makes sense to distinguish “true” and “false” messages.
I note that “common knowledge that all players are perfectly rational” does not (I think) logically entail perfect knowledge of everyone’s strategy, since a game can have more than one Nash equilibrium. So technically neither of us stated “perfect knowledge of everyone’s strategy” as an assumption in the first place, though I admit I sort of hand-waved towards it when I talked about what players would infer from your failure to say something.
I still think that if we don’t assume “perfect knowledge of everyone’s strategy” then lying is potentially beneficial.
Given that clarification, I’m not sure if your numbered chain of reasoning is a crux for either of us, but for the record I found that chain extremely confusing to read, I think step 3 is invalid, and your final paragraph (after the numbered list) was the only part of the comment I found helpful.
In step 3, you seem to be trying to treat the groups and as if they were each a single player so that you can apply the conclusions from two-player games, but I don’t think that’s valid. The two-player result was based on an implicit assumption that transmitting a message from Beth to Adam cannot have any effect on the game except through Adam’s reaction, but that’s not true here because isn’t a unified agent, so transmitting can change the game (by affecting other members of ) even if refuses to react to it. does not get a veto on changing the game, like Adam does. Chris does not need to be listening in order for Adam and Beth to strike a mutually-beneficial deal at his expense.
So the inference that as a group cannot be profiting is invalid.
(Also note that your claim proves too much: If this were accepted, you haven’t proven that false messages are useless, you’ve proven that all messages are useless.)
I do not have a formal proof, but here is an outline:
Consider a message
If it is beneficial to transmit
The subsets
Thus, for
Also, if there are several coalition-forming messages, and one of them leads to the highest payoff whether or not it is true, it is always beneficial to transmit that message, so it is cheap talk.
Therefore, the highest-payoff coalition-forming message is either true or silence.
The third step is a little tricky. What if the coalition forming itself benefits everyone in
Your final sentence clarified some things for me:
In that sentence, you are not arguing that the lie is no better than silence, you are arguing that it is no better than some truthful message. (This is technically still a falsification of my previously-stated interpretation of your claim.)
This argument is based on the assumption that the other players already know all circumstances under which you would transmit this message, so there’s no harm in admitting them.
I now realize that if all players have perfect knowledge of the exact conditions under which you would transmit some message, then the actual informational payload of every message is that those conditions are true. (Even with a randomized strategy, you can just interpret the RNG output as part of the conditions.) You might as well literally say “message #27”. Classifying the message itself as truth or lie becomes academic, because no one is expected or intended to pay attention to its face-value claim, and in fact there’s no reason for it to make a face-value claim at all. (Under this very strong assumption.)
So if we’re going to assume players have perfect knowledge of each others’ strategies (including what messages they send under what circumstances), I no longer think it makes sense to distinguish “true” and “false” messages.
I note that “common knowledge that all players are perfectly rational” does not (I think) logically entail perfect knowledge of everyone’s strategy, since a game can have more than one Nash equilibrium. So technically neither of us stated “perfect knowledge of everyone’s strategy” as an assumption in the first place, though I admit I sort of hand-waved towards it when I talked about what players would infer from your failure to say something.
I still think that if we don’t assume “perfect knowledge of everyone’s strategy” then lying is potentially beneficial.
Given that clarification, I’m not sure if your numbered chain of reasoning is a crux for either of us, but for the record I found that chain extremely confusing to read, I think step 3 is invalid, and your final paragraph (after the numbered list) was the only part of the comment I found helpful.
In step 3, you seem to be trying to treat the groups
So the inference that
(Also note that your claim proves too much: If this were accepted, you haven’t proven that false messages are useless, you’ve proven that all messages are useless.)