Also, with rational agents silence is just as good as dishonesty.
I don’t think this claim particularly matters to the thrust of your post, since I think we agree that you’re not playing with perfectly rational agents, but I’m interested in the claim as a matter of game theory.
To be clear, I’m interpreting this as saying something at least as strong as: “In a game of Catan where there is common knowledge that all players are perfectly rational, speaking a falsehood is never more advantageous for the speaker than remaining silent.”
After pondering this for about 20 minutes, I’m pretty convinced the claim is false, and I suspect you are over-generalizing from two-player games.
If Adam and Beth are playing a two-player zero-sum game, and Adam knows that Beth is perfectly rational, then:
If Beth reacts in any way to anything Adam says, then that reaction must be beneficial to Beth (since she is assumed perfectly rational), which means it must be harmful to Adam (because the game is zero-sum), which means Adam shouldn’t have said it.
By similar reasoning, if Beth says anything (that’s not required by the rules), then the fact that she said it can’t be harmful to herself, which means it can’t be beneficial to Adam, which means whatever Adam does in response won’t be (predictably) better than what he would have done anyway.
Therefore, Adam can safely adopt a policy of never saying anything and ignoring whatever Beth says, and this will be no worse than any other policy.
But Catan is played with at least 3 players. The game as a whole is zero-sum, but it’s possible for an action to benefit both Adam and Beth at the same time, provided it harms Chris.[1]
In a non-zero-sum negotiation, it is sometimes helpful to share information in order to coordinate on a mutually-beneficial action. So silence is not, in general, a global optimum.
But if there are situations where you would share some information if it were true, and the other player is aware of this, then silence becomes a tacit admission that it’s not true. So it might become necessary to lie in order to avoid passively leaking secrets.
The lie will only be believable if it’s a claim you would have made if it were true, which sharply limits what lies you can tell. But it does not, in general, limit it to the empty set.
This is not a proof, since I have not constructed an example game position where I can mathematically demonstrate that all of the relevant properties apply at the same time. It is conceivable there’s some reason that hasn’t occurred to me that they can’t all apply at the same time. But I have no candidates for what such a reason would be, and my brief Internet searches have failed to turn up any known result that matches the original claim.
I’m not sure if this is known art, but I’ve found it helpful to think of zero-sum-ness as applying to a set of players rather than to a game. In a 3-player Catan game with Adam, Beth, and Chris, the set (Adam, Beth, Chris) is zero-sum, but the set (Adam, Beth) is non-zero-sum. Note that any non-zero-sum game can be converted to a strategically-equivalent zero-sum game by adding a dummy player whose score is the negative sum of all other players’ scores (or vice versa, by adding a dummy player whose score isn’t that), so it cannot be strategically important whether “the whole game” is zero-sum if we haven’t changed the zero-sum-ness of any particular subset of players.
Consider the 2-player game where A is allowed to broadcast a public message, then B is allowed to press one of 9 buttons or pass, and then A and B receive a result. Add a dummy player C as you suggested if you wish to make the game “zero sum” among 3 players rather than non-zero-sum among 2 players.
Rules: * 10% of the time, all buttons are red, while 90% of the time, a uniform random single one of the buttons is blue while all other buttons are red. * If B presses a blue button the resulting utility for (A,B) is (1, 1). If B presses a red button the result is (0, 0). If B passes, the result is (-1, 0.5). * A has private information—only A can see the color of the buttons. B (and C, if C exists) is colorblind/blindfolded/whatever, but aside from that the rules of the game are common knowledge.
The following is a Nash equilibrium: * If one of the buttons is blue, A always says truthfully which button is blue. * If no button is blue, A picks one of the buttons uniformly at random and lies and says that button is blue. * B always trusts A and presses the button that A claims was blue.
This is a Nash equilibrium because no player can do better in expectation by unilaterally deviating from this protocol. (A is receiving the maximum possible utility they can in every scenario so A cannot improve by unilaterally deviating. B doesn’t see the button colors so it boils down to trust A and get expected utility 0.9, or pass and get utility 0.5, so B should continue to trust A even though A lies sometimes).
Does this provide the kind of example you were thinking of?
Yes, that is the sort of example I meant. Though of course this particular example does not prove that the game of Catan, in particular, has situations like this.
Based on his other reply, I expect James would want to point out that there is an equivalent equilibrium where player A, instead of saying “button N is blue”, says “either button N is blue or no button is”, which produces the same outcome without technically lying.
I’m coming to think that there should be some other distinction we can draw that rhymes with the truthful/lying distinction but that talks about consequences instead of semantics, and therefore can’t be dodged by relabeling the signals. Still thinking about it.
Though of course this particular example does not prove that the game of Catan, in particular, has situations like this.
A has 7 points, “Year of Plenty” card, 1 brick and 3 wood. A can get Longest Road either by building 4 roads or by breaking B’s road with a settlement, but to build this settlement A has to first build one road.
B has 9 points including 2 points from Longest Road and enough resources so they can build a settlement in one turn unless 7 is rolled.
C has 9 points, 1 brick and can maybe win in one turn depending on dice rolls.
A’s turn.
A: “I have Road Builder card, 3 wood, but only 1 brick. C, can you sell me brick for wood? I will build 4 roads, get Longest Road, B will not win in their turn and then we both have a chance.”
C: “I don’t really need wood, but I see that B probably wins if we don’t do it, so OK.”
A plays “Year of Plenty”, takes grain and wool, builds one road and a settlement, wins the game.
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.)
Been thinking more about this claim:
I don’t think this claim particularly matters to the thrust of your post, since I think we agree that you’re not playing with perfectly rational agents, but I’m interested in the claim as a matter of game theory.
To be clear, I’m interpreting this as saying something at least as strong as: “In a game of Catan where there is common knowledge that all players are perfectly rational, speaking a falsehood is never more advantageous for the speaker than remaining silent.”
After pondering this for about 20 minutes, I’m pretty convinced the claim is false, and I suspect you are over-generalizing from two-player games.
If Adam and Beth are playing a two-player zero-sum game, and Adam knows that Beth is perfectly rational, then:
If Beth reacts in any way to anything Adam says, then that reaction must be beneficial to Beth (since she is assumed perfectly rational), which means it must be harmful to Adam (because the game is zero-sum), which means Adam shouldn’t have said it.
By similar reasoning, if Beth says anything (that’s not required by the rules), then the fact that she said it can’t be harmful to herself, which means it can’t be beneficial to Adam, which means whatever Adam does in response won’t be (predictably) better than what he would have done anyway.
Therefore, Adam can safely adopt a policy of never saying anything and ignoring whatever Beth says, and this will be no worse than any other policy.
But Catan is played with at least 3 players. The game as a whole is zero-sum, but it’s possible for an action to benefit both Adam and Beth at the same time, provided it harms Chris.[1]
In a non-zero-sum negotiation, it is sometimes helpful to share information in order to coordinate on a mutually-beneficial action. So silence is not, in general, a global optimum.
But if there are situations where you would share some information if it were true, and the other player is aware of this, then silence becomes a tacit admission that it’s not true. So it might become necessary to lie in order to avoid passively leaking secrets.
The lie will only be believable if it’s a claim you would have made if it were true, which sharply limits what lies you can tell. But it does not, in general, limit it to the empty set.
This is not a proof, since I have not constructed an example game position where I can mathematically demonstrate that all of the relevant properties apply at the same time. It is conceivable there’s some reason that hasn’t occurred to me that they can’t all apply at the same time. But I have no candidates for what such a reason would be, and my brief Internet searches have failed to turn up any known result that matches the original claim.
Do you think I’ve missed something?
I’m not sure if this is known art, but I’ve found it helpful to think of zero-sum-ness as applying to a set of players rather than to a game. In a 3-player Catan game with Adam, Beth, and Chris, the set (Adam, Beth, Chris) is zero-sum, but the set (Adam, Beth) is non-zero-sum.
Note that any non-zero-sum game can be converted to a strategically-equivalent zero-sum game by adding a dummy player whose score is the negative sum of all other players’ scores (or vice versa, by adding a dummy player whose score isn’t that), so it cannot be strategically important whether “the whole game” is zero-sum if we haven’t changed the zero-sum-ness of any particular subset of players.
Consider the 2-player game where A is allowed to broadcast a public message, then B is allowed to press one of 9 buttons or pass, and then A and B receive a result. Add a dummy player C as you suggested if you wish to make the game “zero sum” among 3 players rather than non-zero-sum among 2 players.
Rules:
* 10% of the time, all buttons are red, while 90% of the time, a uniform random single one of the buttons is blue while all other buttons are red.
* If B presses a blue button the resulting utility for (A,B) is (1, 1). If B presses a red button the result is (0, 0). If B passes, the result is (-1, 0.5).
* A has private information—only A can see the color of the buttons. B (and C, if C exists) is colorblind/blindfolded/whatever, but aside from that the rules of the game are common knowledge.
The following is a Nash equilibrium:
* If one of the buttons is blue, A always says truthfully which button is blue.
* If no button is blue, A picks one of the buttons uniformly at random and lies and says that button is blue.
* B always trusts A and presses the button that A claims was blue.
This is a Nash equilibrium because no player can do better in expectation by unilaterally deviating from this protocol. (A is receiving the maximum possible utility they can in every scenario so A cannot improve by unilaterally deviating. B doesn’t see the button colors so it boils down to trust A and get expected utility 0.9, or pass and get utility 0.5, so B should continue to trust A even though A lies sometimes).
Does this provide the kind of example you were thinking of?
Yes, that is the sort of example I meant. Though of course this particular example does not prove that the game of Catan, in particular, has situations like this.
Based on his other reply, I expect James would want to point out that there is an equivalent equilibrium where player A, instead of saying “button N is blue”, says “either button N is blue or no button is”, which produces the same outcome without technically lying.
I’m coming to think that there should be some other distinction we can draw that rhymes with the truthful/lying distinction but that talks about consequences instead of semantics, and therefore can’t be dodged by relabeling the signals. Still thinking about it.
A has 7 points, “Year of Plenty” card, 1 brick and 3 wood. A can get Longest Road either by building 4 roads or by breaking B’s road with a settlement, but to build this settlement A has to first build one road.
B has 9 points including 2 points from Longest Road and enough resources so they can build a settlement in one turn unless 7 is rolled.
C has 9 points, 1 brick and can maybe win in one turn depending on dice rolls.
A’s turn.
A: “I have Road Builder card, 3 wood, but only 1 brick. C, can you sell me brick for wood? I will build 4 roads, get Longest Road, B will not win in their turn and then we both have a chance.”
C: “I don’t really need wood, but I see that B probably wins if we don’t do it, so OK.”
A plays “Year of Plenty”, takes grain and wool, builds one road and a settlement, wins the game.
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.)