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.
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.