I assume that my friend and I have common knowledge of the rules of the game, and that we have a common interest in seeing him maximise his winnings or minimise his losses. Here is my analysis (with final conclusions rot13′d).
My first thought on seeing this game is that “truth” and “lie” are not accurate descriptions of the actions “name the actual suit” and “name a different suit”. The real rules are that my friend and I both know that we can use the two bits of information available in my response in whatever way we can manage to agree on. To name a different suit is no more a lie than is a conventional bid in the game of bridge. The requirement of not having a pre-arranged strategy, as bridge partners do, complicates things somewhat but does not affect this essential point, that an agreed convention is not a lie.
To simplify the matter, I shall assume that my friend and I are not preternaturally adept at Schelling games, and cannot magically independently pluck a common strategy out of the space of all strategies (otherwise the no collaboration rule is rendered meaningless). I do assume we are logically omniscient, so if there is a unique optimal strategy, we will both discover it and have common knowledge that the other has also discovered it.
The space of all my possible strategies consists of my responses to each of three situations: my friend’s guess is correct, it is the right suit but the wrong rank, or it is a different suit. Although I have four possible responses in each situation, my response can communicate only one bit to my friend, because all he receives from me is a suit that is either the same as his guess or different. The three suits that are different are not distinguishable in the absence of magical Schelling abilities. So the information I can communicate reduces to saying the same suit as he guessed or saying a different suit.
Given three possible situations and a binary choice to make in each one, I have 8 strategies.
My friend’s action is one of three: stick with the original guess, guess another card in the same suit, or guess a card of a different suit. (When I name a suit different to my friend’s guess, the last of these strategies could be split into two: guess a card in the suit I mentioned, or guess one in a suit not equal to either his first guess or my response. But this makes no difference to the payoffs.) He must make his choice one way if I name the same suit as he guessed, and one way if I name a different suit, so he has 9 strategies.
Of the 8*9 = 72 joint strategies, is there a single one which maximises his winnings? If so, that is common knowledge to us both and that is the strategy to use.
But before brute-forcing this with the computer, there is a symmetry to notice. If in my strategy I swap the actions “say the same suit” and “say a different suit”, and my friend also swaps his responses to those two actions, the payoff remains the same. Choosing between these must require a Schelling-type decision, and the only relevant information that could be used to prize one of them over the other is the everyday ideas of truth and lies, according to which truth is better than lies. Therefore, other things being equal, we might decide to favour that strategy with the greatest probability of telling the “truth”. If that still does not decide the strategy uniquely, a further consideration could be that trust in a friend is also good, therefore we should favour a strategy which most often results in the same action by the friend as taking my statement as actually truthful would.
The results of my computer-aided calculation: gurer ner sbhe fgengrtvrf juvpu nyy cebqhpr gur fnzr rkcrpgrq tnva bs 7⁄52. Gurfr ner:
1 naq 2. V gryy gur “gehgu” vs zl sevraq unf thrffrq gur evtug pneq be gur jebat fhvg, naq “yvr” vs ur unf thrffrq gur jebat pneq bs gur evtug fhvg.
Zl sevraq fubhyq fgvpx gb uvf thrff vs gur fhvg V naabhapr vf gur fnzr nf uvf thrff, bgurejvfr ur fubhyq fjvgpu gb nal bgure pneq. (Guvf vf gjb qvssrerag fgengrtvrf jvgu gur fnzr rssrpg, nf ur unf gur pubvpr bs fjvgpuvat gb nabgure pneq bs gur fnzr fhvg, be n pneq bs nabgure fhvg.)
3 naq 4. Gur fnzr nf 1 naq 4, jvgu “gehgu” naq “yvr” fjvgpurq va zl fgengrtl, naq “fhvg == thrff” naq “fhvg != thrff” fjvgpurq va zl sevraq’f.
Fvapr 1 naq 2 unir yrff rkcrpgrq “ylvat” guna 3 naq 4, V “yvr” va gur fpranevb jurer zl sevraq’f thrff vf 7Q naq gur gbc pneq vf 3Q.
ETA: I missed the fact that the friend has the option of doubling if he sticks with the same card, and analysed the game on the basis that he must always double his bet if he stays with the same card. But I expect the choice of strategy that results will be the same. ETA2: it is.
I assume that my friend and I have common knowledge of the rules of the game, and that we have a common interest in seeing him maximise his winnings or minimise his losses. Here is my analysis (with final conclusions rot13′d).
My first thought on seeing this game is that “truth” and “lie” are not accurate descriptions of the actions “name the actual suit” and “name a different suit”. The real rules are that my friend and I both know that we can use the two bits of information available in my response in whatever way we can manage to agree on. To name a different suit is no more a lie than is a conventional bid in the game of bridge. The requirement of not having a pre-arranged strategy, as bridge partners do, complicates things somewhat but does not affect this essential point, that an agreed convention is not a lie.
To simplify the matter, I shall assume that my friend and I are not preternaturally adept at Schelling games, and cannot magically independently pluck a common strategy out of the space of all strategies (otherwise the no collaboration rule is rendered meaningless). I do assume we are logically omniscient, so if there is a unique optimal strategy, we will both discover it and have common knowledge that the other has also discovered it.
The space of all my possible strategies consists of my responses to each of three situations: my friend’s guess is correct, it is the right suit but the wrong rank, or it is a different suit. Although I have four possible responses in each situation, my response can communicate only one bit to my friend, because all he receives from me is a suit that is either the same as his guess or different. The three suits that are different are not distinguishable in the absence of magical Schelling abilities. So the information I can communicate reduces to saying the same suit as he guessed or saying a different suit.
Given three possible situations and a binary choice to make in each one, I have 8 strategies.
My friend’s action is one of three: stick with the original guess, guess another card in the same suit, or guess a card of a different suit. (When I name a suit different to my friend’s guess, the last of these strategies could be split into two: guess a card in the suit I mentioned, or guess one in a suit not equal to either his first guess or my response. But this makes no difference to the payoffs.) He must make his choice one way if I name the same suit as he guessed, and one way if I name a different suit, so he has 9 strategies.
Of the 8*9 = 72 joint strategies, is there a single one which maximises his winnings? If so, that is common knowledge to us both and that is the strategy to use.
But before brute-forcing this with the computer, there is a symmetry to notice. If in my strategy I swap the actions “say the same suit” and “say a different suit”, and my friend also swaps his responses to those two actions, the payoff remains the same. Choosing between these must require a Schelling-type decision, and the only relevant information that could be used to prize one of them over the other is the everyday ideas of truth and lies, according to which truth is better than lies. Therefore, other things being equal, we might decide to favour that strategy with the greatest probability of telling the “truth”. If that still does not decide the strategy uniquely, a further consideration could be that trust in a friend is also good, therefore we should favour a strategy which most often results in the same action by the friend as taking my statement as actually truthful would.
The results of my computer-aided calculation: gurer ner sbhe fgengrtvrf juvpu nyy cebqhpr gur fnzr rkcrpgrq tnva bs 7⁄52. Gurfr ner:
1 naq 2. V gryy gur “gehgu” vs zl sevraq unf thrffrq gur evtug pneq be gur jebat fhvg, naq “yvr” vs ur unf thrffrq gur jebat pneq bs gur evtug fhvg.
Zl sevraq fubhyq fgvpx gb uvf thrff vs gur fhvg V naabhapr vf gur fnzr nf uvf thrff, bgurejvfr ur fubhyq fjvgpu gb nal bgure pneq. (Guvf vf gjb qvssrerag fgengrtvrf jvgu gur fnzr rssrpg, nf ur unf gur pubvpr bs fjvgpuvat gb nabgure pneq bs gur fnzr fhvg, be n pneq bs nabgure fhvg.)
3 naq 4. Gur fnzr nf 1 naq 4, jvgu “gehgu” naq “yvr” fjvgpurq va zl fgengrtl, naq “fhvg == thrff” naq “fhvg != thrff” fjvgpurq va zl sevraq’f.
Fvapr 1 naq 2 unir yrff rkcrpgrq “ylvat” guna 3 naq 4, V “yvr” va gur fpranevb jurer zl sevraq’f thrff vf 7Q naq gur gbc pneq vf 3Q.
ETA: I missed the fact that the friend has the option of doubling if he sticks with the same card, and analysed the game on the basis that he must always double his bet if he stays with the same card. But I expect the choice of strategy that results will be the same. ETA2: it is.