Can you solve the 6 Cards game?

Originally published at: https://boingboing.net/2020/07/16/can-you-solve-the-6-cards-game.html

This argument assumes that your opponent is perfectly rational and familiar with game theory, which usually isn’t the case. A more naive strategy might be to want to trade if you have 1-3 as you then have less than 50% chance of having the high card. If you think your opponent thinks like that it’s 50/50 to trade or not since he might have a one or a three.

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Here’s a simpler game.

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The bits where you have the 6 or where you have the 5, I understand those instances just fine. Where I get fuzzy, is where you have the 4, and fuzzier still where you have the 3.

If you have the three, you might assume I have the 4, and I might want to trade hoping to get a 5 or a six.

In a purely rational environment, I suppose, it would be unrealistic for me to assume you’d be willing to trade a 5 or a 6. But when republicans are in charge of the senate, anything seems possible.

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This feels more like a variation of the unexpected hangman puzzle …

@anansi133 Imagine you have a 4.

If the other player has a 6, they won’t trade, you lose.

If the other player has a 5, they won’t ask for a trade, because either you have a 6 and will say no, or you have a lower number and will say yes.

So if you have a 4 and they ask for a trade, you should say no and you win.

Same for a 3 … 6 no trade, 5 trade, and 4 no trade (because now they have a 4 in the above example, so if you ask, you don’t have a 5 or a 6.)

So if they ask, you say no and you win.

Same for 2 by the same logic … if they ask, they don’t have a high enough card to not bother asking.

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This seems to be a variant of the “unexpected hanging” problem to me - the logic is sound right up to the point that you try to use it.

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Yeah, the whole concept of game theory is founded on the obviously-false premise of perfectly rational players. Which doesn’t mean it’s not worthwhile – or even that it doesn’t have real-world applications – but to sell it as some kind of psychohistory-like physics of human behavior is a level of academic malpractice that would make an economist blush.

Also, in this case it’s a bit pretentious, because it just concludes what any fool would see if they played this game for stakes: if the other person wants to be in your position, then you shouldn’t accept.

In fact, when I paused the video I assumed there must be some trick, like he was going to reveal that when player 2 asks “would you like to trade”, then if you say “yes”, P2 is allowed to take the offer back.

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What that means is that if both players are entirely rational, and understand the Dominated Strategy principle, no trading will ever happen. That’s a pretty dumb game.

I can clearly not drink the wine in front of me.

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The upfront explanation of the rules could have used more depth. Which player can offer to trade? If both, who has the opportunity to do it first? Is one player obligated to offer? (No, it turns out, but that’s not clear until the revelation of the answer.)

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You have to go one step deeper than that. Imagine that the other player has a 2. That means that as far as he knows you probably have a better number so he’d like to be in your position. It’s just that if you are in a better position you aren’t going to accept the trade. Only if you have a 1 will you accept, and thus making the offer is only going to turn a bad position into an even worse one.

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I know the solution makes sense using game theory, but it still seems like a dumb “game”. If you draw anything but a 1, you get what you get, because no one would trade anything but a 1. I guess you’re taking advantage of the lack of game theory knowledge of your opponent? Which some people find fun, I guess? I don’t know, maybe I’m just distracted from having to listen to “stradgy” over and over.

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I wonder if we can do better. What if bluffing is introduced as element? I suspect this would reduced to a different GTO strategy where one player bluffs a certain percentage of the time and the other calls the bluff a certain percentage based on the cards that they are holding.

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Especially because he begins the video stating that most people won’t play this way, making the whole point moot. In the reverse, if everyone plays this way, only one person is willing to trade, also mooting the whole game.

I feel like a sucker every time I bite on one of these mind trick posts.

But suppose your card shows a door with a goat behind it? Should you offer to trade?

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image

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It also doesn’t specify the number of players!

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Exactly. Before opening the video I had inferred that there would be six players and as such thought “What’s the frickin’ point - “Six” will never trade…

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“I take a look and see the number on my card. I then ask ‘Would you like to trade cards? The higher card wins the game.’”

So he only decided that the higher card would win AFTER he looked at his card? His conclusion alleges his card must be the 1. But if he had the 1 he would have chosen not to trade and simply announced “The lowest card wins the game.”

So, yet another problem setter can’t distinguish between what is known about the game rules before the game starts and what is decided once the game is in motion.

Most people trying to explain the badly named “Monty Hall problem” make the same type of error. They say things like “The contestant choose a door and then the quizmaster opened a different door”, which allows the possibility that the quizmaster only decided that he would open a door after seeing which door the contestant chose.

I hate it when these “Youtube experts” can’t even express the problem clearly.

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I get that “6” never trades.
I even get that “5” never trades, because “6” will never trade with him.
I even get that “4” never trades, because the only possible person he can trade with is “5”, but “5” knows that his only trade-up is “6”-- and “6” ain’t trading. So, “4” cannot trade.

OK.

Now “3”… “3” knows that “4”, “5”, “6” are not trading, so “3” shuts down.
Likewise, somehow, “2” (?)

Except. Does this really extend to, say, 100 cards? I think the nuances of this game exist because it’s such a small deck. Are you really telling me, that in a 100-card game, that no one except “1” wants to trade? “22” never trades for “21”?

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I think it does, even with soft logic…
a)
You both pick cards. The other guy offers to swap before you have worked out what to do, because he can see how shit his card is. Stick with yours.
b)
You are both offering to swap cards at the same time. Swapping is symmetric, so there is no known gain unless you are actually looking at the ace. You might offer to swap in a hundred-card game at 20 but stick at 21, but the same is true for the other guy.
c)
You are looking at the ace. The other guy has not read his BoingBoing. Swap his ass!
d)
Wait…are aces high in this game?

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