Here’s a card game: You lay out the A,2,3 of Spades, Diamonds, Clubs in random order on the table face up. So that’s 9 cards in total. There are two players and they take turns picking up cards from the table, one at a time. The winner is the first to collect a triplet where a triplet is any one of the following sets of three:

- Three cards of the same suit
- Three cards of the same value
- Ace of Spaces, 2 of Diamonds, 3 of Clubs
- Ace of Clubs, 2 of Diamonds, 3 of Spades

Got it? Ok, this game can be solved and the solution is that with best play the result is a draw, neither player can collect a triplet. See if you can figure out why. (Drew Fudenberg got it almost immediately [spoiler.]) Answer and more discussion are after the jump.

The game is isomorphic to Tic-Tac-Toe. The 9 cards represent a 3×3 matrix where the value of the card is the column and the suit is the row and we put Diamonds in the center row. To collect a triplet then is equivalent to getting three in a row in Tic-Tac-Toe.

I’ll bet that you would have no trouble saving a draw in Tic Tac Toe but, before you saw the isomorphism, might actually have lost a game or two of this version. (Win bets at parties: beat your friends at this game, then bet them you can beat them at Tic Tac Toe.) I don’t know of any theory of “bounded rationality” in game theory that would predict differences in play between this game and Tic Tac Toe.

I think that having subjects play, unbenknownst to them, several games that are isomorphic to another could be the basis for some interesting experiments. Keeping the same basic strategic elements but changing the framing could allow you to isolate differences in levels of sophistication that are different than the usual “cognitive hierachies.” For example Tic Tac Toe emphasizes geometry (trivially) whereas the variant emphasizes more abstract thinking.

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February 21, 2012 at 8:16 am

Michael WebsterI like this, it is interesting.

1. If the illusion had been stubborn, like the Allais Paradox, you would have been on to something very interesting about backwards induction.

2. The term Tversky and Kahneman used for this was “invariance”. Unfortunately, they are known more for providing counter-examples to axioms of rationality than raising the more important question of when we can expect invariance in decision making, what to do when we cannot, and should/ought improve decision making by relying on techniques which require invariance.

3. All experiments in game theory should be testing invariance and not whether agents are predictable by game theory standards. Take a group of people, get x% acting as if they were game theorists. Modify the game with something game theory says is irrelevant, and see how many people now act as if they were game theorists, %y. If x is much different from y, ask Tom Schelling why what game theory thought was not relevant turned out to be relevant. Publish and repeat – with different iterations of Tom Schelling.

February 21, 2012 at 9:12 am

M. AllenThis paper, if I recall correctly, deals a bit with experimental results and learning in game theory, suggesting that the initial rounds of play are prone to error/off-equilibrium path behavior: http://www.soc.utu.fi/sivustot/pcrc/files/Sieberg.pdf

February 21, 2012 at 10:39 am

Michael Webster@M. Allen, thanks. I will take a look.

February 21, 2012 at 1:40 pm

emirThis does not apply to tic-tac-toe, but the impact of frames on games will be particularly tricky to identify if there is any possibility that players might exhibit social preferences.

Suppose I give you a problem where you pick a number x in [0,10] trying to maximize your utility which is increasing in x. This is very easy. But now suppose you have to pick any real number x and your utility is increasing in f(x) where f:R->[0,10] is some surjective but super complicated function. Clearly it is the “same problem” but if f is sufficiently complicated we will get a very different outcome.

Now, suppose we play one of two games.

Game A: you pick x in [0,10], you get x, I get 10-x.

Game B: you pick x in R, you get f(x), I get 10-f(x).

These two versions of the Dictator game vary along two dimensions:

1. They are different for the same reason that the two decision making problems above are different. This has nothing do with games per se.

2. They are different because picking a large f(x) does not display bad manners the way that picking a large x does (I stipulate). This is related to the framing results by Ariel Rubinstein (http://arielrubinstein.tau.ac.il/papers/73.pdf)

February 21, 2012 at 5:51 pm

Art VandelayThis is a very cool variant! Thanks! It is hilarious that Drew figured it out immediately :D

You might want to check out this: http://acad88.sahs.uth.tmc.edu/research/publications/TTT-cs98.pdf

February 21, 2012 at 6:14 pm

jeffGreat link. Thanks.

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